From: bangerth <bangerth@0785d39b-7218-0410-832d-ea1e28bc413d>
Date: Sat, 1 Mar 2008 13:45:53 +0000 (+0000)
Subject: Add David Neckel's compressible Euler flow tutorial program.
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=4d9e9156a92082e9649c6fc6131edf2ad22dfa6f;p=dealii-svn.git

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
new file mode 100644
index 0000000000..2102793621
--- /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
new file mode 100644
index 0000000000..e720513c32
--- /dev/null
+++ b/deal.II/examples/step-33/doc/intro.dox
@@ -0,0 +1,152 @@
+<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
new file mode 100644
index 0000000000..4e4ad15b43
--- /dev/null
+++ b/deal.II/examples/step-33/doc/results.dox
@@ -0,0 +1,134 @@
+<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
new file mode 100644
index 0000000000..1b35b6b886
Binary files /dev/null and b/deal.II/examples/step-33/doc/step-33.slide.gif differ
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
new file mode 100644
index 0000000000..afefcac800
Binary files /dev/null and b/deal.II/examples/step-33/doc/step-33.slide_adapt.gif differ
diff --git a/deal.II/examples/step-33/slide.inp b/deal.II/examples/step-33/slide.inp
new file mode 100644
index 0000000000..66684bc8b6
--- /dev/null
+++ b/deal.II/examples/step-33/slide.inp
@@ -0,0 +1,3560 @@
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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
index 0000000000..bb2389e563
--- /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 */
+/* Author: David Neckels, Boulder Colorado 2007  */
+/*    step-12.cc,v 1.33 2005/08/08 16:41:40 wolf Exp       */
+/*    Version: Version-5-2-0                                          */
+/*                                                                */
+/*    Copyright (C) 2001, 2002, 2003, 2004, 2005 by the deal.II authors */
+/*                                                                */
+/*    This file is subject to QPL and may not be  distributed     */
+/*    without copyright and license information. Please refer     */
+/*    to the file deal.II/doc/license.html for the  text  and     */
+/*    further information on this license.                        */
+
+                                  // This program solves the Euler equations
+                                  // of gas dynamics for a given configuration
+                                  // file.  It uses a standard Galerkin approach
+                                  // with weakly applied boundary conditions.
+ 
+                                  // @sect3{Include files}
+
+                                  // Aztecoo require mpi (even though we run on only
+                                  // one processor in this example).
+#include <mpi.h>
+
+                                 // Here we have the necessary TRILINOS includes.
+                                 //
+                                 // Epetra is the basic trilinos vector/matrix library.
+#include <Epetra_MpiComm.h>
+#include <Epetra_Map.h>
+#include <Epetra_CrsGraph.h>
+#include <Epetra_CrsMatrix.h>
+#include <Epetra_Vector.h>
+                                 // Teuchos is a Trilinos utility library that is used
+                                 // to set parameters within the Aztec solver library.
+#include "Teuchos_ParameterList.hpp"
+                                 // Aztec is the iterative solver library.
+#include <AztecOO.h>
+#include <AztecOO_Operator.h>
+#define HAVE_IFPACK_TEUCHOS
+#include <Ifpack.h>
+
+                                 // Amesos is a direct solver package within Trilinos.
+#include <Amesos.h>
+                                 // Sacado is the automatic differentiation package, which
+                                 // is used to find the jacobian for a fully implicit Newton
+                                 // iteration.
+#include <Sacado.hpp>
+
+                                 // A standard set of dealii includes.  Nothing special to
+                                 // comment on here.
+#include <base/quadrature_lib.h>
+#include <base/function.h>
+#include <base/parameter_handler.h>
+#include <base/function_parser.h>
+
+#include <lac/vector.h>
+#include <lac/sparse_matrix.h>
+#include <lac/vector_memory.h>
+
+#include <grid/tria.h>
+#include <grid/grid_generator.h>
+#include <grid/grid_out.h>
+#include <grid/grid_refinement.h>
+#include <grid/tria_accessor.h>
+#include <grid/tria_iterator.h>
+#include <grid/grid_in.h>
+
+#include <fe/fe_values.h>
+#include <fe/fe_system.h>
+
+#include <dofs/dof_handler.h>
+#include <dofs/dof_accessor.h>
+#include <dofs/dof_tools.h>
+
+#include <numerics/data_out.h>
+#include <numerics/vectors.h>
+#include <numerics/solution_transfer.h>
+
+#include <fe/mapping_q1.h>
+#include <fe/fe_q.h>
+#include <numerics/derivative_approximation.h>
+				 // And this again is C++:
+#include <iostream>
+#include <fstream>
+#include <vector>
+
+                                // Introduce the dealii library into the current namespace.
+using namespace dealii;
+
+                                // We define a shorter name for the automatic differentiation
+                                // type.
+typedef Sacado::Fad::DFad<double> fad_double;
+typedef unsigned int UInt;
+                                // The Epetra library requires a 'communicator', which describes
+                                // the layout of a parallel (or serial) set of processors.
+Epetra_MpiComm *Comm;
+
+                                //@sect3{Flux function definition}
+                                // Here we define the flux function for this system of conservation
+                                // laws.  Note: it would be terribly difficult to use this example
+                                // to solve some other system of conservation laws.
+                                //
+                                // We define the number of components in the system.  Euler's has
+                                // one entry for momenta in each spatial direction, plus the energy
+                                // and density components.
+#define N_COMP (2 + DIMENSION)
+                                // Define a handle to the density and energy indices.  We have arrange
+                                // the momenta to be first, then density, and, lastly, energy.
+#define DENS_IDX DIMENSION
+#define ENERGY_IDX (DIMENSION+1)
+
+                                // The gas constant.  This value is representative of air.
+const double GAMMA = 1.4;
+                                // We define the flux functions as one large matrix.  Each row of this
+                                // matrix represents a scalar conservation law for the component in
+                                // that row.  We template the numerical type of the flux function
+                                // so that we may use the automatic differentiation type here.
+                                // The flux functions are defined in terms of the
+                                // conserved variables $\rho w_0, \dots, \rho w_{d-1}, \rho, E$,
+                                // so they do not look exactly like the Euler equations one is
+                                // used to seeing.  We evaluate the flux at a single quadrature
+                                // point.
