Add David Neckel's compressible Euler flow tutorial program.

git-svn-id: https://svn.dealii.org/trunk@15823 0785d39b-7218-0410-832d-ea1e28bc413d

diff --git a/deal.II/examples/step-33/Makefile b/deal.II/examples/step-33/Makefile
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--- /dev/null
+++ b/deal.II/examples/step-33/Makefile
@@ -0,0 +1,154 @@
+# $Id: Makefile 14008 2006-10-17 04:05:33Z bangerth $
+
+
+# For the small projects Makefile, you basically need to fill in only
+# four fields.
+#
+# The first is the name of the application. It is assumed that the
+# application name is the same as the base file name of the single C++
+# file from which the application is generated.
+target = $(basename $(shell echo step-*.cc))
+
+# The second field determines whether you want to run your program in
+# debug or optimized mode. The latter is significantly faster, but no
+# run-time checking of parameters and internal states is performed, so
+# you should set this value to `on' while you develop your program,
+# and to `off' when running production computations.
+debug-mode = on
+
+
+# As third field, we need to give the path to the top-level deal.II
+# directory. You need to adjust this to your needs. Since this path is
+# probably the most often needed one in the Makefile internals, it is
+# designated by a single-character variable, since that can be
+# reference using $D only, i.e. without the parentheses that are
+# required for most other parameters, as e.g. in $(target).
+D = ../../
+
+
+# The last field specifies the names of data and other files that
+# shall be deleted when calling `make clean'. Object and backup files,
+# executables and the like are removed anyway. Here, we give a list of
+# files in the various output formats that deal.II supports.
+clean-up-files = *gmv *gnuplot *gpl *eps *pov
+
+
+
+
+#
+#
+# Usually, you will not need to change something beyond this point.
+#
+#
+# The next statement tell the `make' program where to find the
+# deal.II top level directory and to include the file with the global
+# settings
+include $D/common/Make.global_options
+
+
+# Since the whole project consists of only one file, we need not
+# consider difficult dependencies. We only have to declare the
+# libraries which we want to link to the object file, and there need
+# to be two sets of libraries: one for the debug mode version of the
+# application and one for the optimized mode. Here we have selected
+# the versions for 2d. Note that the order in which the libraries are
+# given here is important and that your applications won't link
+# properly if they are given in another order.
+#
+# You may need to augment the lists of libraries when compiling your
+# program for other dimensions, or when using third party libraries
+libs.g   = $(lib-deal2-2d.g) \
+          $(lib-lac.g)      \
+           $(lib-base.g)
+libs.o   = $(lib-deal2-2d.o) \
+          $(lib-lac.o)      \
+           $(lib-base.o)
+
+
+# We now use the variable defined above which switch between debug and
+# optimized mode to select the set of libraries to link with. Included
+# in the list of libraries is the name of the object file which we
+# will produce from the single C++ file. Note that by default we use
+# the extension .g.o for object files compiled in debug mode and .o for
+# object files in optimized mode (or whatever the local default on your
+# system is instead of .o).
+ifeq ($(debug-mode),on)
+  libraries = $(target).g.$(OBJEXT) $(libs.g)
+else
+  libraries = $(target).$(OBJEXT) $(libs.o)
+endif
+
+
+# Now comes the first production rule: how to link the single object
+# file produced from the single C++ file into the executable. Since
+# this is the first rule in the Makefile, it is the one `make' selects
+# if you call it without arguments.
+$(target) : $(libraries)
+       @echo ============================ Linking $@
+       @$(CXX) -o $@$(EXEEXT) $^ $(LIBS) $(LDFLAGS)
+
+
+# To make running the application somewhat independent of the actual
+# program name, we usually declare a rule `run' which simply runs the
+# program. You can then run it by typing `make run'. This is also
+# useful if you want to call the executable with arguments which do
+# not change frequently. You may then want to add them to the
+# following rule:
+run: $(target)
+       @echo ============================ Running $<
+       @./$(target)$(EXEEXT)
+
+
+# As a last rule to the `make' program, we define what to do when
+# cleaning up a directory. This usually involves deleting object files
+# and other automatically created files such as the executable itself,
+# backup files, and data files. Since the latter are not usually quite
+# diverse, you needed to declare them at the top of this file.
+clean:
+       -rm -f *.$(OBJEXT) *~ Makefile.dep $(target)$(EXEEXT) $(clean-up-files)
+
+
+# Since we have not yet stated how to make an object file from a C++
+# file, we should do so now. Since the many flags passed to the
+# compiler are usually not of much interest, we suppress the actual
+# command line using the `at' sign in the first column of the rules
+# and write the string indicating what we do instead.
+./%.g.$(OBJEXT) :
+       @echo ==============debug========= $(<F)
+       @$(CXX) $(CXXFLAGS.g) -c $< -o $@
+./%.$(OBJEXT) :
+       @echo ==============optimized===== $(<F)
+       @$(CXX) $(CXXFLAGS.o) -c $< -o $@
+
+
+# The following statement tells make that the rules `run' and `clean'
+# are not expected to produce files of the same name as Makefile rules
+# usually do.
+.PHONY: run clean
+
+
+# Finally there is a rule which you normally need not care much about:
+# since the executable depends on some include files from the library,
+# besides the C++ application file of course, it is necessary to
+# re-generate the executable when one of the files it depends on has
+# changed. The following rule to created a dependency file
+# `Makefile.dep', which `make' uses to determine when to regenerate
+# the executable. This file is automagically remade whenever needed,
+# i.e. whenever one of the cc-/h-files changed. Make detects whether
+# to remake this file upon inclusion at the bottom of this file.
+#
+# If the creation of Makefile.dep fails, blow it away and fail
+Makefile.dep: $(target).cc Makefile \
+              $(shell echo $D/*/include/*/*.h)
+       @echo ============================ Remaking $@
+       @$D/common/scripts/make_dependencies  $(INCLUDE) -B. $(target).cc \
+               > $@ \
+         || (rm -f $@ ; false)
+       @if test -s $@ ; then : else rm $@ ; fi
+
+
+# To make the dependencies known to `make', we finally have to include
+# them:
+include Makefile.dep
+
+

diff --git a/deal.II/examples/step-33/doc/intro.dox b/deal.II/examples/step-33/doc/intro.dox
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+<a name="Intro"></a> <h1>Introduction</h1>
+
+This program was written for fun by David Neckels (NCAR) while working
+at Sandia (on the Wyoming Express bus to and from Corrales each day).
+The main purpose was to better understand Euler flow.  
+The code solves the basic Euler equations of gas dynamics, by using a
+fully implicit Newton iteration (inspired by Sandia's Aria code).  The
+code may be configured by an input deck to run different simulations
+on different meshes, with differing boundary conditions.
+
+The program also uses the Trilinos linear solvers (Aztec/Amesos) and
+an automatic differentiation package, Sacado.
+
+<h2>Euler flow</h2>
+
+The equations for a compressible, inviscid gas (the Euler equations) are
+a basic system of conservation laws, in spatial dimension $d$, 
+@f[
+\partial_t \mathbf{w} + \nabla \cdot \mathbf{F}(\mathbf{w}) = \mathbf{0},
+@f]
+with $\mathbf{w}=(\rho,\rho v_1,\dots,\rho v_d, E)^{\top}$ for $v_i$ equal to the
+flow velocity in spatial direction $i$, $\rho$ equal to the fluid density, and
+$E$ the energy of the gas.  The flux matrix (or system of flux functions)
+are defined such that the entire system of equations are
+@f{eqnarray*}
+  \partial_t \rho + \sum_{s=1}^d \frac{\partial(\rho v_s)}{\partial x_s} &=& 0  \\
+  \partial_t (\rho v_i) + \sum_{s=1}^d \frac{\partial(\rho v_i v_s + \delta_{is} p)}{\partial x_s} &=& 0, i=1,\dots,d \\
+  \partial_t E + \sum_{s=1}^d \frac{\partial((E+p)v_s)}{\partial x_s} &=& 0 \\
+  p = (\gamma -1)(E-\frac{1}{2} \rho |v|^2).
+@f}
+
+For air, $\gamma=1.4$.
+
+<h2>Discretization</h2>
+
+We choose a finite element space $V_h$, and integrate our conservation law against
+our (vector) test function $\mathbf{v} \in V_h$.  We integrate by parts and approximate the
+boundary flux with a <i> numerical </i> flux $\mathbf{H}$,
+@f{eqnarray*}
+\int_{\Omega} (\partial_t \mathbf{w}, \mathbf{v}) + (\nabla \cdot \mathbf{F}(\mathbf{w}), \mathbf{v}) & \approx & \\
+\int_{\Omega} (\partial_t \mathbf{w}, \mathbf{v}) + (\mathbf{F}(\mathbf{w}), \nabla \mathbf{v}) + h^{\eta}(\nabla \mathbf{w} , \nabla \mathbf{v}) + \int_{\partial \Omega} (\mathbf{H}(\mathbf{w}^+, \mathbf{w}^-, \mathbf{n}), \mathbf{v}^+),
+@f}
+where $+$ is the interior trace of a function, and $-$ represents the outer trace.
+The diffusion term $h^{\eta}(\nabla \mathbf{w} , \nabla \mathbf{v})$ is introduced strictly for stability,
+ where $h$ is the mesh size and $\eta$ is a parameter prescribing how much diffusion to add.
+Depending on the boundary condition, we prescribe the outer trace $\mathbf{w}^-$,
+<ul>
+<li> inflow boundary: $\mathbf{w}^-$ is prescribed to be the desired value.  
+<li> supersonic outflow boundary: $\mathbf{w}^- = \mathbf{w}^+$
+<li> subsonic outflow boundary: $\mathbf{w}^- = \mathbf{w}^+$ except that the energy variable
+is modified to support a prescribed pressure $p_o$, i.e. 
+$\mathbf{w}^- =(\rho^+, \rho v_1^+, \dots, \rho v_d^+, p_o/(\gamma -1) + 0.5 \rho |\mathbf{v}^+|^2)$
+<li> reflective boundary: we set $\mathbf{w}^-$ so that $(\mathbf{v}^+ + \mathbf{v}^-) \cdot \mathbf{n} = 0$ and
+$\rho^- = \rho^+,E^-=E^+$.
+</ul>
+
+For more information, please consult Ralf Hartmann's thesis TODO:Ref.
+
+
+Our full discretization is thus
+@f{eqnarray*}
+R(\mathbf{W}_{n+1}) = \\
+\int_{\Omega} (\frac{\mathbf{w}_{n+1} - \mathbf{w}_n}{\delta t}, \mathbf{v}) + \int_{\Omega} (\mathbf{F}(\tilde{\mathbf{w}}), \mathbf{v}) +  h^{\eta}(\nabla \mathbf{w} , \nabla \mathbf{v}) + \int_{\partial \Omega} (\mathbf{H}(\tilde{\mathbf{w}^+}), \mathbf{w}^-((\tilde{\mathbf{w}^+})), \mathbf{n}), \mathbf{v}) & = & 0
+@f}
+where $\tilde{\mathbf{w}} = \theta \mathbf{w}_{n+1} + (1-\theta) \mathbf{w}_n$ for $0 \leq \theta \leq 1$ and
+$\mathbf{w}_i = \sum_k \mathbf{W}_i^k \mathbf{\phi}_k$.
+
+We choose the Lax-Friedrich's flux, 
+$\mathbf{H}(\mathbf{a},\mathbf{b},\mathbf{n}) = \frac{1}{2}(\mathbf{F}(\mathbf{a})\cdot \mathbf{n} + \mathbf{F}(\mathbf{b})\cdot \mathbf{n} + \alpha (\mathbf{a} - \mathbf{b}))$.
+
+We solve the nonlinear system by a Newton iteration, i.e. by iterating
+@f{eqnarray*}
+\partial R(\mathbf{W}^k) \delta \mathbf{W} & = & - R(\mathbf{W}^{k}) \\
+\mathbf{W}^{k+1} &=& \mathbf{W}^k + \delta \mathbf{W},
+@f}
+until $|R(\mathbf{W}^k)|$ (the residual) is sufficiently small.
+
+<h2> Auto-Differentiation </h2>
+
+Since computing the Jacobian $\partial R$ is a terrible beast, we use an automatic differentiation package,
+Sacado, to do this.  Sacado is a C++ type that supports basic arithmetic operators and functions
+such as <code> sqrt, sin, cos, pow,  </code> etc.  One declares a collection Sacado type and then declares
+some of this collection as degrees of freedom.  These variables are used in an algorithm, and as the variables
+are used, their sensitivities with respect to these degrees of freedom are continuously updated.
+One can imagine that for the full Jacobian, this could be prohibitively expensive.  However, we do
+not use the Sacado type for the entire computation, but only element by element.  The author has
+used this approach side by side with a hand coded Jacobian for the Incompressible Navier-Stokes problem
+and found the Sacado approach to be just as fast as using a hand coded Jacobian.  Since using the
+auto-differentiation requires only that one code the Residual $R(\mathbf{W})$, ensuring code correctness
+and maintaining code becomes tremendously more simple.  We show a very simple Sacado example:
+
+@code
+#include <Sacado.hpp>
+
+typedef Sacado::Fad::DFad<double> fad_double;
+
+main() {
+
+  fad_double a,b,c;
+
+  a.diff(0,2);  // Set a to be dof 0, in a 2-dof system.
+
+  b.diff(1,2);  // Set b to be dof 1, in a 2-dof system.
+
+  a = 1; b = 2;
+
+  c = 2*a+cos(a*b);
+
+  double *derivs = c.fastAccessDx(0); // Access derivatives
+
+  std::cout << "dc/da = " << derivs[0] << ", dc/db=" << derivs[1] << std::endl;
+
+}
+@endcode
+
+It should be noted that Sacado provides more auto-differentation capabilities than the small subset
+that I use in this program.  However, if one understands the example above, they will understand
+how I use Sacado in this Euler flow program.