+template <typename number, int dim>
+void Flux(std::vector<std::vector<number> >  &flux, 
+          const Point<dim> &point, 
+          const std::vector<number> &W)
+{
+
+                               // Pressure is a dependent variable: $p = 
+                               // (\gamma - 1)(E-\frac{1}{2} \rho |v|^2)$.
+    number rho_normVsqr;
+    for (int d0 = 0; d0 < dim; d0++) rho_normVsqr += W[d0]*W[d0];
+                               // Since W are $\rho v$, we get a $\rho^2$ in the
+                               // numerator, so dividing a $\rho$ out gives the desired $ \rho |v|^2$.
+    rho_normVsqr /= W[DENS_IDX];
+
+    number pressure = (GAMMA-1.0)*(W[ENERGY_IDX] - number(0.5)*(rho_normVsqr));
+
+                               // We compute the momentum terms.  We divide by the
+                               // density here to get $v_i \rho v_j$
+    for (int d = 0; d < dim; d++) {
+      for (int d1 = 0; d1 < dim; d1++) {
+        flux[d][d1] = W[d]*W[d1]/W[DENS_IDX];
+      }
+                              // The pressure contribution, along the diagonal:
+      flux[d][d] += pressure;
+                              // Advection/conservation of density:
+      flux[DENS_IDX][d] = W[d]; 
+                              // And, lastly, conservation of energy.
+      flux[ENERGY_IDX][d] = W[d]/W[DENS_IDX]*
+                               (W[ENERGY_IDX] + pressure); // energy
+    }
+}
+
+                              // On the boundaries of the domain and across `hanging nodes` we use
+                              // a numerical flux function to enforce boundary conditions.  This routine
+                              // is the basic Lax-Friedrich's flux with a stabilization parameter
+                              // $\alpha$.
+template <typename number, int dim>
+void LFNumFlux(
+            std::vector<std::vector<fad_double> > &nflux,
+            const std::vector<Point<dim> > &points, 
+            const std::vector<Point<dim> > &normals,
+            const std::vector<std::vector<number> > &Wplus,
+            const std::vector<std::vector<number> > &Wminus,
+            double alpha)
+{
+  int n_q_points = points.size();
+
+                             // We evaluate the flux at each of the quadrature points.
+  for (int q = 0; q < n_q_points; q++) {
+    std::vector<std::vector<fad_double> > iflux(N_COMP,
+                                std::vector<fad_double>(dim, 0));
+    std::vector<std::vector<fad_double> > oflux(N_COMP,
+                                std::vector<fad_double>(dim, 0));
+
+    Flux<number, dim>(iflux, points[q], Wplus[q]);
+    Flux<number, dim>(oflux, points[q], Wminus[q]);
+
+    for (int di = 0; di < N_COMP; di++) {
+      nflux[q][di] = 0;
+      for (int d = 0; d < dim; d++) {
+        nflux[q][di] += 0.5*(iflux[di][d] + oflux[di][d])*normals[q](d);
+      }
+        nflux[q][di] += 0.5*alpha*(Wplus[q][di] - Wminus[q][di]);
+    }
+  }
+
+}
+
+                            // @sect3{Initial and side condition parsing}
+                            // For the initial condition we use the expression parser function
+                            // object.
+template <int dim>
+class InitialCondition :  public FunctionParser<dim> 
+{
+  public:
+    InitialCondition ();
+    
+                            // This function should be called after parsing, but before using
+                            // the object.  It formalizes the expressions and initializes the
+                            // function parser with the appropriate expressions.
+    void Init();
+
+                            // During parsing we call this function as the initial condition
+                            // for one of the $\mathbf{w}$ variables is encountered.
+
+    void set_ic(int _row, std::string &expr) {
+      expressions[_row] = expr;
+    }
+
+    virtual void vector_value_list (const std::vector<Point<dim> > &points,
+				    std::vector<Vector<double> >   &value_list) const;
+ private:
+ std::vector<std::string> expressions;
+};
+
+template <int dim>
+InitialCondition<dim>::InitialCondition () :
+		FunctionParser<dim> (N_COMP),
+                expressions(N_COMP, "0.0")
+{}
+
+                            // Here we set up x,y,z as the variables that one should use in the input
+                            // deck to describe their initial condition.
+template<int dim>
+void InitialCondition<dim>::Init() {
+ std::map<std::string, double> constants;
+ constants["M_PI"] =  M_PI;
+ std::string variables = (dim == 2 ? "x,y" : "x,y,z");
+
+ FunctionParser<dim>::initialize(variables, expressions, constants);
+
+}
+
+template <int dim>
+void InitialCondition<dim>::vector_value_list (const std::vector<Point<dim> > &points,
+					    std::vector<Vector<double> >   &value_list) const 
+{
+  const unsigned int n_points = points.size();
+
+  Assert (value_list.size() == n_points, 
+	  ExcDimensionMismatch (value_list.size(), n_points));
+
+  for (unsigned int p=0; p<n_points; ++p)
+    InitialCondition<dim>::vector_value (points[p],
+				      value_list[p]);
+}
+
+                             // As above, we use the expression function parser for boundary conditions.
+template <int dim>
+class SideCondition :  public FunctionParser<dim> 
+{
+  public:
+    SideCondition (int ncomp);
+    ~SideCondition ();
+
+                            // As above.
+    void Init();
+                           // As above.
+    void set_coeff_row(int _row_n, std::string &expr);
+    
+    virtual void vector_value_list (const std::vector<Point<dim> > &points,
+				    std::vector<Vector<double> >   &value_list) const;
+  private:
+  std::vector<std::string> expressions;
+};
+
+template <int dim>
+SideCondition<dim>::SideCondition (int ncomp) :
+		FunctionParser<dim> (ncomp),
+                expressions(ncomp, "0.0")
+{
+}
+template <int dim>
+void SideCondition<dim>::set_coeff_row (int _row_n, std::string &expr) 
+{
+ expressions[_row_n] = expr;
+}
+
+template <int dim>
+void SideCondition<dim>::Init() {
+ std::map<std::string, double> constants;
+ constants["M_PI"] =  M_PI;
+ std::string variables = (dim == 2 ? "x,y" : "x,y,z");
+
+ FunctionParser<dim>::initialize(variables, expressions, constants);
+
+}
+
+template <int dim>
+SideCondition<dim>::~SideCondition () 
+{
+}
+
+template <int dim>
+void SideCondition<dim>::vector_value_list (const std::vector<Point<dim> > &points,
+					    std::vector<Vector<double> >   &value_list) const 
+{
+  const unsigned int n_points = points.size();
+
+  Assert (value_list.size() == n_points, 
+	  ExcDimensionMismatch (value_list.size(), n_points));
+
+  for (unsigned int p=0; p<n_points; ++p)
+    SideCondition<dim>::vector_value (points[p],
+				      value_list[p]);
+}
+                            //@sect3{Conservation Law class}
+                           // Here we define a Conservation Law class that helps group
+                           // operations and data for our Euler equations into a manageable
+                           // entity.  Functions will be described as their definitions appear.
+template <int dim>
+class ConsLaw
+{
+  public:
+    ConsLaw ();
+    ~ConsLaw ();
+
+    void run ();
+    void declare_parameters();
+    void load_parameters(const char *);
+    
+  private:
+    void build_fe();
+    void setup_system ();
+    void initialize_system ();
+    void assemble_system (double &res_norm);
+    void solve (Vector<double> &solution, int &, double &);
+    void refine_grid ();
+    void output_results (const unsigned int cycle) const;
+    void initialize();
+    void zero_matrix();
+    void estimate();
+    void postprocess();
+    void compute_predictor();
+    
+    Triangulation<dim>   triangulation;
+    const MappingQ1<dim> mapping;
+    
+    
+    FESystem<dim>        *fe_ptr;
+
+    DoFHandler<dim>      dof_handler;
+
+    SparsityPattern      sparsity_pattern;
+    const QGauss<dim>   quadrature;
+    const QGauss<dim-1> face_quadrature;
+    
+                                     // The actual solution to the Euler equation
+    Vector<double>       solution;
+                                     // The current value of the solution during the Newton iteration
+    Vector<double>       nlsolution;
+                                     // An estimate of the next time value; used for adaptivity and as a
+                                     // guess for the next Newton iteration.
+    Vector<double>       predictor;
+                                     // Values after post-processing (used to output the physical variables).
+    Vector<double>       ppsolution;
+                                     // The solution to the linear problem during the Newton iteration
+    Vector<double>       dsolution;
+    Vector<double>       right_hand_side;
+    
+  public:
+
+    void assemble_cell_term(const FEValues<dim>& fe_v,
+                            std::vector<unsigned int> &dofs,
+                            unsigned int cell_no
+                            );
+    
+    void assemble_face_term(
+                            int face_no,
+                            const FEFaceValuesBase<dim>& fe_v,
+			     const FEFaceValuesBase<dim>& fe_v_neighbor,
+                             std::vector<unsigned int> &dofs,
+                             std::vector<unsigned int> &dofs_neighbor,
+                             int boundary = -1
+                             );
+
+    unsigned int get_n_components() const { return N_COMP;}
+
+  private:
+                                    // T = current time, dT = time step, TF = final time.
+    double T, dT, TF;
+    double face_diameter;
+    double cell_diameter;
+                                    // An object to handle parsing the input deck.
+    ParameterHandler prm;
+                                    // Name of the mesh to read in.
+    string mesh;
+    InitialCondition<dim> ic;
+
+                                    // Enums for the various supported boundary conditions.
+    typedef enum {INFLOW_BC = 1, OUTFLOW_BC=2, NO_PENETRATION_BC=3, PRESSURE_BC=4} bc_type;
+
+                                    // For each boundary we store a map from boundary # to the type
+                                    // of boundary condition.  If the boundary condition is prescribed,
+                                    // we store a pointer to a function object that will hold the expression
+                                    // for that boundary condition.
+    typedef typename std::map<unsigned int, std::pair<std::vector<bc_type>, Function<dim>*> > bdry_map_type;
+    bdry_map_type bdry_map;
+
+    void add_boundary(unsigned int bd, std::vector<bc_type>& flags, Function<dim> *bf);
+
+                                    // An object to store parameter information about the Aztec solver.
+    typedef struct {
+      int LIN_OUTPUT;
+      typedef enum { GMRES = 0, DIRECT = 1} solver_type;
+      solver_type SOLVER;
+      typedef enum { QUIET = 0, VERBOSE = 1 } output_type;
+      output_type OUTPUT;
+                                    // Linear residual tolerance.
+      double RES;
+      int MAX_ITERS;
+                                    // We use the ILUT preconditioner.  This is similar
+                                    // to the ILU.  FILL is the number of extra entries
+                                    // to add when forming the ILU decomposition.
+      double ILUT_FILL;
+                                    // When forming the preconditioner, for certain problems
+                                    // bad conditioning (or just bad luck) can cause the
+                                    // preconditioner to be very poorly conditioned.  Hence
+                                    // it can help to add diagonal perturbations to the
+                                    // original matrix and form the preconditioner for this
+                                    // slightly better matrix.  ATOL is an absolute perturbation
+                                    // that is added to the diagonal before forming the
+                                    // prec, and RTOL is a scaling factor $rtol >= 1$.
+      double ILUT_ATOL;
+      double ILUT_RTOL;
+                                    // The ILUT will drop any values that have magnitude less
+                                    // than this value.  This is a way to
+                                    // manage the amount of memory used by this preconditioner.
+      double ILUT_DROP;
+    } solver_params_type;
+
+    solver_params_type solver_params;
+
+                                    // Some refinement parameters.
+    typedef struct {
+      typedef enum { NONE = 0, FIXED_NUMBER = 1, SHOCK = 2} refine_type;
+      double high_frac;
+      double low_frac;
+      refine_type refine;
+      double high_frac_sav;
+      double max_cells;
+      double shock_val;
+      double shock_levels;
+    } refinement_params_type;
+
+    refinement_params_type refinement_params;
+
+                                    // The user can set the stabilization parameter $\alpha$ 
+                                    // in the Lax-Friedrich's flux.  