+
+<h2> Trilinos solvers </h2>
+The program uses either the Aztec iterative solvers, or the Amesos direct solver, as provided by
+the Trilinos package.  This package is inherently designed to be used in a parallel program, however,
+it may be used in serial just as easily, as is done here.  The Epetra package is the basic
+vector/matrix library upon which the solvers are built.  This very powerful package can be used
+to describe the parallel distribution of a vector, and to define sparse matrices that operate
+on these vectors.  Please view the commented code for more details on how these solvers are used
+within the example.
+
+<h2> Adaptivity </h2>
+The example uses an ad-hoc refinement indicator that shows some usefullness in shock-type problems, and
+in the downhill flow example included.  We refine according to the squared gradient of the density.  
+Hanging nodes are handled by using computing the numerical flux across cells that are of differing
+refinement levels.  In this way, the example combines the continuous and DG methodologies.
+
+Further, we enforce a maximum number of refinement levels to keep refinement under check.  It is the
+author's experience that adaptivity in a time dependent problem, refinement can easily lead the simulation to
+a screeching halt if care is not taken.  The amount of refinement is
+limited in the example by letting the user specify the
+maximum level of refinement that will be present anywhere in the mesh.  In this way, refinement
+tends not to slow the simulation to a halt.  This, of course, is purely a heuristic strategy, and
+if the author's advisor heard about it, the author would likely be exiled forever from the finite
+ element error estimation community.
+
+<h2> Input Deck </h2>
+
+We use an input file deck to drive the simulation.  In this way, we can alter the boundary conditions
+and other important properties of the simulation without having to recompile.  For more information on 
+the format, look at the results section, where we describe an example file in more detail.
+
+
+
+

diff --git a/deal.II/examples/step-33/doc/results.dox b/deal.II/examples/step-33/doc/results.dox
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+++ b/deal.II/examples/step-33/doc/results.dox
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+<h1>Results</h1>
+
+We run the problem with the mesh slide.inp and the following input deck:
+@verbatim
+# Listing of Parameters
+# ---------------------
+
+# The input grid 
+set mesh=slide.inp
+
+# Stabilization parameter
+set diffusion power = 2.0
+
+# Scaled value for gravity.  Positive means gravity points down.
+set gravity = 1.0
+
+# Boundary conditions
+# We may specify boundary conditions for up to MAX_BD boundaries.
+# Your .inp file should have these boundaries designated.
+#
+subsection boundary_1
+  set no penetration = true # reflective boundary condition
+end
+
+subsection boundary_2
+# outflow boundary
+  #set w_3 = pressure
+  #set w_3 value = 1.5 - y
+end
+
+subsection boundary_3
+  set no penetration = true # reflective
+  #set w_3 = pressure
+  #set w_3 value = 1.0
+end
+subsection boundary_4
+  set no penetration = true #reflective
+end
+
+# Initial Conditions
+# We set the initial conditions of the conservative variables.  These lines
+# are passed to the expression parsing function.  You should use x,y,z for
+# the coordinate variables.
+#
+subsection initial condition
+  set w_0 value = 0
+  set w_1 value = 0
+  set w_2 value   = 10*(x<-0.7)*(y> 0.3)*(y< 0.45) + (1-(x<-0.7)*(y> 0.3)*(y< 0.45))*1.0
+  set w_3 value = (1.5-(1.0*1.0*y))/0.4
+end
+
+# Time stepping control
+#
+subsection time stepping
+  set final time = 10.0 # simulation end time
+  set time step  = 0.02 # simulation time step
+end
+
+subsection linear solver
+  set output         = quiet
+  set method         = gmres
+  set ilut fill      = 1.5
+  set ilut drop tolerance = 1e-6
+  set ilut absolute tolerance = 1e-6
+  set ilut relative tolerance = 1.0
+end
+
+# Output frequency.
+# You may wish to set this > time step if you dont want output at every step
+subsection output
+  set step = 0.01
+end
+
+# Refinement control
+subsection refinement
+  set refinement = shock # none only other option
+  set shock value = 1.5
+  set shock levels = 1 # how many levels of refinement to allow
+end
+
+# Flux parameters
+subsection flux
+ set stab = mesh
+ #set stab value = 1.0
+end
+@endverbatim
+
+When we run the program, we get the following kind of output:
+@code
+T=3.14,    Number of active cells:       2617
+   Number of degrees of freedom: 11184
+NonLin Res:       Lin Iter     Lin Res
+______________________________________
+* 1.684e-02        0007        1.84e-13
+* 4.414e-05        0006        3.82e-15
+* 1.750e-09        0006        1.56e-19
+* 6.831e-16        0000        1.56e-19
+max_df:124
+T=3.16,    Number of active cells:       2626
+   Number of degrees of freedom: 11220
+NonLin Res:       Lin Iter     Lin Res
+______________________________________
+* 1.673e-02        0007        9.64e-14
+* 4.230e-05        0006        3.83e-15
+* 8.748e-10        0006        7.67e-20
+* 6.804e-16        0000        7.67e-20
+max_df:124
+T=3.18,    Number of active cells:       2644
+   Number of degrees of freedom: 11296
+NonLin Res:       Lin Iter     Lin Res
+______________________________________
+* 1.674e-02        0007        9.29e-14
+* 3.306e-05        0007        8.07e-17
+* 4.660e-10        0006        4.05e-20
+* 6.898e-16        0000        4.05e-20
+max_df:128
+T=3.2,    Number of active cells:       2647
+   Number of degrees of freedom: 11312
+NonLin Res:       Lin Iter     Lin Res
+______________________________________
+...
+@endcode
+
+This output reports the progress of the Newton iterations and the time stepping.
+
+@image html step-33.slide.gif
+
+As we see, the heavy mass of fluid is drawn down the slope by gravity, where it collides
+with the ski lodge and is flung into the air!  Hopefully everyone escapes!
+
+@image html step-33.slide_adapt.gif
+
+The adaptivity follows and preceeds the flow pattern, based on this heuristic refinement scheme.
+

diff --git a/deal.II/examples/step-33/doc/step-33.slide.gif b/deal.II/examples/step-33/doc/step-33.slide.gif
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diff --git a/deal.II/examples/step-33/doc/step-33.slide_adapt.gif b/deal.II/examples/step-33/doc/step-33.slide_adapt.gif
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diff --git a/deal.II/examples/step-33/slide.inp b/deal.II/examples/step-33/slide.inp
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+++ b/deal.II/examples/step-33/slide.inp
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+

diff --git a/deal.II/examples/step-33/step-33.cc b/deal.II/examples/step-33/step-33.cc
new file mode 100644 (file)
index 0000000..bb2389e
--- /dev/null
+++ b/deal.II/examples/step-33/step-33.cc
@@ -0,0 +1,2114 @@
+/* step-12.cc,v 1.33 2005/08/08 16:41:40 wolf Exp */\r
+/* Author: David Neckels, Boulder Colorado 2007  */\r
+/*    step-12.cc,v 1.33 2005/08/08 16:41:40 wolf Exp       */\r
+/*    Version: Version-5-2-0                                          */\r
+/*                                                                */\r
+/*    Copyright (C) 2001, 2002, 2003, 2004, 2005 by the deal.II authors */\r
+/*                                                                */\r
+/*    This file is subject to QPL and may not be  distributed     */\r
+/*    without copyright and license information. Please refer     */\r
+/*    to the file deal.II/doc/license.html for the  text  and     */\r
+/*    further information on this license.                        */\r
+\r
+                                  // This program solves the Euler equations\r
+                                  // of gas dynamics for a given configuration\r
+                                  // file.  It uses a standard Galerkin approach\r
+                                  // with weakly applied boundary conditions.\r
+ \r
+                                  // @sect3{Include files}\r
+\r
+                                  // Aztecoo require mpi (even though we run on only\r
+                                  // one processor in this example).\r
+#include <mpi.h>\r
+\r
+                                 // Here we have the necessary TRILINOS includes.\r
+                                 //\r
+                                 // Epetra is the basic trilinos vector/matrix library.\r
+#include <Epetra_MpiComm.h>\r
+#include <Epetra_Map.h>\r
+#include <Epetra_CrsGraph.h>\r
+#include <Epetra_CrsMatrix.h>\r
+#include <Epetra_Vector.h>\r
+                                 // Teuchos is a Trilinos utility library that is used\r
+                                 // to set parameters within the Aztec solver library.\r
+#include "Teuchos_ParameterList.hpp"\r
+                                 // Aztec is the iterative solver library.\r
+#include <AztecOO.h>\r
+#include <AztecOO_Operator.h>\r
+#define HAVE_IFPACK_TEUCHOS\r
+#include <Ifpack.h>\r
+\r
+                                 // Amesos is a direct solver package within Trilinos.\r
+#include <Amesos.h>\r
+                                 // Sacado is the automatic differentiation package, which\r
+                                 // is used to find the jacobian for a fully implicit Newton\r
+                                 // iteration.\r
+#include <Sacado.hpp>\r
+\r
+                                 // A standard set of dealii includes.  Nothing special to\r
+                                 // comment on here.\r
+#include <base/quadrature_lib.h>\r
+#include <base/function.h>\r
+#include <base/parameter_handler.h>\r
+#include <base/function_parser.h>\r
+\r
+#include <lac/vector.h>\r
+#include <lac/sparse_matrix.h>\r
+#include <lac/vector_memory.h>\r
+\r
+#include <grid/tria.h>\r
+#include <grid/grid_generator.h>\r
+#include <grid/grid_out.h>\r
+#include <grid/grid_refinement.h>\r
+#include <grid/tria_accessor.h>\r
+#include <grid/tria_iterator.h>\r
+#include <grid/grid_in.h>\r
+\r
+#include <fe/fe_values.h>\r
+#include <fe/fe_system.h>\r
+\r
+#include <dofs/dof_handler.h>\r
+#include <dofs/dof_accessor.h>\r
+#include <dofs/dof_tools.h>\r
+\r
+#include <numerics/data_out.h>\r
+#include <numerics/vectors.h>\r
+#include <numerics/solution_transfer.h>\r
+\r
+#include <fe/mapping_q1.h>\r
+#include <fe/fe_q.h>\r
+#include <numerics/derivative_approximation.h>\r
+                                // And this again is C++:\r
+#include <iostream>\r
+#include <fstream>\r
+#include <vector>\r
+\r
+                                // Introduce the dealii library into the current namespace.\r
+using namespace dealii;\r
+\r
+                                // We define a shorter name for the automatic differentiation\r
+                                // type.\r
+typedef Sacado::Fad::DFad<double> fad_double;\r
+typedef unsigned int UInt;\r
+                                // The Epetra library requires a 'communicator', which describes\r
+                                // the layout of a parallel (or serial) set of processors.\r
+Epetra_MpiComm *Comm;\r
+\r
+                                //@sect3{Flux function definition}\r
+                                // Here we define the flux function for this system of conservation\r
+                                // laws.  Note: it would be terribly difficult to use this example\r
+                                // to solve some other system of conservation laws.\r
+                                //\r
+                                // We define the number of components in the system.  Euler's has\r
+                                // one entry for momenta in each spatial direction, plus the energy\r
+                                // and density components.\r
+#define N_COMP (2 + DIMENSION)\r
+                                // Define a handle to the density and energy indices.  We have arrange\r
+                                // the momenta to be first, then density, and, lastly, energy.\r
+#define DENS_IDX DIMENSION\r
+#define ENERGY_IDX (DIMENSION+1)\r
+\r
+                                // The gas constant.  This value is representative of air.\r
+const double GAMMA = 1.4;\r
+                                // We define the flux functions as one large matrix.  Each row of this\r
+                                // matrix represents a scalar conservation law for the component in\r
+                                // that row.  We template the numerical type of the flux function\r
+                                // so that we may use the automatic differentiation type here.\r
+                                // The flux functions are defined in terms of the\r
+                                // conserved variables $\rho w_0, \dots, \rho w_{d-1}, \rho, E$,\r
+                                // so they do not look exactly like the Euler equations one is\r
+                                // used to seeing.  We evaluate the flux at a single quadrature\r
+                                // point.\r
+template <typename number, int dim>\r
+void Flux(std::vector<std::vector<number> >  &flux, \r
+          const Point<dim> &point, \r
+          const std::vector<number> &W)\r
+{\r
+\r
+                               // Pressure is a dependent variable: $p = \r
+                               // (\gamma - 1)(E-\frac{1}{2} \rho |v|^2)$.\r
+    number rho_normVsqr;\r
+    for (int d0 = 0; d0 < dim; d0++) rho_normVsqr += W[d0]*W[d0];\r
+                               // Since W are $\rho v$, we get a $\rho^2$ in the\r
+                               // numerator, so dividing a $\rho$ out gives the desired $ \rho |v|^2$.\r
+    rho_normVsqr /= W[DENS_IDX];\r
+\r
+    number pressure = (GAMMA-1.0)*(W[ENERGY_IDX] - number(0.5)*(rho_normVsqr));\r
+\r
+                               // We compute the momentum terms.  We divide by the\r
+                               // density here to get $v_i \rho v_j$\r
+    for (int d = 0; d < dim; d++) {\r
+      for (int d1 = 0; d1 < dim; d1++) {\r
+        flux[d][d1] = W[d]*W[d1]/W[DENS_IDX];\r
+      }\r
+                              // The pressure contribution, along the diagonal:\r
+      flux[d][d] += pressure;\r
+                              // Advection/conservation of density:\r
+      flux[DENS_IDX][d] = W[d]; \r
+                              // And, lastly, conservation of energy.\r
+      flux[ENERGY_IDX][d] = W[d]/W[DENS_IDX]*\r
+                               (W[ENERGY_IDX] + pressure); // energy\r
+    }\r
+}\r
+\r
+                              // On the boundaries of the domain and across `hanging nodes` we use\r
+                              // a numerical flux function to enforce boundary conditions.  This routine\r
+                              // is the basic Lax-Friedrich's flux with a stabilization parameter\r
+                              // $\alpha$.\r
+template <typename number, int dim>\r
+void LFNumFlux(\r
+            std::vector<std::vector<fad_double> > &nflux,\r
+            const std::vector<Point<dim> > &points, \r
+            const std::vector<Point<dim> > &normals,\r
+            const std::vector<std::vector<number> > &Wplus,\r
+            const std::vector<std::vector<number> > &Wminus,\r
+            double alpha)\r
+{\r
+  int n_q_points = points.size();\r
+\r
+                             // We evaluate the flux at each of the quadrature points.\r
+  for (int q = 0; q < n_q_points; q++) {\r
+    std::vector<std::vector<fad_double> > iflux(N_COMP,\r
+                                std::vector<fad_double>(dim, 0));\r
+    std::vector<std::vector<fad_double> > oflux(N_COMP,\r
+                                std::vector<fad_double>(dim, 0));\r
+\r
+    Flux<number, dim>(iflux, points[q], Wplus[q]);\r
+    Flux<number, dim>(oflux, points[q], Wminus[q]);\r
+\r
+    for (int di = 0; di < N_COMP; di++) {\r
+      nflux[q][di] = 0;\r
+      for (int d = 0; d < dim; d++) {\r
+        nflux[q][di] += 0.5*(iflux[di][d] + oflux[di][d])*normals[q](d);\r
+      }\r
+        nflux[q][di] += 0.5*alpha*(Wplus[q][di] - Wminus[q][di]);\r
+    }\r
+  }\r
+\r
+}\r
+\r
+                            // @sect3{Initial and side condition parsing}\r
+                            // For the initial condition we use the expression parser function\r
+                            // object.\r
+template <int dim>\r
+class InitialCondition :  public FunctionParser<dim> \r
+{\r
+  public:\r
+    InitialCondition ();\r
+    \r
+                            // This function should be called after parsing, but before using\r
+                            // the object.  It formalizes the expressions and initializes the\r
+                            // function parser with the appropriate expressions.\r
+    void Init();\r
+\r
+                            // During parsing we call this function as the initial condition\r
+                            // for one of the $\mathbf{w}$ variables is encountered.\r
+\r
+    void set_ic(int _row, std::string &expr) {\r
+      expressions[_row] = expr;\r
+    }\r
+\r
+    virtual void vector_value_list (const std::vector<Point<dim> > &points,\r
+                                   std::vector<Vector<double> >   &value_list) const;\r
+ private:\r
+ std::vector<std::string> expressions;\r
+};\r
+\r
+template <int dim>\r
+InitialCondition<dim>::InitialCondition () :\r
+               FunctionParser<dim> (N_COMP),\r
+                expressions(N_COMP, "0.0")\r
+{}\r
+\r
+                            // Here we set up x,y,z as the variables that one should use in the input\r
+                            // deck to describe their initial condition.\r
+template<int dim>\r
+void InitialCondition<dim>::Init() {\r
+ std::map<std::string, double> constants;\r
+ constants["M_PI"] =  M_PI;\r
+ std::string variables = (dim == 2 ? "x,y" : "x,y,z");\r
+\r
+ FunctionParser<dim>::initialize(variables, expressions, constants);\r
+\r
+}\r
+\r
+template <int dim>\r
+void InitialCondition<dim>::vector_value_list (const std::vector<Point<dim> > &points,\r
+                                           std::vector<Vector<double> >   &value_list) const \r
+{\r
+  const unsigned int n_points = points.size();\r
+\r
+  Assert (value_list.size() == n_points, \r
+         ExcDimensionMismatch (value_list.size(), n_points));\r
+\r
+  for (unsigned int p=0; p<n_points; ++p)\r
+    InitialCondition<dim>::vector_value (points[p],\r
+                                     value_list[p]);\r
+}\r
+\r
+                             // As above, we use the expression function parser for boundary conditions.\r
+template <int dim>\r
+class SideCondition :  public FunctionParser<dim> \r
+{\r
+  public:\r
+    SideCondition (int ncomp);\r
+    ~SideCondition ();\r
+\r
+                            // As above.\r
+    void Init();\r
+                           // As above.\r
+    void set_coeff_row(int _row_n, std::string &expr);\r
+    \r
+    virtual void vector_value_list (const std::vector<Point<dim> > &points,\r
+                                   std::vector<Vector<double> >   &value_list) const;\r
+  private:\r
+  std::vector<std::string> expressions;\r
+};\r
+\r
+template <int dim>\r
+SideCondition<dim>::SideCondition (int ncomp) :\r
+               FunctionParser<dim> (ncomp),\r
+                expressions(ncomp, "0.0")\r
+{\r
+}\r
+template <int dim>\r
+void SideCondition<dim>::set_coeff_row (int _row_n, std::string &expr) \r
+{\r
+ expressions[_row_n] = expr;\r
+}\r
+\r
+template <int dim>\r
+void SideCondition<dim>::Init() {\r
+ std::map<std::string, double> constants;\r
+ constants["M_PI"] =  M_PI;\r
+ std::string variables = (dim == 2 ? "x,y" : "x,y,z");\r
+\r
+ FunctionParser<dim>::initialize(variables, expressions, constants);\r
+\r
+}\r
+\r
+template <int dim>\r
+SideCondition<dim>::~SideCondition () \r
+{\r
+}\r
+\r
+template <int dim>\r
+void SideCondition<dim>::vector_value_list (const std::vector<Point<dim> > &points,\r
+                                           std::vector<Vector<double> >   &value_list) const \r
+{\r
+  const unsigned int n_points = points.size();\r
+\r
+  Assert (value_list.size() == n_points, \r
+         ExcDimensionMismatch (value_list.size(), n_points));\r
+\r
+  for (unsigned int p=0; p<n_points; ++p)\r
+    SideCondition<dim>::vector_value (points[p],\r
+                                     value_list[p]);\r
+}\r
+                            //@sect3{Conservation Law class}\r
+                           // Here we define a Conservation Law class that helps group\r
+                           // operations and data for our Euler equations into a manageable\r
+                           // entity.  Functions will be described as their definitions appear.\r
+template <int dim>\r
+class ConsLaw\r
+{\r
+  public:\r
+    ConsLaw ();\r
+    ~ConsLaw ();\r
+\r
+    void run ();\r
+    void declare_parameters();\r
+    void load_parameters(const char *);\r
+    \r
+  private:\r
+    void build_fe();\r
+    void setup_system ();\r
+    void initialize_system ();\r
+    void assemble_system (double &res_norm);\r
+    void solve (Vector<double> &solution, int &, double &);\r
+    void refine_grid ();\r
+    void output_results (const unsigned int cycle) const;\r
+    void initialize();\r
+    void zero_matrix();\r
+    void estimate();\r
+    void postprocess();\r
+    void compute_predictor();\r
+    \r
+    Triangulation<dim>   triangulation;\r
+    const MappingQ1<dim> mapping;\r
+    \r
+    \r
+    FESystem<dim>        *fe_ptr;\r
+\r
+    DoFHandler<dim>      dof_handler;\r
+\r
+    SparsityPattern      sparsity_pattern;\r
+    const QGauss<dim>   quadrature;\r
+    const QGauss<dim-1> face_quadrature;\r
+    \r
+                                     // The actual solution to the Euler equation\r
+    Vector<double>       solution;\r
+                                     // The current value of the solution during the Newton iteration\r
+    Vector<double>       nlsolution;\r
+                                     // An estimate of the next time value; used for adaptivity and as a\r
+                                     // guess for the next Newton iteration.\r
+    Vector<double>       predictor;\r
+                                     // Values after post-processing (used to output the physical variables).\r
+    Vector<double>       ppsolution;\r
+                                     // The solution to the linear problem during the Newton iteration\r
+    Vector<double>       dsolution;\r
+    Vector<double>       right_hand_side;\r
+    \r
+  public:\r
+\r
+    void assemble_cell_term(const FEValues<dim>& fe_v,\r
+                            std::vector<unsigned int> &dofs,\r
+                            unsigned int cell_no\r
+                            );\r
+    \r
+    void assemble_face_term(\r
+                            int face_no,\r
+                            const FEFaceValuesBase<dim>& fe_v,\r
+                            const FEFaceValuesBase<dim>& fe_v_neighbor,\r
+                             std::vector<unsigned int> &dofs,\r
+                             std::vector<unsigned int> &dofs_neighbor,\r
+                             int boundary = -1\r
+                             );\r
+\r
+    unsigned int get_n_components() const { return N_COMP;}\r
+\r
+  private:\r
+                                    // T = current time, dT = time step, TF = final time.\r
+    double T, dT, TF;\r
+    double face_diameter;\r
+    double cell_diameter;\r
+                                    // An object to handle parsing the input deck.\r
+    ParameterHandler prm;\r
+                                    // Name of the mesh to read in.\r
+    string mesh;\r
+    InitialCondition<dim> ic;\r
+\r
+                                    // Enums for the various supported boundary conditions.\r
+    typedef enum {INFLOW_BC = 1, OUTFLOW_BC=2, NO_PENETRATION_BC=3, PRESSURE_BC=4} bc_type;\r
+\r
+                                    // For each boundary we store a map from boundary # to the type\r
+                                    // of boundary condition.  If the boundary condition is prescribed,\r
+                                    // we store a pointer to a function object that will hold the expression\r
+                                    // for that boundary condition.\r
+    typedef typename std::map<unsigned int, std::pair<std::vector<bc_type>, Function<dim>*> > bdry_map_type;\r
+    bdry_map_type bdry_map;\r
+\r
+    void add_boundary(unsigned int bd, std::vector<bc_type>& flags, Function<dim> *bf);\r
+\r
+                                    // An object to store parameter information about the Aztec solver.\r
+    typedef struct {\r
+      int LIN_OUTPUT;\r
+      typedef enum { GMRES = 0, DIRECT = 1} solver_type;\r
+      solver_type SOLVER;\r
+      typedef enum { QUIET = 0, VERBOSE = 1 } output_type;\r
+      output_type OUTPUT;\r
+                                    // Linear residual tolerance.\r
+      double RES;\r
+      int MAX_ITERS;\r
+                                    // We use the ILUT preconditioner.  This is similar\r
+                                    // to the ILU.  FILL is the number of extra entries\r
+                                    // to add when forming the ILU decomposition.\r
+      double ILUT_FILL;\r
+                                    // When forming the preconditioner, for certain problems\r
+                                    // bad conditioning (or just bad luck) can cause the\r
+                                    // preconditioner to be very poorly conditioned.  Hence\r
+                                    // it can help to add diagonal perturbations to the\r
+                                    // original matrix and form the preconditioner for this\r
+                                    // slightly better matrix.  ATOL is an absolute perturbation\r
+                                    // that is added to the diagonal before forming the\r
+                                    // prec, and RTOL is a scaling factor $rtol >= 1$.\r
+      double ILUT_ATOL;\r
+      double ILUT_RTOL;\r
+                                    // The ILUT will drop any values that have magnitude less\r
+                                    // than this value.  This is a way to\r
+                                    // manage the amount of memory used by this preconditioner.\r
+      double ILUT_DROP;\r
+    } solver_params_type;\r
+\r
+    solver_params_type solver_params;\r
+\r
+                                    // Some refinement parameters.\r
+    typedef struct {\r
+      typedef enum { NONE = 0, FIXED_NUMBER = 1, SHOCK = 2} refine_type;\r
+      double high_frac;\r
+      double low_frac;\r
+      refine_type refine;\r
+      double high_frac_sav;\r
+      double max_cells;\r
+      double shock_val;\r
+      double shock_levels;\r
+    } refinement_params_type;\r
+\r
+    refinement_params_type refinement_params;\r
+\r
+                                    // The user can set the stabilization parameter $\alpha$ \r
+                                    // in the Lax-Friedrich's flux.  \r
+    typedef struct {\r
+      typedef enum {CONSTANT=1,MESH=2} LF_stab_type;\r
+      LF_stab_type LF_stab;\r
+      double LF_stab_value;\r
+    } flux_params_type;\r
+\r
+    flux_params_type flux_params;\r
+\r
+    bool is_stationary;\r
+\r
+                                    // Power for the mesh stabilization term.\r
+    double diffusion_power;\r
+    double gravity;\r
+                                    // If true, we output the squared gradient of the\r
+                                    // density instead of density.  Using this one can\r
+                                    // create shock plots.\r
+    bool schlieren_plot;\r
+                                    // How often to create an output file.\r
+    double output_step;\r
+\r
+    Epetra_CrsMatrix   *Matrix;\r
+    Epetra_Map         *Map;\r
+    Vector<double>      indicator;\r
+ \r
+                                   // Crank-Nicolson value\r
+    const double        theta; \r
+\r
+};\r
+\r
+\r
+                                    // Asign a row of the conservation law a specified\r
+                                    // boundary type and (possibly) function.\r
+template <int dim>\r
+void ConsLaw<dim>::add_boundary(unsigned int bd,\r
+        std::vector<bc_type> &flags, Function<dim> *bf) {\r
+\r
+  std::pair<std::vector<bc_type>, Function<dim> *> entry(flags, bf);\r
+  bdry_map[bd] = entry;\r
+}\r
+\r
+\r
+                                    // Apply the initialial condition.  Simultaneously\r