+    typedef struct {
+      typedef enum {CONSTANT=1,MESH=2} LF_stab_type;
+      LF_stab_type LF_stab;
+      double LF_stab_value;
+    } flux_params_type;
+
+    flux_params_type flux_params;
+
+    bool is_stationary;
+
+                                    // Power for the mesh stabilization term.
+    double diffusion_power;
+    double gravity;
+                                    // If true, we output the squared gradient of the
+                                    // density instead of density.  Using this one can
+                                    // create shock plots.
+    bool schlieren_plot;
+                                    // How often to create an output file.
+    double output_step;
+
+    Epetra_CrsMatrix   *Matrix;
+    Epetra_Map         *Map;
+    Vector<double>      indicator;
+ 
+                                   // Crank-Nicolson value
+    const double        theta; 
+
+};
+
+
+                                    // Asign a row of the conservation law a specified
+                                    // boundary type and (possibly) function.
+template <int dim>
+void ConsLaw<dim>::add_boundary(unsigned int bd,
+        std::vector<bc_type> &flags, Function<dim> *bf) {
+
+  std::pair<std::vector<bc_type>, Function<dim> *> entry(flags, bf);
+  bdry_map[bd] = entry;
+}
+
+
+                                    // Apply the initialial condition.  Simultaneously
+                                    // initialize the non-linear solution.
+template <int dim>
+void ConsLaw<dim>::initialize() {
+ VectorTools::interpolate(dof_handler,
+                           ic, solution);
+ VectorTools::interpolate(dof_handler,
+                           ic, nlsolution);
+
+}
+
+				 // @sect3{Assembly}
+				 // @sect4{Function: assemble_cell_term}
+				 //
+                                 // Assembles the cell term, adding minus the residual
+                                 // to the right hand side, and adding in the Jacobian
+                                 // contributions.
+template <int dim>
+void ConsLaw<dim>::assemble_cell_term(
+  const FEValues<dim> &fe_v,
+  std::vector<unsigned int> &dofs,
+  unsigned int cell_no
+  ) 
+{
+                                 // The residual for each row (i) will be accumulating 
+                                 // into this fad variable.  At the end of the assembly
+                                 // for this row, we will query for the sensitivities
+                                 // to this variable and add them into the Jacobian.
+  fad_double F_i;
+  int dofs_per_cell = fe_v.dofs_per_cell;
+  int n_q_points = fe_v.n_quadrature_points;
+
+                                 // We will define the dofs on this cell in these fad variables.
+  std::vector<fad_double> DOF(dofs_per_cell);
+
+                                 // Values of the conservative variables at the quadrature points.
+  std::vector<std::vector<fad_double > > W (n_q_points,
+                                               std::vector<fad_double >(get_n_components()));
+
+                                 // Values at the last time step of the conservative variables.
+                                 // Note that these do not use fad variables, since they do
+                                 // not depend on the 'variables to be sought'=DOFS.
+  std::vector<std::vector<double > > Wl (n_q_points,
+                                               std::vector<double >(get_n_components()));
+
+                                 // Here we will hold the averaged values of the conservative
+                                 // variables that we will linearize around (cn=Crank Nicholson).
+  std::vector<std::vector<fad_double > > Wcn (n_q_points,
+                                               std::vector<fad_double >(get_n_components()));
+
+                                 // Gradients of the current variables.  It is a
+                                 // bit of a shame that we have to compute these; we almost don't.
+                                 // The nice thing about a simple conservation law is that the
+                                 // the flux doesn't generally involve any gradients.  We do
+                                 // need these, however, for the diffusion stabilization. 
+   std::vector<std::vector<std::vector<fad_double> > > Wgrads (n_q_points,
+            std::vector<std::vector<fad_double> >(get_n_components(),
+                            std::vector<fad_double>(dim)));
+
+
+  const std::vector<double> &JxW = fe_v.get_JxW_values ();
+
+  
+                                  // Here is the magical point where we declare a subset
+                                  // of the fad variables as degrees of freedom.  All 
+                                  // calculations that reference these variables (either
+                                  // directly or indirectly) will accumulate sensitivies
+                                  // with respect to these dofs.
+  for (int in = 0; in < dofs_per_cell; in++) {
+      DOF[in] = nlsolution(dofs[in]);
+      DOF[in].diff(in, dofs_per_cell);
+  }
+
+                                  // Here we compute the shape function values and gradients
+                                  // at the quadrature points.  Ideally, we could call into 
+                                  // something like get_function_values, get_function_grads,
+                                  // but since we don't want to make the entire solution vector
+                                  // fad types, only the local cell variables, we explicitly
+                                  // code this loop;
+  for (int q = 0; q < n_q_points; q++) {
+    for (int di = 0; di < get_n_components(); di++) {
+      W[q][di] = 0;
+      Wl[q][di] = 0;
+      Wcn[q][di] = 0;
+      for (int d = 0; d < dim; d++) {
+        Wgrads[q][di][d] = 0;
+      }
+    }
+    for (int sf = 0; sf < dofs_per_cell; sf++) {
+     int di = fe_v.get_fe().system_to_component_index(sf).first;
+     W[q][di] +=
+                DOF[sf]*fe_v.shape_value_component(sf, q, di);
+     Wl[q][di] +=
+                solution(dofs[sf])*fe_v.shape_value_component(sf, q, di);
+     Wcn[q][di] +=
+                (theta*DOF[sf]+(1-theta)*solution(dofs[sf]))*fe_v.shape_value_component(sf, q, di);
+
+     for (int d = 0; d < dim; d++) {
+       Wgrads[q][di][d] += DOF[sf]*
+                 fe_v.shape_grad_component(sf, q, di)[d];
+     } // for d
+
+    }
+
+  } // for q
+
+                                   // Gather the flux values for all components at
+                                   // all of the quadrature points.  This also
+                                   // computes the matrix of sensitivities.  Perhaps
+                                   // this could be done in a better way, since this
+                                   // could be a rather large object, but for now it 
+                                   // seems to work just fine.