+                                    // initialize the non-linear solution.\r
+template <int dim>\r
+void ConsLaw<dim>::initialize() {\r
+ VectorTools::interpolate(dof_handler,\r
+                           ic, solution);\r
+ VectorTools::interpolate(dof_handler,\r
+                           ic, nlsolution);\r
+\r
+}\r
+\r
+                                // @sect3{Assembly}\r
+                                // @sect4{Function: assemble_cell_term}\r
+                                //\r
+                                 // Assembles the cell term, adding minus the residual\r
+                                 // to the right hand side, and adding in the Jacobian\r
+                                 // contributions.\r
+template <int dim>\r
+void ConsLaw<dim>::assemble_cell_term(\r
+  const FEValues<dim> &fe_v,\r
+  std::vector<unsigned int> &dofs,\r
+  unsigned int cell_no\r
+  ) \r
+{\r
+                                 // The residual for each row (i) will be accumulating \r
+                                 // into this fad variable.  At the end of the assembly\r
+                                 // for this row, we will query for the sensitivities\r
+                                 // to this variable and add them into the Jacobian.\r
+  fad_double F_i;\r
+  int dofs_per_cell = fe_v.dofs_per_cell;\r
+  int n_q_points = fe_v.n_quadrature_points;\r
+\r
+                                 // We will define the dofs on this cell in these fad variables.\r
+  std::vector<fad_double> DOF(dofs_per_cell);\r
+\r
+                                 // Values of the conservative variables at the quadrature points.\r
+  std::vector<std::vector<fad_double > > W (n_q_points,\r
+                                               std::vector<fad_double >(get_n_components()));\r
+\r
+                                 // Values at the last time step of the conservative variables.\r
+                                 // Note that these do not use fad variables, since they do\r
+                                 // not depend on the 'variables to be sought'=DOFS.\r
+  std::vector<std::vector<double > > Wl (n_q_points,\r
+                                               std::vector<double >(get_n_components()));\r
+\r
+                                 // Here we will hold the averaged values of the conservative\r
+                                 // variables that we will linearize around (cn=Crank Nicholson).\r
+  std::vector<std::vector<fad_double > > Wcn (n_q_points,\r
+                                               std::vector<fad_double >(get_n_components()));\r
+\r
+                                 // Gradients of the current variables.  It is a\r
+                                 // bit of a shame that we have to compute these; we almost don't.\r
+                                 // The nice thing about a simple conservation law is that the\r
+                                 // the flux doesn't generally involve any gradients.  We do\r
+                                 // need these, however, for the diffusion stabilization. \r
+   std::vector<std::vector<std::vector<fad_double> > > Wgrads (n_q_points,\r
+            std::vector<std::vector<fad_double> >(get_n_components(),\r
+                            std::vector<fad_double>(dim)));\r
+\r
+\r
+  const std::vector<double> &JxW = fe_v.get_JxW_values ();\r
+\r
+  \r
+                                  // Here is the magical point where we declare a subset\r
+                                  // of the fad variables as degrees of freedom.  All \r
+                                  // calculations that reference these variables (either\r
+                                  // directly or indirectly) will accumulate sensitivies\r
+                                  // with respect to these dofs.\r
+  for (int in = 0; in < dofs_per_cell; in++) {\r
+      DOF[in] = nlsolution(dofs[in]);\r
+      DOF[in].diff(in, dofs_per_cell);\r
+  }\r
+\r
+                                  // Here we compute the shape function values and gradients\r
+                                  // at the quadrature points.  Ideally, we could call into \r
+                                  // something like get_function_values, get_function_grads,\r
+                                  // but since we don't want to make the entire solution vector\r
+                                  // fad types, only the local cell variables, we explicitly\r
+                                  // code this loop;\r
+  for (int q = 0; q < n_q_points; q++) {\r
+    for (int di = 0; di < get_n_components(); di++) {\r
+      W[q][di] = 0;\r
+      Wl[q][di] = 0;\r
+      Wcn[q][di] = 0;\r
+      for (int d = 0; d < dim; d++) {\r
+        Wgrads[q][di][d] = 0;\r
+      }\r
+    }\r
+    for (int sf = 0; sf < dofs_per_cell; sf++) {\r
+     int di = fe_v.get_fe().system_to_component_index(sf).first;\r
+     W[q][di] +=\r
+                DOF[sf]*fe_v.shape_value_component(sf, q, di);\r
+     Wl[q][di] +=\r
+                solution(dofs[sf])*fe_v.shape_value_component(sf, q, di);\r
+     Wcn[q][di] +=\r
+                (theta*DOF[sf]+(1-theta)*solution(dofs[sf]))*fe_v.shape_value_component(sf, q, di);\r
+\r
+     for (int d = 0; d < dim; d++) {\r
+       Wgrads[q][di][d] += DOF[sf]*\r
+                 fe_v.shape_grad_component(sf, q, di)[d];\r
+     } // for d\r
+\r
+    }\r
+\r
+  } // for q\r
+\r
+                                   // Gather the flux values for all components at\r
+                                   // all of the quadrature points.  This also\r
+                                   // computes the matrix of sensitivities.  Perhaps\r
+                                   // this could be done in a better way, since this\r
+                                   // could be a rather large object, but for now it \r
+                                   // seems to work just fine.\r
+   std::vector<std::vector<std::vector<fad_double> > > flux(n_q_points, \r
+                                std::vector<std::vector<fad_double> >(get_n_components(),\r
+                                std::vector<fad_double>(dim, 0)));\r
+\r
+    for (unsigned int q=0; q < n_q_points; ++q) {\r
+      Flux<fad_double, dim>(flux[q], fe_v.get_quadrature_points()[q], Wcn[q]);\r
+    }\r
+\r
+                                    // We now have all of the function values/grads/fluxes,\r
+                                    // so perform the assembly.  We have an outer loop\r
+                                    // through the components of the system, and an\r
+                                    // inner loop over the quadrature points, where we\r
+                                    // accumulate contributions to the ith residual.\r
+                                    //\r
+                                    // We initialy sum all contributions of the residual\r
+                                    // in the positive sense, so that we don't need to\r
+                                    // negative the Jacobian entries.  Then, when we sum\r
+                                    // into the <code> right_hand_side </code> vector,\r
+                                    // we negate this residual.\r
+    for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i) \r
+      {\r
+                                    // Find which component this dof contributes to.\r
+        const unsigned int\r
+          component_i = fe_v.get_fe().system_to_component_index(i).first;\r
+\r
+                                    // Initialize the fad residual to zero (removes\r
+                                    // any previous sensitivities.\r
+        F_i = 0;\r
+\r
+                                    // Loop quadrature points.\r
+        for (unsigned int point=0; point<fe_v.n_quadrature_points; ++point) {\r
+\r
+          fad_double fdotgv = 0;\r
+\r
+                                    // Integrate the flux times gradient of the test function\r
+          for (int d = 0; d < dim; d++) \r
+            fdotgv += flux[point][component_i][d]*fe_v.shape_grad_component(i, point, component_i)[d];\r
+           \r
+          F_i -= fdotgv*JxW[point];\r
+\r
+                                    // The mass term (if the simulation is non-stationary).\r
+          fad_double delta_t= 1.0/dT*(W[point][component_i] - Wl[point][component_i]);\r
+          if (!is_stationary) F_i += delta_t*\r
+                 fe_v.shape_value_component(i, point, component_i)*JxW[point];\r
+\r
+                                   // Stabilization (cell wise diffusion)\r
+          fad_double guv = 0;\r
+          for (int d = 0; d < dim; d++) {\r
+            guv += fe_v.shape_grad_component(i, point, component_i)[d]*\r
+                      Wgrads[point][component_i][d];\r
+          }\r
+\r
+            F_i += 1.0*std::pow(cell_diameter, diffusion_power)*guv*JxW[point];\r
+          \r
+                                   // The gravity component only enters into the energy \r
+                                   // equation and into the vertical component of the \r
+                                   // velocity.\r
+          if (component_i == dim - 1) {\r
+            F_i += gravity*Wcn[point][DENS_IDX]*fe_v.shape_value_component(i,point, component_i)*JxW[point];\r
+          } else if (component_i == ENERGY_IDX) {\r
+            F_i += gravity*Wcn[point][DENS_IDX]*Wcn[point][dim-1]*\r
+                   fe_v.shape_value_component(i,point, component_i)*JxW[point];\r
+          }\r
+        } // for q\r
+\r
+                                   // Here we gain access to the array of sensitivities\r
+                                   // of the residual.  We then sum these into the\r
+                                   // Epetra matrix.\r
+        double *values = &(F_i.fastAccessDx(0));\r
+        Matrix->SumIntoGlobalValues(dofs[i],\r
+            dofs_per_cell, &values[0], reinterpret_cast<int*>(&dofs[0]));\r
+ \r
+                                   // Add minus the residual to the right hand side.\r
+        right_hand_side(dofs[i]) -= F_i.val();\r
+\r
+      } // for i\r
+}\r
+                                   // @sect4{Function: assemble_face_term}\r
+                                   // These are either\r
+                                   // boundary terms or terms across differing \r
+                                   // levels of refinement.  In the first case,\r
+                                   // fe_v==fe_v_neighbor and dofs==dofs_neighbor.\r
+                                   // The int boundary < 0 if not at a boundary,\r
+                                   // otherwise it is the boundary indicator.\r
+template <int dim>\r
+void ConsLaw<dim>::assemble_face_term(\r
+  int face_no,\r
+  const FEFaceValuesBase<dim>& fe_v,\r
+  const FEFaceValuesBase<dim>& fe_v_neighbor,      \r
+  std::vector<unsigned int> &dofs,\r
+  std::vector<unsigned int> &dofs_neighbor,\r
+  int boundary\r
+  ) \r
+{\r
+  fad_double F_i;\r
+  const unsigned int n_q_points = fe_v.n_quadrature_points;\r
+  const unsigned int dofs_per_cell = fe_v.get_fe().dofs_per_cell;\r
+  const unsigned int ndofs_per_cell = fe_v_neighbor.get_fe().dofs_per_cell;\r
+  Assert(dofs_per_cell == ndofs_per_cell,\r
+        ExcDimensionMismatch(dofs_per_cell, ndofs_per_cell));\r
+\r
+                                  // As above, the fad degrees of freedom\r
+  std::vector<fad_double> DOF(dofs_per_cell+ndofs_per_cell);\r
+\r
+                                  // The conservative variables for this cell,\r
+                                  // and for \r
+  std::vector<std::vector<fad_double > > Wplus (n_q_points,\r
+                                               std::vector<fad_double >(get_n_components()));\r
+  std::vector<std::vector<fad_double > > Wminus (n_q_points,\r
+                                               std::vector<fad_double >(get_n_components()));\r
+\r
+\r
+  const std::vector<double> &JxW = fe_v.get_JxW_values ();\r
+  const std::vector<Point<dim> > &normals = fe_v.get_normal_vectors ();\r
+\r
+\r
+                                  // If we are at a boundary, then dofs_neighbor are\r
+                                  // the same as dofs, so we do not want to duplicate them.\r
+                                  // If there is a neighbor cell, then we want to include \r
+                                  // them.\r
+  int ndofs = (boundary < 0 ? dofs_per_cell + ndofs_per_cell : dofs_per_cell);\r
+                                  // Set the local DOFS.\r
+  for (int in = 0; in < dofs_per_cell; in++) {\r
+      DOF[in] = nlsolution(dofs[in]);\r
+      DOF[in].diff(in, ndofs);\r
+  }\r
+                                  // If present, set the neighbor dofs.\r
+  if (boundary < 0)\r
+  for (int in = 0; in < ndofs_per_cell; in++) {\r
+      DOF[in+dofs_per_cell] = nlsolution(dofs_neighbor[in]);\r
+      DOF[in+dofs_per_cell].diff(in+dofs_per_cell, ndofs);\r
+  }\r
+\r
+                                  // Set the values of the local conservative variables.\r
+                                  // Initialize all variables to zero.\r
+  for (int q = 0; q < n_q_points; q++) {\r
+    for (int di = 0; di < get_n_components(); di++) {\r
+           Wplus[q][di] = 0;\r
+           Wminus[q][di] = 0;\r
+    }\r
+    for (int sf = 0; sf < dofs_per_cell; sf++) {\r
+     int di = fe_v.get_fe().system_to_component_index(sf).first;\r
+     Wplus[q][di] +=\r
+                (theta*DOF[sf]+(1.0-theta)*solution(dofs[sf]))*fe_v.shape_value_component(sf, q, di);\r
+    }\r
+\r
+\r
+                                 // If there is a cell across, then initialize\r
+                                 // the exterior trace as a function of the other\r
+                                 // cell degrees of freedom.\r
+    if (boundary < 0) {\r
+      for (int sf = 0; sf < ndofs_per_cell; sf++) {\r
+       int di = fe_v_neighbor.get_fe().system_to_component_index(sf).first;\r
+       Wminus[q][di] +=\r
+                (theta*DOF[sf+dofs_per_cell]+(1.0-theta)*solution(dofs_neighbor[sf]))*\r
+                fe_v_neighbor.shape_value_component(sf, q, di);\r
+      }\r
+    } \r
+   } // for q\r
+\r
+                               // If this is a boundary, then the values of $W^-$ will\r
+                               // be either functions of $W^+$, or they will be prescribed.\r
+                               // This switch sets them appropriately.  Since we are\r
+                               // using fad variables here, sensitivities will be updated \r
+                               // appropriately.  These sensitivities would be tremendously\r
+                               // difficult to manage without fad!!!\r
+   if (boundary >= 0) {\r
+                               // Get the boundary descriptor.\r
+     typename bdry_map_type::iterator bme = bdry_map.find(boundary);\r
+     assert(bme != bdry_map.end());\r
+\r
+                             // Evaluate the function object.  This is a bit\r
+                             // tricky; a given boundary might have both prescribed\r
+                             // and implicit values.  If a particular component is not\r
+                             // prescribed, the values evaluate to zero and are\r
+                             // ignored, below.\r
+     std::vector<Vector<double> > bvals(n_q_points, Vector<double>(N_COMP));\r
+     bme->second.second->vector_value_list(fe_v.get_quadrature_points(), bvals);\r
+\r
+                             // We loop the quadrature points, and we treat each\r
+                             // component individualy.\r
+     for (int q = 0; q < n_q_points; q++) {\r
+      for (int di = 0; di < get_n_components(); di++) {\r
+\r
+                             // An inflow/dirichlet type of boundary condition\r
+        if (bme->second.first[di] == INFLOW_BC) {\r
+          Wminus[q][di] = bvals[q](di);\r
+        } else if (bme->second.first[di] == PRESSURE_BC) {\r
+                             // A prescribed pressure boundary condition.  This boundary\r
+                             // condition is complicated by the fact that even though\r
+                             // the pressure is prescribed, we really are setting\r
+                             // the energy index here, which will depend on velocity\r
+                             // and pressure. So even though this seems like a dirichlet\r
+                             // type boundary condition, we get sensitivities of\r
+                             // energy to velocity and density (unless these\r
+                             // are also prescribed.\r
+          fad_double rho_vel_sqr = 0;\r
+          fad_double dens;\r
+          \r
+          dens = bme->second.first[DENS_IDX] == INFLOW_BC ? bvals[q](DENS_IDX) :\r
+                 Wplus[q][DENS_IDX];\r
+\r
+          for (int d=0; d < dim; d++) {\r
+            if (bme->second.first[d] == INFLOW_BC)\r
+              rho_vel_sqr += bvals[q](d)*bvals[q](d);\r
+            else\r
+              rho_vel_sqr += Wplus[q][d]*Wplus[q][d];\r
+          }\r
+          rho_vel_sqr /= dens;\r
+                             // Finally set the energy value as determined by the\r
+                             // prescribed pressure and the other variables.\r
+          Wminus[q][di] = bvals[q](di)/(GAMMA-1.0) +\r
+                             0.5*rho_vel_sqr;\r
+\r
+        } else if (bme->second.first[di] == OUTFLOW_BC) {\r
+                            // A free/outflow boundary, very simple.\r
+          Wminus[q][di] = Wplus[q][di];\r
+\r
+        } else { \r
+                            // We must be at a no-penetration boundary.  We\r
+                            // prescribe the velocity (we are dealing with a\r
+                            // particular component here so that the average\r
+                            // of the velocities is orthogonal to the surface\r
+                            // normal.  This creates sensitivies of across\r
+                            // the velocity components.\r
+          fad_double vdotn = 0;\r
+          for (int d = 0; d < dim; d++) {\r
+            vdotn += Wplus[q][d]*normals[q](d);\r
+          }\r
+\r
+          Wminus[q][di] = Wplus[q][di] - 2.0*vdotn*normals[q](di);\r
+        }\r
+      }\r
+     } // for q\r
+   } // b>= 0\r
+   \r
+                           // Determine the Lax-Friedrich's stability parameter,\r
+                           // and evaluate the numerical flux function at the quadrature points\r
+   std::vector<std::vector<fad_double> > nflux(n_q_points, std::vector<fad_double>(get_n_components(), 0));\r
+     double alpha = 1;\r
+\r
+     switch(flux_params.LF_stab) {\r
+       case flux_params_type::CONSTANT:\r
+         alpha = flux_params.LF_stab_value;\r
+       break;\r
+       case flux_params_type::MESH:\r
+         alpha = face_diameter/(2.0*dT);\r
+       break;\r
+     }\r
+\r
+     LFNumFlux<fad_double, dim>(nflux, fe_v.get_quadrature_points(), normals, Wplus, Wminus,\r
+                     alpha);\r
+\r
+                          // Now assemble the face term\r
+     for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i) {\r
+        if (!fe_v.get_fe().has_support_on_face(i, face_no)) continue;\r
+        F_i = 0;\r
+        for (unsigned int point=0; point<n_q_points; ++point)\r
+        {\r
+          const unsigned int\r
+            component_i = fe_v.get_fe().system_to_component_index(i).first;\r
+\r
+          F_i += nflux[point][component_i]*fe_v.shape_value_component(i, point, component_i)*JxW[point];\r
+\r
+         } \r
+\r
+                          // Retrieve a pointer to the jacobian.\r
+        double *values = &(F_i.fastAccessDx(0));\r
+\r
+                          // Honestly, I forget why this can happen, but \r
+                          // for some reason it can!!\r
+        if (!values) continue;\r
+\r
+                          // Update the matrix.  Depending on whether there\r
+                          // is/isn't a neighboring cell, we add more/less\r
+                          // entries.\r
+        Matrix->SumIntoGlobalValues(dofs[i],\r
+          dofs_per_cell, &values[0], reinterpret_cast<int*>(&dofs[0]));\r
+        if (boundary < 0) {\r
+          Matrix->SumIntoGlobalValues(dofs[i],\r
+            dofs_per_cell, &values[dofs_per_cell], reinterpret_cast<int*>(&dofs_neighbor[0]));\r
+        }\r
+\r
+                          // And add into the residual\r
+        right_hand_side(dofs[i]) -= F_i.val();\r
+      } \r
+\r
+}\r
+                                 // @sect4{Assembling the whole system}\r
+                                 // Now we put all of the assembly pieces together\r
+                                 // in a routine that dispatches the correct\r
+                                 // piece for each cell/face.  We keep track of\r
+                                 // the norm of the resdual for the Newton iteration.\r
+template <int dim>\r
+void ConsLaw<dim>::assemble_system (double &res_norm) \r
+{\r
+  FESystem<dim> &fe = *fe_ptr;\r
+  const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;\r
+\r
+                                  // We track the dofs on this cell and (if necessary)\r
+                                  // the adjacent cell.\r
+  std::vector<unsigned int> dofs (dofs_per_cell);\r
+  std::vector<unsigned int> dofs_neighbor (dofs_per_cell);\r
+\r
+                                  // First we create the\r
+                                  // ``UpdateFlags'' for the\r
+                                  // ``FEValues'' and the\r
+                                  // ``FEFaceValues'' objects.\r
+  UpdateFlags update_flags = update_values\r
+                            | update_gradients\r
+                            | update_q_points\r
+                            | update_JxW_values;\r
+\r
+                                  // Note, that on faces we do not\r
+                                  // need gradients but we need\r
+                                  // normal vectors.\r
+  UpdateFlags face_update_flags = update_values\r
+                                 | update_q_points\r
+                                 | update_JxW_values\r
+                                 | update_normal_vectors;\r
+  \r
+                                  // On the neighboring cell we only\r
+                                  // need the shape values. Given a\r
+                                  // specific face, the quadrature\r
+                                  // points and `JxW values' are the\r
+                                  // same as for the current cells,\r
+                                  // the normal vectors are known to\r
+                                  // be the negative of the normal\r
+                                  // vectors of the current cell.\r
+  UpdateFlags neighbor_face_update_flags = update_values;\r
+   \r
+                                  // Then we create the ``FEValues''\r
+                                  // object. Note, that since version\r
+                                  // 3.2.0 of deal.II the constructor\r
+                                  // of this class takes a\r
+                                  // ``Mapping'' object as first\r
+                                  // argument. Although the\r
+                                  // constructor without ``Mapping''\r
+                                  // argument is still supported it\r
+                                  // is recommended to use the new\r
+                                  // constructor. This reduces the\r
+                                  // effect of `hidden magic' (the\r
+                                  // old constructor implicitely\r
+                                  // assumes a ``MappingQ1'' mapping)\r
+                                  // and makes it easier to change\r
+                                  // the mapping object later.\r
+  FEValues<dim> fe_v (\r
+    mapping, fe, quadrature, update_flags);\r
+  \r
+                                  // Similarly we create the\r
+                                  // ``FEFaceValues'' and\r
+                                  // ``FESubfaceValues'' objects for\r
+                                  // both, the current and the\r
+                                  // neighboring cell. Within the\r
+                                  // following nested loop over all\r
+                                  // cells and all faces of the cell\r
+                                  // they will be reinited to the\r
+                                  // current cell and the face (and\r
+                                  // subface) number.\r
+  FEFaceValues<dim> fe_v_face (\r
+    mapping, fe, face_quadrature, face_update_flags);\r
+  FESubfaceValues<dim> fe_v_subface (\r
+    mapping, fe, face_quadrature, face_update_flags);\r
+  FEFaceValues<dim> fe_v_face_neighbor (\r
+    mapping, fe, face_quadrature, neighbor_face_update_flags);\r
+  FESubfaceValues<dim> fe_v_subface_neighbor (\r
+    mapping, fe, face_quadrature, neighbor_face_update_flags);\r
+\r
+                                  // Furthermore we need some cell\r
+                                  // iterators.\r
+  typename DoFHandler<dim>::active_cell_iterator\r
+    cell = dof_handler.begin_active(),\r
+    endc = dof_handler.end();\r
+\r
+                                  // Now we start the loop over all\r
+                                  // active cells.\r
+  int fdofs_per_cell = fe_v.dofs_per_cell;\r
+  int fn_q_points = face_quadrature.n_quadrature_points;\r
+\r
+  unsigned int cell_no = 0;\r
+  for (;cell!=endc; ++cell, ++cell_no) \r
+    {\r
+      \r
+                                      // Now we reinit the ``FEValues''\r
+                                      // object for the current cell\r
+      fe_v.reinit (cell);\r
+\r
+                                       // Collect the local dofs and\r
+                                       // asssemble the cell term.\r
+      cell->get_dof_indices (dofs);\r
+\r
+      cell_diameter = cell->diameter();\r
+\r
+      assemble_cell_term(fe_v,\r
+                         dofs,\r
+                         cell_no);\r
+\r
+                                       // We use the DG style loop through faces\r
+                                       // to determine if we need to apply a\r
+                                       // 'hanging node' flux calculation or a boundary\r
+                                       // computation.\r
+      for (unsigned int face_no=0; face_no<GeometryInfo<dim>::faces_per_cell; ++face_no)\r
+       {\r
+                                          // First we set the face\r
+                                          // iterator\r
+         typename DoFHandler<dim>::face_iterator face=cell->face(face_no);\r
+          face_diameter = face->diameter();\r
+         \r
+         if (face->at_boundary())\r
+           {\r
+                                              // We reinit the\r
+                                              // ``FEFaceValues''\r
+                                              // object to the\r
+                                              // current face\r
+             fe_v_face.reinit (cell, face_no);\r
+\r
+                                              // and assemble the\r
+                                              // corresponding face\r
+                                              // terms.  We send the same\r
+                                               // fe_v and dofs as described\r
+                                               // in the assembly routine.\r
+             assemble_face_term(\r
+                                   face_no, fe_v_face,\r
+                                   fe_v_face,\r
+                                   dofs,\r
+                                   dofs,\r
+                                   face->boundary_indicator());\r
+           }\r
+         else\r
+           {\r
+                                              // Now we are not on\r
+                                              // the boundary of the\r
+                                              // domain, therefore\r
+                                              // there must exist a\r
+                                              // neighboring cell.\r
+             typename DoFHandler<dim>::cell_iterator neighbor=\r
+               cell->neighbor(face_no);;\r
+\r
+             if (face->has_children())\r
+               {\r
+                  // case I: This cell refined compared to neighbor\r
+\r
+                 const unsigned int neighbor2=\r