+   std::vector<std::vector<std::vector<fad_double> > > flux(n_q_points, 
+                                std::vector<std::vector<fad_double> >(get_n_components(),
+                                std::vector<fad_double>(dim, 0)));
+
+    for (unsigned int q=0; q < n_q_points; ++q) {
+      Flux<fad_double, dim>(flux[q], fe_v.get_quadrature_points()[q], Wcn[q]);
+    }
+
+                                    // We now have all of the function values/grads/fluxes,
+                                    // so perform the assembly.  We have an outer loop
+                                    // through the components of the system, and an
+                                    // inner loop over the quadrature points, where we
+                                    // accumulate contributions to the ith residual.
+                                    //
+                                    // We initialy sum all contributions of the residual
+                                    // in the positive sense, so that we don't need to
+                                    // negative the Jacobian entries.  Then, when we sum
+                                    // into the <code> right_hand_side </code> vector,
+                                    // we negate this residual.
+    for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i) 
+      {
+                                    // Find which component this dof contributes to.
+        const unsigned int
+          component_i = fe_v.get_fe().system_to_component_index(i).first;
+
+                                    // Initialize the fad residual to zero (removes
+                                    // any previous sensitivities.
+        F_i = 0;
+
+                                    // Loop quadrature points.
+        for (unsigned int point=0; point<fe_v.n_quadrature_points; ++point) {
+
+          fad_double fdotgv = 0;
+
+                                    // Integrate the flux times gradient of the test function
+          for (int d = 0; d < dim; d++) 
+            fdotgv += flux[point][component_i][d]*fe_v.shape_grad_component(i, point, component_i)[d];
+           
+          F_i -= fdotgv*JxW[point];
+
+                                    // The mass term (if the simulation is non-stationary).
+          fad_double delta_t= 1.0/dT*(W[point][component_i] - Wl[point][component_i]);
+          if (!is_stationary) F_i += delta_t*
+                 fe_v.shape_value_component(i, point, component_i)*JxW[point];
+
+                                   // Stabilization (cell wise diffusion)
+          fad_double guv = 0;
+          for (int d = 0; d < dim; d++) {
+            guv += fe_v.shape_grad_component(i, point, component_i)[d]*
+                      Wgrads[point][component_i][d];
+          }
+
+            F_i += 1.0*std::pow(cell_diameter, diffusion_power)*guv*JxW[point];
+          
+                                   // The gravity component only enters into the energy 
+                                   // equation and into the vertical component of the 
+                                   // velocity.
+          if (component_i == dim - 1) {
+            F_i += gravity*Wcn[point][DENS_IDX]*fe_v.shape_value_component(i,point, component_i)*JxW[point];
+          } else if (component_i == ENERGY_IDX) {
+            F_i += gravity*Wcn[point][DENS_IDX]*Wcn[point][dim-1]*
+                   fe_v.shape_value_component(i,point, component_i)*JxW[point];
+          }
+        } // for q
+
+                                   // Here we gain access to the array of sensitivities
+                                   // of the residual.  We then sum these into the
+                                   // Epetra matrix.
+        double *values = &(F_i.fastAccessDx(0));
+        Matrix->SumIntoGlobalValues(dofs[i],
+            dofs_per_cell, &values[0], reinterpret_cast<int*>(&dofs[0]));
+ 
+                                   // Add minus the residual to the right hand side.
+        right_hand_side(dofs[i]) -= F_i.val();
+
+      } // for i
+}
+                                   // @sect4{Function: assemble_face_term}
+                                   // These are either
+                                   // boundary terms or terms across differing 
+                                   // levels of refinement.  In the first case,
+                                   // fe_v==fe_v_neighbor and dofs==dofs_neighbor.
+                                   // The int boundary < 0 if not at a boundary,
+                                   // otherwise it is the boundary indicator.
+template <int dim>
+void ConsLaw<dim>::assemble_face_term(
+  int face_no,
+  const FEFaceValuesBase<dim>& fe_v,
+  const FEFaceValuesBase<dim>& fe_v_neighbor,      
+  std::vector<unsigned int> &dofs,
+  std::vector<unsigned int> &dofs_neighbor,
+  int boundary
+  ) 
+{
+  fad_double F_i;
+  const unsigned int n_q_points = fe_v.n_quadrature_points;
+  const unsigned int dofs_per_cell = fe_v.get_fe().dofs_per_cell;
+  const unsigned int ndofs_per_cell = fe_v_neighbor.get_fe().dofs_per_cell;
+  Assert(dofs_per_cell == ndofs_per_cell,
+	 ExcDimensionMismatch(dofs_per_cell, ndofs_per_cell));
+
+                                  // As above, the fad degrees of freedom
+  std::vector<fad_double> DOF(dofs_per_cell+ndofs_per_cell);
+
+                                  // The conservative variables for this cell,
+                                  // and for 
+  std::vector<std::vector<fad_double > > Wplus (n_q_points,
+                                               std::vector<fad_double >(get_n_components()));
+  std::vector<std::vector<fad_double > > Wminus (n_q_points,
+                                               std::vector<fad_double >(get_n_components()));
+
+
+  const std::vector<double> &JxW = fe_v.get_JxW_values ();
+  const std::vector<Point<dim> > &normals = fe_v.get_normal_vectors ();
+
+
+                                  // If we are at a boundary, then dofs_neighbor are
+                                  // the same as dofs, so we do not want to duplicate them.
+                                  // If there is a neighbor cell, then we want to include 
+                                  // them.
+  int ndofs = (boundary < 0 ? dofs_per_cell + ndofs_per_cell : dofs_per_cell);
+                                  // Set the local DOFS.
+  for (int in = 0; in < dofs_per_cell; in++) {
+      DOF[in] = nlsolution(dofs[in]);
+      DOF[in].diff(in, ndofs);
+  }
+                                  // If present, set the neighbor dofs.
+  if (boundary < 0)
+  for (int in = 0; in < ndofs_per_cell; in++) {
+      DOF[in+dofs_per_cell] = nlsolution(dofs_neighbor[in]);
+      DOF[in+dofs_per_cell].diff(in+dofs_per_cell, ndofs);
+  }
+
+                                  // Set the values of the local conservative variables.
+                                  // Initialize all variables to zero.