+                   cell->neighbor_of_neighbor(face_no);\r
+                 \r
+                 \r
+                                                  // We loop over\r
+                                                  // subfaces\r
+                 for (unsigned int subface_no=0;\r
+                      subface_no<GeometryInfo<dim>::subfaces_per_face;\r
+                      ++subface_no)\r
+                   {\r
+                     typename DoFHandler<dim>::active_cell_iterator\r
+                        neighbor_child\r
+                        = cell->neighbor_child_on_subface (face_no, subface_no);\r
+\r
+                      face_diameter = neighbor_child->diameter();  // working on subface\r
+                     \r
+                     Assert (neighbor_child->face(neighbor2) == face->child(subface_no),\r
+                             ExcInternalError());\r
+                     Assert (!neighbor_child->has_children(), ExcInternalError());\r
+\r
+                     fe_v_subface.reinit (cell, face_no, subface_no);\r
+                     fe_v_face_neighbor.reinit (neighbor_child, neighbor2);\r
+                     neighbor_child->get_dof_indices (dofs_neighbor);\r
+\r
+                                               // Assemble as if we are working with\r
+                                               // a DG element.\r
+                     assemble_face_term(\r
+                                             face_no, fe_v_subface,\r
+                                            fe_v_face_neighbor,\r
+                                             dofs,\r
+                                             dofs_neighbor);\r
+                     \r
+                   }\r
+                                                  // End of ``if\r
+                                                  // (face->has_children())''\r
+               }\r
+             else\r
+               {\r
+                                                  // We have no children, but \r
+                                                  // the neighbor cell may be refine\r
+                                                  // compared to use\r
+                 neighbor->get_dof_indices (dofs_neighbor);\r
+                 if (neighbor->level() != cell->level()) \r
+                   {\r
+                      // case II: This is refined compared to neighbor\r
+                     Assert(neighbor->level() < cell->level(), ExcInternalError());\r
+                     const std::pair<unsigned int, unsigned int> faceno_subfaceno=\r
+                       cell->neighbor_of_coarser_neighbor(face_no);\r
+                     const unsigned int neighbor_face_no=faceno_subfaceno.first,\r
+                                     neighbor_subface_no=faceno_subfaceno.second;\r
+\r
+                     Assert (neighbor->neighbor_child_on_subface (neighbor_face_no,\r
+                                                                   neighbor_subface_no)\r
+                              == cell,\r
+                              ExcInternalError());\r
+\r
+                                                      // Reinit the\r
+                                                      // appropriate\r
+                                                      // ``FEFaceValues''\r
+                                                      // and assemble\r
+                                                      // the face\r
+                                                      // terms.\r
+                     fe_v_face.reinit (cell, face_no);\r
+                     fe_v_subface_neighbor.reinit (neighbor, neighbor_face_no,\r
+                                                   neighbor_subface_no);\r
+                     \r
+                     assemble_face_term(\r
+                                             face_no, fe_v_face,\r
+                                            fe_v_subface_neighbor,\r
+                                             dofs,\r
+                                             dofs_neighbor);\r
+\r
+                   }\r
+\r
+               } \r
+                                    // End of ``face not at boundary'':\r
+           }\r
+                                    // End of loop over all faces:\r
+       } \r
+      \r
+                                     // End iteration through cells.\r
+    } \r
+\r
+                                     // Notify Epetra that the matrix is done.\r
+    Matrix->FillComplete();\r
+\r
+                                    // Compute the nonlinear residual.\r
+    res_norm = right_hand_side.l2_norm();\r
+    \r
+}\r
+\r
+                                    // Create a conservation law with some defaults.\r
+template <int dim>\r
+ConsLaw<dim>::ConsLaw ()\r
+               :\r
+               mapping (),\r
+                fe_ptr(NULL),\r
+               dof_handler (triangulation),\r
+               quadrature (2),\r
+               face_quadrature (2),\r
+                T(0),\r
+                dT(0.05),\r
+                TF(10),\r
+                Map(NULL),\r
+                Matrix(NULL),\r
+                is_stationary(false),\r
+                theta(0.5)\r
+{}\r
+\r
+                        // At one time this example could work for both DG and\r
+                        // continuous finite elements.  The choice was made here.\r
+template <int dim>\r
+void ConsLaw<dim>::build_fe() {\r
+  fe_ptr = new FESystem<dim>(FE_Q<dim>(1), N_COMP);\r
+}\r
+\r
+                        // Bye bye Conservation law.\r
+template <int dim>\r
+ConsLaw<dim>::~ConsLaw () \r
+{\r
+  dof_handler.clear ();\r
+  delete fe_ptr;\r
+}\r
+\r
+                        // @sect3{Initialize System}\r
+                        // Sizes all of the vectors and sets up the\r
+                        // sparsity patter.  This function is called at\r
+                        // the very beginning of a simulation.  The function\r
+                        // <code> setup_system </code> repeats some of these\r
+                        // chores and is called after adaptivity in leiu\r
+                        // of this function.\r
+template <int dim>\r
+void ConsLaw<dim>::initialize_system ()\r
+{\r
+                                  // First we need to distribute the\r
+                                  // DoFs.\r
+  dof_handler.clear();\r
+  dof_handler.distribute_dofs (*fe_ptr);\r
+  \r
+                                   // Size all of the fields.\r
+  solution.reinit (dof_handler.n_dofs());\r
+  nlsolution.reinit (dof_handler.n_dofs());\r
+  predictor.reinit (dof_handler.n_dofs());\r
+  ppsolution.reinit (dof_handler.n_dofs());\r
+  dsolution.reinit (dof_handler.n_dofs());\r
+  right_hand_side.reinit (dof_handler.n_dofs());\r
+  indicator.reinit(triangulation.n_active_cells());\r
+}\r
+\r
+                                  // @sect3{Setup System}\r
+                                  // We call this function to build the sparsity\r
+                                  // and the matrix.\r
+template <int dim>\r
+void ConsLaw<dim>::setup_system ()\r
+{\r
+\r
+                                  // The DoFs of a cell are coupled\r
+                                  // with all DoFs of all neighboring\r
+                                  // cells.  Therefore the maximum\r
+                                  // number of matrix entries per row\r
+                                  // is needed when all neighbors of\r
+                                  // a cell are once more refined\r
+                                  // than the cell under\r
+                                  // consideration.\r
+  sparsity_pattern.reinit (dof_handler.n_dofs(),\r
+                          dof_handler.n_dofs(),\r
+                          (GeometryInfo<dim>::faces_per_cell\r
+                           *GeometryInfo<dim>::subfaces_per_face+1)*fe_ptr->dofs_per_cell);\r
+  \r
+                                   // Since the continuous sparsity pattern is\r
+                                   // a subset of the DG one, and since we need\r
+                                   // the DG terms for handling hanging nodes, we use\r
+                                   // the flux pattern.\r
+  DoFTools::make_flux_sparsity_pattern (dof_handler, sparsity_pattern);\r
+  \r
+  sparsity_pattern.compress();\r
+  \r
+                                   // Rebuild the map.  In serial this doesn't do much,\r
+                                   // but is needed.  In parallel, this would desribe\r
+                                   // the parallel dof layout.\r
+  if (Map) delete Map;\r
+  Map  = new Epetra_Map(dof_handler.n_dofs(), 0, *Comm);\r
+\r
+                                   // Epetra can build a more efficient matrix if\r
+                                   // one knows ahead of time the maxiumum number of\r
+                                   // columns in any row entry.  We traverse the sparsity\r
+                                   // to discover this.\r
+  int cur_row = 0;\r
+  int cur_col = 0;\r
+  int max_df = -1;\r
+  for (SparsityPattern::iterator s_i = sparsity_pattern.begin(); \r
+       s_i != sparsity_pattern.end(); s_i++) {\r
+    if (s_i->row() != cur_row) {\r
+      cur_col = 0;\r
+      cur_row = s_i->row();\r
+    }\r
+    cur_col++;\r
+   if (cur_col >= max_df) max_df = cur_col;\r
+  }\r
+\r
+  if (cur_col >= max_df) max_df = cur_col;\r
+  std::cout << "max_df:" << max_df << std::endl;\r
+\r
+                                  // Now we build the matrix, using the constructor\r
+                                  // that optimizes with the <code> max_df </code> variable.\r
+  if (Matrix) delete Matrix;\r
+  Matrix = new Epetra_CrsMatrix(Copy, *Map, max_df+1, true);\r
+\r
+                                 // We add the sparsity pattern to the matrix by\r
+                                 // inserting zeros.\r
+  std::vector<double> vals(max_df, 0);\r
+  std::vector<int> row_indices(max_df);\r
+ \r
+  cur_row = 0;\r
+  cur_col = 0;\r
+  for (SparsityPattern::iterator s_i = sparsity_pattern.begin(); \r
+     s_i != sparsity_pattern.end(); s_i++) {\r
+    if (s_i->row() != cur_row) {\r
+      Matrix->InsertGlobalValues(cur_row, cur_col, &vals[0], &row_indices[0]);\r
+      cur_col = 0;\r
+      cur_row = s_i->row();\r
+    }\r
+  row_indices[cur_col++] = s_i->column();\r
+  }\r
+                                 // The last row.\r
+  Matrix->InsertGlobalValues(cur_row, cur_col, &vals[0], &row_indices[0]);\r
+\r
+                                 // Epetra requires this function after building or\r
+                                 // filling a matrix.  It typically does some parallel\r
+                                 // bookeeping; perhaps more.\r
+  Matrix->FillComplete();\r
+\r
+}\r
+\r
+                                 // @sect3{Solving the linear system}\r
+                                 // Actually solve the linear system, using either\r
+                                 // Aztec of Amesos.\r
+template <int dim>\r
+void ConsLaw<dim>::solve (Vector<double> &dsolution, int &niter, double &lin_residual) \r
+{\r
+\r
+                                 // We must hand the solvers Epetra vectors.\r
+                                 // Luckily, they support the concept of a \r
+                                 // 'view', so we just send in a pointer to our\r
+                                 // dealii vectors.\r
+    Epetra_Vector x(View, *Map, dsolution.begin());\r
+    Epetra_Vector b(View, *Map, right_hand_side.begin());\r
+\r
+                                 // The Direct option selects the Amesos solver.\r
+  if (solver_params.SOLVER == solver_params_type::DIRECT) {\r
+   \r
+                                 // Setup for solving with\r
+                                 // Amesos.\r
+     Epetra_LinearProblem prob;\r
+     prob.SetOperator(Matrix);\r
+     Amesos_BaseSolver *solver;\r
+     Amesos Factory;\r
+\r
+                                 // Other solvers are available\r
+                                 // and may be selected by changing this\r
+                                 // string.\r
+     char *stype = "Amesos_Klu";\r
+\r
+     solver = Factory.Create(stype, prob);\r
+\r
+     Assert (solver != NULL, ExcInternalError());\r
+\r
+                                 // There are two parts to the direct solve.\r
+                                 // As I understand, the symbolic part figures\r
+                                 // out the sparsity patterns, and then the\r
+                                 // numerical part actually performs Gaussian\r
+                                 // elimination or whatever the approach is.\r
+     if (solver_params.OUTPUT == solver_params_type::VERBOSE)\r
+       std::cout << "Starting Symbolic fact\n" << std::flush;\r
+\r
+     solver->SymbolicFactorization();\r
+\r
+     if (solver_params.OUTPUT == solver_params_type::VERBOSE)\r
+         std::cout << "Starting Numeric fact\n" << std::flush;\r
+\r
+     solver->NumericFactorization();\r
+\r
+    \r
+                                 // Define the linear problem by setting the\r
+                                 // right hand and left hand sides.\r
+     prob.SetRHS(&b);\r
+     prob.SetLHS(&x);\r
+                                 // And finally solve the problem.\r
+     if (solver_params.OUTPUT == solver_params_type::VERBOSE)\r
+       std::cout << "Starting solve\n" << std::flush;\r
+     solver->Solve();\r
+     niter = 0;\r
+     lin_residual = 0;\r
+\r
+                                 // We must free the solver that was created\r
+                                 // for us.\r
+     delete solver;\r
+\r
+  } else if (solver_params.SOLVER == solver_params_type::GMRES) {\r
+\r
+                                 // For the iterative solvers, we use Aztec.\r
+    AztecOO Solver;\r
+\r
+                                 // Select the appropriate level of verbosity.\r
+    if (solver_params.OUTPUT == solver_params_type::QUIET)\r
+      Solver.SetAztecOption(AZ_output, AZ_none);\r
+\r
+    if (solver_params.OUTPUT == solver_params_type::VERBOSE)\r
+      Solver.SetAztecOption(AZ_output, AZ_all);\r
+\r
+                                 // Select gmres.  Other solvers are available.\r
+    Solver.SetAztecOption(AZ_solver, AZ_gmres);\r
+    Solver.SetRHS(&b);\r
+    Solver.SetLHS(&x);\r
+\r
+                                 // Set up the ILUT preconditioner.  I do not know\r
+                                 // why, but we must pretend like we are in parallel\r
+                                 // using domain decomposition or the preconditioner\r
+                                 // refuses to activate.\r
+    Solver.SetAztecOption(AZ_precond, AZ_dom_decomp);\r
+    Solver.SetAztecOption(AZ_subdomain_solve, AZ_ilut);\r
+    Solver.SetAztecOption(AZ_overlap, 0);\r
+    Solver.SetAztecOption(AZ_reorder, 0);\r
+\r
+                                 // ILUT parameters as described above.\r
+    Solver.SetAztecParam(AZ_drop, solver_params.ILUT_DROP);\r
+    Solver.SetAztecParam(AZ_ilut_fill, solver_params.ILUT_FILL);\r
+    Solver.SetAztecParam(AZ_athresh, solver_params.ILUT_ATOL);\r
+    Solver.SetAztecParam(AZ_rthresh, solver_params.ILUT_RTOL);\r
+    Solver.SetUserMatrix(Matrix);\r
+\r
+                                 // Run the solver iteration.  Collect the number\r
+                                 // of iterations and the residual.\r
+    Solver.Iterate(solver_params.MAX_ITERS, solver_params.RES);\r
+    niter = Solver.NumIters();\r
+    lin_residual = Solver.TrueResidual();\r
+  }\r
+}\r
+\r
+                                 // @sect3{Postprocessing and Output}\r
+                                 // Recover the physical variables from the conservative\r
+                                 // variables so that output will be (perhaps) more\r
+                                 // meaningfull.\r
+template <int dim>\r
+void ConsLaw<dim>::postprocess() {\r
+  const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;\r
+  std::vector<unsigned int> dofs (dofs_per_cell);\r
+  UpdateFlags update_flags = update_values\r
+                            | update_gradients\r
+                            | update_q_points\r
+                            | update_JxW_values;\r
+  UpdateFlags update_flags1 = update_values\r
+                            | update_gradients\r
+                            | update_q_points\r
+                            | update_JxW_values;\r
+\r
+ QGauss<dim>  quadrature_formula(4);\r
+\r
+ const std::vector<Point<dim> > &us = fe_ptr->base_element(0).get_unit_support_points();\r
+\r
+\r
+ Quadrature<dim>  unit_support(us);\r
+\r
+ int n_q_points = quadrature_formula.n_quadrature_points;\r
+ int n_uq_points = unit_support.n_quadrature_points;\r
+\r
+  FEValues<dim> fe_v (\r
+    mapping, *fe_ptr, quadrature_formula, update_flags);\r
+\r
+  FEValues<dim> fe_v_unit (\r
+    mapping, *fe_ptr, unit_support, update_flags1);\r
+\r
+  std::vector<Vector<double> > U(n_uq_points,\r
+                                 Vector<double>(get_n_components()));\r
+  std::vector<Vector<double> > UU(n_q_points,\r
+                                 Vector<double>(get_n_components()));\r
+  std::vector<std::vector<Tensor<1,dim> > > dU(n_uq_points,\r
+                                            std::vector<Tensor<1,dim> >(get_n_components()));\r
+  \r
+  typename DoFHandler<dim>::active_cell_iterator\r
+    cell = dof_handler.begin_active(),\r
+    endc = dof_handler.end();\r
+\r
+                                    // Loop the cells\r
+  for (unsigned int cell_no=0; cell!=endc; ++cell, ++cell_no) {\r
+    cell->get_dof_indices (dofs);\r
+    fe_v_unit.reinit(cell);\r
+    fe_v.reinit(cell);\r
+\r
+    fe_v_unit.get_function_values(solution, U);\r
+    fe_v_unit.get_function_grads(solution, dU);\r
+    fe_v.get_function_values(solution, UU);\r
+\r
+    const std::vector<double> &JxW = fe_v.get_JxW_values ();\r
+\r
+    for (int q = 0; q < fe_v.get_fe().base_element(0).n_dofs_per_cell(); q++) {\r
+      unsigned int didx = fe_v.get_fe().component_to_system_index(DENS_IDX, q);\r
+      unsigned int eidx = fe_v.get_fe().component_to_system_index(ENERGY_IDX, q);\r
+      double rho_normVsqr = 0;\r
+      for (int d = 0; d < dim; d++) {\r
+        unsigned int vidx = fe_v.get_fe().component_to_system_index(d, q);\r
+        ppsolution(dofs[vidx]) = solution(dofs[vidx])/solution(dofs[didx]);\r
+        rho_normVsqr += solution(dofs[vidx])*solution(dofs[vidx]);\r
+      }\r
+      rho_normVsqr /= solution(dofs[didx]);\r
+                                 // Pressure\r
+      ppsolution(dofs[eidx]) = (GAMMA-1.0)*(solution(dofs[eidx]) - 0.5*rho_normVsqr);\r
+\r
+                                 // Either output density or gradient squared of density,\r
+                                 // depending on what the user wants.\r
+      if (!schlieren_plot) {\r
+        ppsolution(dofs[didx]) = solution(dofs[didx]);\r
+      } else {\r
+        double ng = 0;\r
+        for (int i = 0; i < dim; i++) ng += dU[q][DENS_IDX][i]*dU[q][DENS_IDX][i];\r
+        ng = std::sqrt(ng);\r
+        ppsolution(dofs[didx]) = ng;\r
+      }\r
+    }\r
+\r
+  } // cell\r
+\r
+}\r
+\r
+                            // Loop and assign a value for refinement.  We\r
+                            // simply use the density squared, which selects\r
+                            // shocks with some success.\r
+template <int dim>\r
+void ConsLaw<dim>::estimate() {\r
+  \r
+  const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;\r
+  std::vector<unsigned int> dofs (dofs_per_cell);\r
+  UpdateFlags update_flags = update_values\r
+                            | update_gradients\r
+                            | update_q_points\r
+                            | update_JxW_values;\r
+\r
+ QGauss<dim>  quadrature_formula(1);\r
+ int n_q_points = quadrature_formula.n_quadrature_points;\r
+\r
+\r
+  FEValues<dim> fe_v (\r
+    mapping, *fe_ptr, quadrature_formula, update_flags);\r
+\r
+  std::vector<Vector<double> > U(n_q_points,\r
+                                 Vector<double>(get_n_components()));\r
+  std::vector<std::vector<Tensor<1,dim> > > dU(n_q_points,\r
+                                            std::vector<Tensor<1,dim> >(get_n_components()));\r
+  \r
+  typename DoFHandler<dim>::active_cell_iterator\r
+    cell = dof_handler.begin_active(),\r
+    endc = dof_handler.end();\r
+  for (unsigned int cell_no=0; cell!=endc; ++cell, ++cell_no) {\r
+    fe_v.reinit(cell);\r
+\r
+    fe_v.get_function_values(predictor, U);\r
+    fe_v.get_function_grads(predictor, dU);\r
+\r
+    indicator(cell_no) = 0;\r
+    for (int q = 0; q < n_q_points; q++) {\r
+      double ng = 0;\r
+      for (int d = 0; d < dim; d++) ng += dU[q][DENS_IDX][d]*dU[q][DENS_IDX][d];\r
+\r
+      indicator(cell_no) += std::log(1+std::sqrt(ng));\r
+      \r
+    } \r
+    indicator(cell_no) /= n_q_points;\r
+\r
+  } \r
+}\r
+\r
+template <int dim>\r
+void ConsLaw<dim>::refine_grid ()\r
+{\r
+\r
+  SolutionTransfer<dim, double> soltrans(dof_handler);\r
+\r
+  typename DoFHandler<dim>::active_cell_iterator\r
+    cell = dof_handler.begin_active(),\r
+    endc = dof_handler.end();\r
+\r
+                                  // Loop cells.  If the indicator\r
+                                  // for the cell matches the refinement criterion,\r
+                                  // refine, else unrefine.  The unrefinement has\r
+                                  // a slight hysterisis to avoid 'flashing' from refined\r
+                                  // to unrefined.\r
+  for (unsigned int cell_no=0; cell!=endc; ++cell, ++cell_no) {\r
+    cell->clear_coarsen_flag();\r
+    cell->clear_refine_flag();\r
+    if (cell->level() < refinement_params.shock_levels &&\r
+        std::fabs(indicator(cell_no)) > refinement_params.shock_val ) {\r
+      cell->set_refine_flag();\r
+    } else {\r
+      if (cell->level() > 0 &&\r
+         std::fabs(indicator(cell_no)) < 0.75*refinement_params.shock_val)\r
+           cell->set_coarsen_flag();\r
+    }\r
+  }\r
+\r
+                                  // The following code prolongs the solution\r
+                                  // to the new grid and carries out the refinement.\r
+  std::vector<Vector<double> > interp_in;\r
+  std::vector<Vector<double> > interp_out;\r
+\r
+  interp_in.push_back(solution);\r
+  interp_in.push_back(predictor);\r
+\r
+  triangulation.prepare_coarsening_and_refinement();\r
+  soltrans.prepare_for_coarsening_and_refinement(interp_in);\r
+\r
+  triangulation.execute_coarsening_and_refinement ();\r
+\r
+  dof_handler.clear();\r
+  dof_handler.distribute_dofs (*fe_ptr);\r
+\r
+  {\r
+  Vector<double> new_solution(1);\r
+  Vector<double> new_predictor(1);\r
+\r
+  interp_out.push_back(new_solution);\r
+  interp_out.push_back(new_predictor);\r
+  interp_out[0].reinit(dof_handler.n_dofs());\r
+  interp_out[1].reinit(dof_handler.n_dofs());\r
+  }\r
+\r
+  soltrans.interpolate(interp_in, interp_out);\r
+  \r
+                               // Let the vector delete a very small vector\r
+  solution.reinit(1);\r
+  predictor.reinit(1);\r
+  solution.swap(interp_out[0]);\r
+  predictor.swap(interp_out[1]);\r
+\r
+                               // resize these vectors for the new grid.\r
+  nlsolution.reinit(dof_handler.n_dofs());\r
+  ppsolution.reinit(dof_handler.n_dofs());\r
+  nlsolution = solution;\r
+  dsolution.reinit (dof_handler.n_dofs());\r
+  right_hand_side.reinit (dof_handler.n_dofs());\r
+\r
+  indicator.reinit(triangulation.n_active_cells());\r
+\r
+}\r
+\r
+template <int dim>\r
+void ConsLaw<dim>::output_results (const unsigned int cycle) const\r
+{\r
+  char filename[512];\r
+  std::sprintf(filename, "solution-%03d.vtk", cycle);\r
+  std::ofstream output (filename);\r
+\r
+  DataOut<dim> data_out;\r
+  data_out.attach_dof_handler (dof_handler);\r
+  std::vector<std::string> solution_names;\r
+\r
+                         // Rename the output with the physical variable\r
+                         // names.  Send the post-processed values.\r
+  solution_names.clear();\r
+  for (int i = 0; i < dim; i++) {\r
+    char buf[512];\r
+    std::sprintf(buf, "v_%d", i);\r
+    solution_names.push_back (buf);        \r
+  }\r
+  solution_names.push_back("density");\r
+  solution_names.push_back("pressure");\r
+  data_out.add_data_vector (ppsolution, solution_names);\r
+\r
+  data_out.add_data_vector (indicator, "error");\r
+  data_out.build_patches ();\r
+  data_out.write_vtk (output);\r
+\r
+  output.close();\r
+}\r
+\r
+                                   // @sect3{Parsing the Input Deck}\r
+                                   // Declare the parameters for the\r
+                                   // input deck.  We assume a certain\r
+                                   // maximum number of boundaries and process\r
+                                   // any boundary the user supplies up to\r
+                                   // that maximum number.  We\r
+                                   // leave a detailed explanation of these \r
+                                   // parameters to our description of the input\r
+                                   // sample file.\r
+const UInt MAX_BD = 10;\r
+template <int dim>\r
+void ConsLaw<dim>::declare_parameters() {\r
+\r
+                                   // Global scope parameters/\r
+  prm.declare_entry("mesh", "grid.inp",\r
+                    Patterns::Anything(),\r
+                    "intput file");\r
+\r
+  prm.declare_entry("diffusion power", "2.0",\r
+                     Patterns::Double(),\r
+                     "power of mesh size for diffusion");\r
+\r
+  prm.declare_entry("gravity", "0.0",\r
+                     Patterns::Double(),\r
+                     "gravity forcing");\r
+\r
+                                   // Time stepping block\r
+  prm.enter_subsection("time stepping");\r
+    prm.declare_entry("time step", "0.1",\r
+                     Patterns::Double(),\r
+                     "simulation time step");\r
+    prm.declare_entry("final time", "10.0",\r
+                     Patterns::Double(),\r
+                     "simulation end time");\r
+  prm.leave_subsection();\r
+\r
+\r
+                                  // Declare the boundary parameters\r
+  for (int b = 0; b < MAX_BD; b++) {\r
+    char bd[512];\r
+    std::sprintf(bd, "boundary_%d", b);\r
+    prm.enter_subsection(bd);\r
+    prm.declare_entry("no penetration", "false",\r
+                       Patterns::Selection("true|false"),\r
+                       "<true|false>");\r
+                                  // declare a slot for each of the conservative\r
+                                  // variables.\r
+    for (int di = 0; di < N_COMP; di++) {\r
+      char var[512];\r
+      std::sprintf(var, "w_%d", di);\r
+      prm.declare_entry(var, "outflow",\r
+                     Patterns::Selection(\r
+        "inflow|outflow|pressure"),\r
+        "<inflow|outflow|pressure>");\r
+      \r
+                                   // for dirichlet, a function in x,y,z\r
+      std::sprintf(var, "w_%d value", di);\r
+      prm.declare_entry(var, "0.0",\r
+                     Patterns::Anything(),\r
+                 "expression in x,y,z");\r
+    }\r
+\r
+    prm.leave_subsection();\r
+  }\r
+\r
+                                // Initial condition block.\r
+  prm.enter_subsection("initial condition");\r
+    for (int di = 0; di < N_COMP; di++) {\r
+      char var[512];\r
+      std::sprintf(var, "w_%d", di);\r
+      \r
+      // for dirichlet, a function in x,y,z\r
+      std::sprintf(var, "w_%d value", di);\r
+      prm.declare_entry(var, "0.0",\r
+                     Patterns::Anything(),\r
+                 "expression in x,y,z");\r
+    }\r
+  prm.leave_subsection();\r
+\r
+                              // The linear solver block.\r
+  prm.enter_subsection("linear solver");\r
+    prm.declare_entry("output", "quiet",\r
+                     Patterns::Selection(\r
+                     "quiet|verbose"),\r
+                      "<quiet|verbose>");\r
+    prm.declare_entry("method", "gmres",\r
+                     Patterns::Selection(\r
+                     "gmres|direct"),\r
+                      "<gmres|direct>");\r
+    prm.declare_entry("residual", "1e-10",\r
+                     Patterns::Double(),\r
+                     "linear solver residual");\r
+    prm.declare_entry("max iters", "300",\r
+                     Patterns::Double(),\r
+                     "maximum solver iterations");\r
+    prm.declare_entry("ilut fill", "2",\r
+                     Patterns::Double(),\r