+  for (int q = 0; q < n_q_points; q++) {
+    for (int di = 0; di < get_n_components(); di++) {
+           Wplus[q][di] = 0;
+           Wminus[q][di] = 0;
+    }
+    for (int sf = 0; sf < dofs_per_cell; sf++) {
+     int di = fe_v.get_fe().system_to_component_index(sf).first;
+     Wplus[q][di] +=
+                (theta*DOF[sf]+(1.0-theta)*solution(dofs[sf]))*fe_v.shape_value_component(sf, q, di);
+    }
+
+
+                                 // If there is a cell across, then initialize
+                                 // the exterior trace as a function of the other
+                                 // cell degrees of freedom.
+    if (boundary < 0) {
+      for (int sf = 0; sf < ndofs_per_cell; sf++) {
+       int di = fe_v_neighbor.get_fe().system_to_component_index(sf).first;
+       Wminus[q][di] +=
+                (theta*DOF[sf+dofs_per_cell]+(1.0-theta)*solution(dofs_neighbor[sf]))*
+                fe_v_neighbor.shape_value_component(sf, q, di);
+      }
+    } 
+   } // for q
+
+                               // If this is a boundary, then the values of $W^-$ will
+                               // be either functions of $W^+$, or they will be prescribed.
+                               // This switch sets them appropriately.  Since we are
+                               // using fad variables here, sensitivities will be updated 
+                               // appropriately.  These sensitivities would be tremendously
+                               // difficult to manage without fad!!!
+   if (boundary >= 0) {
+                               // Get the boundary descriptor.
+     typename bdry_map_type::iterator bme = bdry_map.find(boundary);
+     assert(bme != bdry_map.end());
+
+                             // Evaluate the function object.  This is a bit
+                             // tricky; a given boundary might have both prescribed
+                             // and implicit values.  If a particular component is not
+                             // prescribed, the values evaluate to zero and are
+                             // ignored, below.
+     std::vector<Vector<double> > bvals(n_q_points, Vector<double>(N_COMP));
+     bme->second.second->vector_value_list(fe_v.get_quadrature_points(), bvals);
+
+                             // We loop the quadrature points, and we treat each
+                             // component individualy.
+     for (int q = 0; q < n_q_points; q++) {
+      for (int di = 0; di < get_n_components(); di++) {
+
+                             // An inflow/dirichlet type of boundary condition
+        if (bme->second.first[di] == INFLOW_BC) {
+          Wminus[q][di] = bvals[q](di);
+        } else if (bme->second.first[di] == PRESSURE_BC) {
+                             // A prescribed pressure boundary condition.  This boundary
+                             // condition is complicated by the fact that even though
+                             // the pressure is prescribed, we really are setting
+                             // the energy index here, which will depend on velocity
+                             // and pressure. So even though this seems like a dirichlet
+                             // type boundary condition, we get sensitivities of
+                             // energy to velocity and density (unless these
+                             // are also prescribed.
+          fad_double rho_vel_sqr = 0;
+          fad_double dens;
+          
+          dens = bme->second.first[DENS_IDX] == INFLOW_BC ? bvals[q](DENS_IDX) :
+                 Wplus[q][DENS_IDX];
+
+          for (int d=0; d < dim; d++) {
+            if (bme->second.first[d] == INFLOW_BC)
+              rho_vel_sqr += bvals[q](d)*bvals[q](d);
+            else
+              rho_vel_sqr += Wplus[q][d]*Wplus[q][d];
+          }
+          rho_vel_sqr /= dens;
+                             // Finally set the energy value as determined by the
+                             // prescribed pressure and the other variables.
+          Wminus[q][di] = bvals[q](di)/(GAMMA-1.0) +
+                             0.5*rho_vel_sqr;
+
+        } else if (bme->second.first[di] == OUTFLOW_BC) {
+                            // A free/outflow boundary, very simple.
+          Wminus[q][di] = Wplus[q][di];
+
+        } else { 
+                            // We must be at a no-penetration boundary.  We
+                            // prescribe the velocity (we are dealing with a
+                            // particular component here so that the average
+                            // of the velocities is orthogonal to the surface
+                            // normal.  This creates sensitivies of across
+                            // the velocity components.
+          fad_double vdotn = 0;
+          for (int d = 0; d < dim; d++) {
+            vdotn += Wplus[q][d]*normals[q](d);
+          }
+
+          Wminus[q][di] = Wplus[q][di] - 2.0*vdotn*normals[q](di);
+        }
+      }
+     } // for q
+   } // b>= 0
+   
+                           // Determine the Lax-Friedrich's stability parameter,
+                           // and evaluate the numerical flux function at the quadrature points
+   std::vector<std::vector<fad_double> > nflux(n_q_points, std::vector<fad_double>(get_n_components(), 0));
+     double alpha = 1;
+
+     switch(flux_params.LF_stab) {
+       case flux_params_type::CONSTANT:
+         alpha = flux_params.LF_stab_value;
+       break;
+       case flux_params_type::MESH:
+         alpha = face_diameter/(2.0*dT);
+       break;
+     }
+
+     LFNumFlux<fad_double, dim>(nflux, fe_v.get_quadrature_points(), normals, Wplus, Wminus,
+                     alpha);
+
+                          // Now assemble the face term
+     for (unsigned int i=0; i<fe_v.dofs_per_cell; ++i) {
+        if (!fe_v.get_fe().has_support_on_face(i, face_no)) continue;
+        F_i = 0;
+        for (unsigned int point=0; point<n_q_points; ++point)
+        {
+          const unsigned int
+            component_i = fe_v.get_fe().system_to_component_index(i).first;
+
+          F_i += nflux[point][component_i]*fe_v.shape_value_component(i, point, component_i)*JxW[point];
+
+         } 
+
+                          // Retrieve a pointer to the jacobian.
+        double *values = &(F_i.fastAccessDx(0));
+
+                          // Honestly, I forget why this can happen, but 
+                          // for some reason it can!!
+        if (!values) continue;
+
+                          // Update the matrix.  Depending on whether there
+                          // is/isn't a neighboring cell, we add more/less
+                          // entries.
+        Matrix->SumIntoGlobalValues(dofs[i],
+          dofs_per_cell, &values[0], reinterpret_cast<int*>(&dofs[0]));
+        if (boundary < 0) {
+          Matrix->SumIntoGlobalValues(dofs[i],
+            dofs_per_cell, &values[dofs_per_cell], reinterpret_cast<int*>(&dofs_neighbor[0]));
+        }
+
+                          // And add into the residual
+        right_hand_side(dofs[i]) -= F_i.val();
+      } 
+
+}
+                                 // @sect4{Assembling the whole system}
+                                 // Now we put all of the assembly pieces together
+                                 // in a routine that dispatches the correct
+                                 // piece for each cell/face.  We keep track of
+                                 // the norm of the resdual for the Newton iteration.
+template <int dim>
+void ConsLaw<dim>::assemble_system (double &res_norm) 
+{
+  FESystem<dim> &fe = *fe_ptr;
+  const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;
+
+                                  // We track the dofs on this cell and (if necessary)
+                                  // the adjacent cell.
+  std::vector<unsigned int> dofs (dofs_per_cell);
+  std::vector<unsigned int> dofs_neighbor (dofs_per_cell);
+
+				   // First we create the
+				   // ``UpdateFlags'' for the
+				   // ``FEValues'' and the
+				   // ``FEFaceValues'' objects.
+  UpdateFlags update_flags = update_values
+			     | update_gradients
+			     | update_q_points
+			     | update_JxW_values;
+
+				   // Note, that on faces we do not
+				   // need gradients but we need
+				   // normal vectors.
+  UpdateFlags face_update_flags = update_values
+				  | update_q_points
+				  | update_JxW_values
+				  | update_normal_vectors;
+  
+				   // On the neighboring cell we only
+				   // need the shape values. Given a
+				   // specific face, the quadrature
+				   // points and `JxW values' are the
+				   // same as for the current cells,
+				   // the normal vectors are known to
+				   // be the negative of the normal
+				   // vectors of the current cell.
+  UpdateFlags neighbor_face_update_flags = update_values;
+   
+				   // Then we create the ``FEValues''
+				   // object. Note, that since version
+				   // 3.2.0 of deal.II the constructor
+				   // of this class takes a
+				   // ``Mapping'' object as first
+				   // argument. Although the
+				   // constructor without ``Mapping''
+				   // argument is still supported it
+				   // is recommended to use the new
+				   // constructor. This reduces the
+				   // effect of `hidden magic' (the
+				   // old constructor implicitely
+				   // assumes a ``MappingQ1'' mapping)
+				   // and makes it easier to change
+				   // the mapping object later.
+  FEValues<dim> fe_v (
+    mapping, fe, quadrature, update_flags);
+  
+				   // Similarly we create the
+				   // ``FEFaceValues'' and
+				   // ``FESubfaceValues'' objects for
+				   // both, the current and the
+				   // neighboring cell. Within the
+				   // following nested loop over all
+				   // cells and all faces of the cell
+				   // they will be reinited to the
+				   // current cell and the face (and
+				   // subface) number.
+  FEFaceValues<dim> fe_v_face (
+    mapping, fe, face_quadrature, face_update_flags);
+  FESubfaceValues<dim> fe_v_subface (
+    mapping, fe, face_quadrature, face_update_flags);
+  FEFaceValues<dim> fe_v_face_neighbor (
+    mapping, fe, face_quadrature, neighbor_face_update_flags);
+  FESubfaceValues<dim> fe_v_subface_neighbor (
+    mapping, fe, face_quadrature, neighbor_face_update_flags);
+
+				   // Furthermore we need some cell
+				   // iterators.
+  typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+
+				   // Now we start the loop over all
+				   // active cells.
+  int fdofs_per_cell = fe_v.dofs_per_cell;
+  int fn_q_points = face_quadrature.n_quadrature_points;
+
+  unsigned int cell_no = 0;
+  for (;cell!=endc; ++cell, ++cell_no) 
+    {
+      
+				       // Now we reinit the ``FEValues''
+				       // object for the current cell
+      fe_v.reinit (cell);
+
+                                       // Collect the local dofs and
+                                       // asssemble the cell term.
+      cell->get_dof_indices (dofs);
+
+      cell_diameter = cell->diameter();
+
+      assemble_cell_term(fe_v,
+                         dofs,
+                         cell_no);
+
+                                       // We use the DG style loop through faces
+                                       // to determine if we need to apply a
+                                       // 'hanging node' flux calculation or a boundary
+                                       // computation.
+      for (unsigned int face_no=0; face_no<GeometryInfo<dim>::faces_per_cell; ++face_no)
+	{
+					   // First we set the face
+					   // iterator
+	  typename DoFHandler<dim>::face_iterator face=cell->face(face_no);
+          face_diameter = face->diameter();
+	  
+	  if (face->at_boundary())
+	    {
+					       // We reinit the
+					       // ``FEFaceValues''
+					       // object to the
+					       // current face
+	      fe_v_face.reinit (cell, face_no);
+
+					       // and assemble the
+					       // corresponding face
+					       // terms.  We send the same
+                                               // fe_v and dofs as described
+                                               // in the assembly routine.
+	      assemble_face_term(
+                                   face_no, fe_v_face,
+                                   fe_v_face,
+                                   dofs,
+                                   dofs,
+                                   face->boundary_indicator());
+	    }
+	  else
+	    {
+					       // Now we are not on
+					       // the boundary of the
+					       // domain, therefore
+					       // there must exist a
+					       // neighboring cell.
+	      typename DoFHandler<dim>::cell_iterator neighbor=
+		cell->neighbor(face_no);;
+
+	      if (face->has_children())
+		{
+                  // case I: This cell refined compared to neighbor
+
+		  const unsigned int neighbor2=
+		    cell->neighbor_of_neighbor(face_no);
+		  
+		  
+						   // We loop over
+						   // subfaces
+		  for (unsigned int subface_no=0;
+		       subface_no<GeometryInfo<dim>::subfaces_per_face;
+		       ++subface_no)
+		    {
+		      typename DoFHandler<dim>::active_cell_iterator
+                        neighbor_child
+                        = cell->neighbor_child_on_subface (face_no, subface_no);
+
+                      face_diameter = neighbor_child->diameter();  // working on subface
+		      
+		      Assert (neighbor_child->face(neighbor2) == face->child(subface_no),
+			      ExcInternalError());
+		      Assert (!neighbor_child->has_children(), ExcInternalError());
+
+		      fe_v_subface.reinit (cell, face_no, subface_no);
+		      fe_v_face_neighbor.reinit (neighbor_child, neighbor2);
+		      neighbor_child->get_dof_indices (dofs_neighbor);
+
+                                               // Assemble as if we are working with
+                                               // a DG element.