+                     "ilut preconditioner fill");\r
+    prm.declare_entry("ilut absolute tolerance", "1e-9",\r
+                     Patterns::Double(),\r
+                     "ilut preconditioner tolerance");\r
+    prm.declare_entry("ilut relative tolerance", "1.1",\r
+                     Patterns::Double(),\r
+                     "rel tol");\r
+    prm.declare_entry("ilut drop tolerance", "1e-10",\r
+                     Patterns::Double(),\r
+                     "ilut drop tol");\r
+  prm.leave_subsection();\r
+\r
+\r
+                           // A refinement controller block.\r
+  prm.enter_subsection("refinement");\r
+    prm.declare_entry("refinement", "none",\r
+                     Patterns::Selection(\r
+                     "none|fixed number|shock"),\r
+                      "<on|off>");\r
+    prm.declare_entry("refinement fraction", "0.1",\r
+                     Patterns::Double(),\r
+                     "Fraction of high refinement");\r
+    prm.declare_entry("unrefinement fraction", "0.1",\r
+                     Patterns::Double(),\r
+                     "Fraction of low unrefinement");\r
+    prm.declare_entry("max elements", "1000000",\r
+                     Patterns::Double(),\r
+                     "maximum number of elements");\r
+    prm.declare_entry("shock value", "4.0",\r
+                     Patterns::Double(),\r
+                     "value for shock indicator");\r
+    prm.declare_entry("shock levels", "3.0",\r
+                     Patterns::Double(),\r
+                     "number of shock refinement levels");\r
+  prm.leave_subsection();\r
+\r
+                      // Output control.\r
+  prm.enter_subsection("output");\r
+    prm.declare_entry("density", "standard",\r
+                     Patterns::Selection(\r
+                     "standard|schlieren"),\r
+                      "<standard|schlieren>");\r
+    prm.declare_entry("step", "-1",\r
+                     Patterns::Double(),\r
+                     "output once per this period");\r
+  prm.leave_subsection();\r
+\r
+                      // Flux control\r
+  prm.enter_subsection("flux");\r
+    prm.declare_entry("stab", "alpha",\r
+                     Patterns::Selection(\r
+                     "alpha|constant|mesh"),\r
+                      "<alpha|constant|mesh>");\r
+    prm.declare_entry("stab value", "1",\r
+                     Patterns::Double(),\r
+                     "alpha stabilization");\r
+  prm.leave_subsection();\r
+\r
+\r
+}\r
+\r
+                     // Code to actually parse an input file.  This function\r
+                     // matches the declarations above.\r
+template <int dim>\r
+void ConsLaw<dim>::load_parameters(const char *infile){\r
+\r
+  prm.read_input(infile);\r
+\r
+                     // The global parameters.\r
+  mesh = prm.get("mesh");\r
+\r
+  diffusion_power = prm.get_double("diffusion power");\r
+\r
+  gravity = prm.get_double("gravity");\r
+\r
+                    // The time stepping.\r
+  prm.enter_subsection("time stepping");\r
+  dT = prm.get_double("time step");\r
+  std::cout << "dT=" << dT << std::endl;\r
+  if (dT == 0) {\r
+    is_stationary = true;\r
+    dT = 1.0;\r
+    TF = 1.0;\r
+    std::cout << "Stationary mode" << std::endl;\r
+  }\r
+  TF = prm.get_double("final time");\r
+  std::cout << "TF=" << TF << std::endl;\r
+  prm.leave_subsection();\r
+\r
+                   // The boundary info\r
+  for (int b = 0; b < MAX_BD; b++) {\r
+    std::vector<bc_type> flags(N_COMP, OUTFLOW_BC);\r
+\r
+                   // Define a parser for every boundary, though it may be\r
+                   // unused.\r
+    SideCondition<dim> *sd = new SideCondition<dim>(N_COMP);\r
+    char bd[512];\r
+    std::sprintf(bd, "boundary_%d", b);\r
+    prm.enter_subsection(bd);\r
+\r
+    const std::string &nopen = prm.get("no penetration");\r
+\r
+                    // Determine how each component is handled.\r
+    for (int di = 0; di < N_COMP; di++) {\r
+      char var[512];\r
+      std::sprintf(var, "w_%d", di);\r
+      std::string btype = prm.get(var);\r
+      std::sprintf(var, "w_%d value", di);\r
+      std::string var_value = prm.get(var);\r
+\r
+      if (di < dim && nopen == "true") {\r
+        flags[di] = NO_PENETRATION_BC;\r
+      } else if (btype == "inflow") {\r
+        flags[di] = INFLOW_BC;\r
+        sd->set_coeff_row(di, var_value);  \r
+      } else if (btype == "pressure") {\r
+        flags[di] = PRESSURE_BC;\r
+        sd->set_coeff_row(di, var_value);  \r
+      }\r
+    } \r
+    prm.leave_subsection();\r
+\r
+                     // Add the boundary condition to the law.\r
+    sd->Init();\r
+    add_boundary(b, flags, sd);\r
+   }\r
+\r
+                     // Initial conditions.\r
+   prm.enter_subsection("initial condition");\r
+    for (int di = 0; di < N_COMP; di++) {\r
+      char var[512];\r
+\r
+      std::sprintf(var, "w_%d value", di);\r
+      std::string var_value = prm.get(var);\r
+      ic.set_ic(di, var_value);  \r
+    }\r
+    ic.Init();\r
+   prm.leave_subsection();\r
+\r
+                    // The linear solver.\r
+ prm.enter_subsection("linear solver");\r
+    const std::string &op = prm.get("output");\r
+    if (op == "verbose") solver_params.OUTPUT = solver_params_type::VERBOSE;\r
+    if (op == "quiet") solver_params.OUTPUT = solver_params_type::QUIET;\r
+    const std::string &sv = prm.get("method");\r
+    if (sv == "direct") {\r
+      solver_params.SOLVER = solver_params_type::DIRECT;\r
+    } else if (sv == "gmres") {\r
+      solver_params.SOLVER = solver_params_type::GMRES;\r
+    } \r
+\r
+    solver_params.RES = prm.get_double("residual");\r
+    solver_params.MAX_ITERS = (int) prm.get_double("max iters");\r
+    solver_params.ILUT_FILL = prm.get_double("ilut fill");\r
+    solver_params.ILUT_ATOL = prm.get_double("ilut absolute tolerance");\r
+    solver_params.ILUT_RTOL = prm.get_double("ilut relative tolerance");\r
+    solver_params.ILUT_DROP = prm.get_double("ilut drop tolerance");\r
+    solver_params.RES = prm.get_double("residual");\r
+  prm.leave_subsection();\r
+\r
+\r
+                       // And refiement.\r
+  prm.enter_subsection("refinement");\r
+    const std::string &ref = prm.get("refinement");\r
+    if (ref == "none") {\r
+      refinement_params.refine = refinement_params_type::NONE;\r
+    } else if (ref == "fixed number") {\r
+      refinement_params.refine = refinement_params_type::FIXED_NUMBER;\r
+    } else if (ref == "shock") {\r
+      refinement_params.refine = refinement_params_type::SHOCK;\r
+    } else\r
+    refinement_params.high_frac = prm.get_double("refinement fraction");\r
+    refinement_params.high_frac_sav = refinement_params.high_frac;\r
+    refinement_params.low_frac = prm.get_double("unrefinement fraction");\r
+    refinement_params.max_cells = prm.get_double("max elements");\r
+    refinement_params.shock_val = prm.get_double("shock value");\r
+    refinement_params.shock_levels = prm.get_double("shock levels");\r
+  prm.leave_subsection();\r
+    \r
+                           // Output control.\r
+  prm.enter_subsection("output");\r
+    const std::string &dens = prm.get("density");\r
+    schlieren_plot = dens == "schlieren" ? true : false;\r
+    output_step = prm.get_double("step");\r
+  prm.leave_subsection();\r
+\r
+                           // Flux control.\r
+  prm.enter_subsection("flux");\r
+    const std::string &stab = prm.get("stab");\r
+    if (stab == "constant") {\r
+      flux_params.LF_stab = flux_params_type::CONSTANT;\r
+    } else if (stab == "mesh ") {\r
+      flux_params.LF_stab = flux_params_type::MESH;\r
+    }\r
+    flux_params.LF_stab_value = prm.get_double("stab value");\r
+  prm.leave_subsection();\r
+\r
+\r
+}\r
+\r
+template<int dim>\r
+void ConsLaw<dim>::zero_matrix() {\r
+  Matrix->PutScalar(0); Matrix->FillComplete();\r
+}\r
+\r
+                          // We use a predictor to try and make adaptivity\r
+                          // work better.  The idea is to try and refine ahead\r
+                          // of a front, rather than stepping into a coarse\r
+                          // set of elements and smearing the solution.  This\r
+                          // simple time extrapolator does the job.\r
+template<int dim>\r
+void ConsLaw<dim>::compute_predictor() {\r
+  predictor = nlsolution;\r
+  predictor.sadd(3/2.0, -1/2.0, solution);\r
+}\r
+\r
+                          // @sect3{Run the simulation}  Contains the initialization,\r
+                          // the time loop, and the inner Newton iteration.\r
+template <int dim>\r
+void ConsLaw<dim>::run () \r
+{\r
+\r
+                          // Open and load the mesh.\r
+  GridIn<dim> grid_in;\r
+  grid_in.attach_triangulation(triangulation);\r
+  std::cout << "Opening mesh <" << mesh << ">" << std::endl;\r
+  std::ifstream input_file(mesh.c_str(), std::ios::in);\r
+\r
+  Assert (infile,\r
+         ExcFileNotOpen());\r
+\r
+  grid_in.read_ucd(input_file);   \r
+  input_file.close();\r
+  \r
+  build_fe();\r
+\r
+  unsigned int nstep = 0;\r
+  \r
+                           // Initialize fields and matrices.\r
+  initialize_system (); \r
+  setup_system();\r
+  initialize(); \r
+  predictor = solution;\r
+\r
+                          // Initial refinement.  We apply the ic,\r
+                          // estimate, refine, and repeat until\r
+                          // happy.\r
+  if (refinement_params.refine != refinement_params_type::NONE)\r
+  for (int i = 0; i < refinement_params.shock_levels; i++) {\r
+    estimate();\r
+    refine_grid();\r
+    setup_system();\r
+    initialize(); \r
+    predictor = solution;\r
+  }\r
+  postprocess();\r
+  output_results (nstep);\r
+\r
+                           // Determine when we will output next.\r
+  double next_output = T + output_step;\r
+\r
+                           // @sect4{Main time stepping loop}\r
+  predictor = solution;\r
+  while(T < TF)\r
+    {\r
+      std::cout << "T=" << T << ", ";\r
+\r
+\r
+      std::cout << "   Number of active cells:       "\r
+               << triangulation.n_active_cells()\r
+               << std::endl;\r
+\r
+\r
+      std::cout << "   Number of degrees of freedom: "\r
+               << dof_handler.n_dofs()\r
+               << std::endl;\r
+\r
+      bool nonlin_done = false;\r
+      double res_norm;\r
+      int lin_iter;\r
+\r
+                              // Print some relevant information during the\r
+                              // Newton iteration.\r
+      std::cout << "NonLin Res:       Lin Iter     Lin Res" << std::endl;\r
+      std::cout << "______________________________________" << std::endl;\r
+\r
+      int max_nonlin = 7;\r
+      int nonlin_iter = 0;\r
+      double lin_res;\r
+\r
+                             // @sect5{Newton iteration}\r
+      nlsolution = predictor;\r
+      while (!nonlin_done) {\r
+        lin_iter = 0;\r
+        zero_matrix();\r
+        right_hand_side = 0;\r
+        assemble_system (res_norm);\r
+                            // Flash a star to the screen so one can\r
+                            // know when the assembly has stopped and the linear\r
+                            // solution is starting.\r
+        std::cout << "* " << std::flush;\r
+\r
+                            // Test against a (hardcoded) nonlinear tolderance.\r
+                            // Do not solve the linear system at the last step \r
+                            // (since it would be a waste).\r
+                      \r
+        if (fabs(res_norm) < 1e-10) {\r
+          nonlin_done = true;\r
+        } else {\r
+                            // Solve the linear system and update with the\r
+                            // delta.\r
+           dsolution = 0;\r
+           solve (dsolution, lin_iter, lin_res);\r
+           nlsolution.add(1.0, dsolution);\r
+        }\r
+\r
+                            // Print the residuals.\r
+        std::printf("%-16.3e %04d        %-5.2e\n",\r
+              res_norm, lin_iter, lin_res);\r
+\r
+        nonlin_iter++;\r
+      } \r
+\r
+                           // Various post convergence tasks.\r
+      compute_predictor();\r
+\r
+      solution = nlsolution;\r
+\r
+\r
+      estimate();\r
+\r
+      postprocess();\r
+\r
+      T += dT;\r
+\r
+                          // Output if it is time.\r
+      if (output_step < 0) {\r
+        output_results (++nstep);\r
+      } else if (T >= next_output) {\r
+        output_results (++nstep);\r
+        next_output += output_step;\r
+      }\r
+\r
+                          // Refine, if refinement is selected.\r
+      if (refinement_params.refine != refinement_params_type::NONE) {\r
+        refine_grid();\r
+        setup_system();\r
+      }\r
+    }\r
+}\r
+\r
+                                // The following ``main'' function is\r
+                                // similar to previous examples and\r
+                                // need not to be commented on.\r
+int main (int argc, char *argv[]) \r
+{\r
+\r
+  MPI_Init(&argc, &argv);\r
+  Comm = new Epetra_MpiComm(MPI_COMM_WORLD);\r
+\r
+  if (argc != 2) {\r
+    std::cout << "Usage:" << argv[0] << " infile" << std::endl;\r
+    std::exit(1);\r
+  }\r
+  try\r
+    {\r
+      ConsLaw<DIMENSION> cons;\r
+      cons.declare_parameters();\r
+      cons.load_parameters(argv[1]);\r
+      cons.run ();\r
+    }\r
+  catch (std::exception &exc)\r
+    {\r
+      std::cerr << std::endl << std::endl\r
+               << "----------------------------------------------------"\r
+               << std::endl;\r
+      std::cerr << "Exception on processing: " << std::endl\r
+               << exc.what() << std::endl\r
+               << "Aborting!" << std::endl\r
+               << "----------------------------------------------------"\r
+               << std::endl;\r
+      return 1;\r
+    }\r
+  catch (...) \r
+    {\r
+      std::cerr << std::endl << std::endl\r
+               << "----------------------------------------------------"\r
+               << std::endl;\r
+      std::cerr << "Unknown exception!" << std::endl\r
+               << "Aborting!" << std::endl\r
+               << "----------------------------------------------------"\r
+               << std::endl;\r
+      return 1;\r
+    };\r
+  \r
+  return 0;\r
+}\r
+\r