+		      assemble_face_term(
+                                             face_no, fe_v_subface,
+					     fe_v_face_neighbor,
+                                             dofs,
+                                             dofs_neighbor);
+		      
+		    }
+						   // End of ``if
+						   // (face->has_children())''
+		}
+	      else
+		{
+                                                  // We have no children, but 
+                                                  // the neighbor cell may be refine
+                                                  // compared to use
+		  neighbor->get_dof_indices (dofs_neighbor);
+		  if (neighbor->level() != cell->level()) 
+		    {
+                      // case II: This is refined compared to neighbor
+		      Assert(neighbor->level() < cell->level(), ExcInternalError());
+		      const std::pair<unsigned int, unsigned int> faceno_subfaceno=
+			cell->neighbor_of_coarser_neighbor(face_no);
+		      const unsigned int neighbor_face_no=faceno_subfaceno.first,
+				      neighbor_subface_no=faceno_subfaceno.second;
+
+		      Assert (neighbor->neighbor_child_on_subface (neighbor_face_no,
+                                                                   neighbor_subface_no)
+                              == cell,
+                              ExcInternalError());
+
+						       // Reinit the
+						       // appropriate
+						       // ``FEFaceValues''
+						       // and assemble
+						       // the face
+						       // terms.
+		      fe_v_face.reinit (cell, face_no);
+		      fe_v_subface_neighbor.reinit (neighbor, neighbor_face_no,
+						    neighbor_subface_no);
+		      
+		      assemble_face_term(
+                                             face_no, fe_v_face,
+					     fe_v_subface_neighbor,
+                                             dofs,
+                                             dofs_neighbor);
+
+		    }
+
+		} 
+				     // End of ``face not at boundary'':
+	    }
+				     // End of loop over all faces:
+	} 
+      
+                                     // End iteration through cells.
+    } 
+
+                                     // Notify Epetra that the matrix is done.
+    Matrix->FillComplete();
+
+                                    // Compute the nonlinear residual.
+    res_norm = right_hand_side.l2_norm();
+    
+}
+
+                                    // Create a conservation law with some defaults.
+template <int dim>
+ConsLaw<dim>::ConsLaw ()
+		:
+		mapping (),
+                fe_ptr(NULL),
+		dof_handler (triangulation),
+		quadrature (2),
+		face_quadrature (2),
+                T(0),
+                dT(0.05),
+                TF(10),
+                Map(NULL),
+                Matrix(NULL),
+                is_stationary(false),
+                theta(0.5)
+{}
+
+                        // At one time this example could work for both DG and
+                        // continuous finite elements.  The choice was made here.
+template <int dim>
+void ConsLaw<dim>::build_fe() {
+  fe_ptr = new FESystem<dim>(FE_Q<dim>(1), N_COMP);
+}
+
+                        // Bye bye Conservation law.
+template <int dim>
+ConsLaw<dim>::~ConsLaw () 
+{
+  dof_handler.clear ();
+  delete fe_ptr;
+}
+
+                        // @sect3{Initialize System}
+                        // Sizes all of the vectors and sets up the
+                        // sparsity patter.  This function is called at
+                        // the very beginning of a simulation.  The function
+                        // <code> setup_system </code> repeats some of these
+                        // chores and is called after adaptivity in leiu
+                        // of this function.
+template <int dim>
+void ConsLaw<dim>::initialize_system ()
+{
+				   // First we need to distribute the
+				   // DoFs.
+  dof_handler.clear();
+  dof_handler.distribute_dofs (*fe_ptr);
+  
+                                   // Size all of the fields.
+  solution.reinit (dof_handler.n_dofs());
+  nlsolution.reinit (dof_handler.n_dofs());
+  predictor.reinit (dof_handler.n_dofs());
+  ppsolution.reinit (dof_handler.n_dofs());
+  dsolution.reinit (dof_handler.n_dofs());
+  right_hand_side.reinit (dof_handler.n_dofs());
+  indicator.reinit(triangulation.n_active_cells());
+}
+
+                                  // @sect3{Setup System}
+                                  // We call this function to build the sparsity
+                                  // and the matrix.
+template <int dim>
+void ConsLaw<dim>::setup_system ()
+{
+
+				   // The DoFs of a cell are coupled
+				   // with all DoFs of all neighboring
+				   // cells.  Therefore the maximum
+				   // number of matrix entries per row
+				   // is needed when all neighbors of
+				   // a cell are once more refined
+				   // than the cell under
+				   // consideration.
+  sparsity_pattern.reinit (dof_handler.n_dofs(),
+			   dof_handler.n_dofs(),
+			   (GeometryInfo<dim>::faces_per_cell
+			    *GeometryInfo<dim>::subfaces_per_face+1)*fe_ptr->dofs_per_cell);
+  
+                                   // Since the continuous sparsity pattern is
+                                   // a subset of the DG one, and since we need
+                                   // the DG terms for handling hanging nodes, we use
+                                   // the flux pattern.
+  DoFTools::make_flux_sparsity_pattern (dof_handler, sparsity_pattern);
+  
+  sparsity_pattern.compress();
+  
+                                   // Rebuild the map.  In serial this doesn't do much,
+                                   // but is needed.  In parallel, this would desribe
+                                   // the parallel dof layout.
+  if (Map) delete Map;
+  Map  = new Epetra_Map(dof_handler.n_dofs(), 0, *Comm);
+
+                                   // Epetra can build a more efficient matrix if
+                                   // one knows ahead of time the maxiumum number of
+                                   // columns in any row entry.  We traverse the sparsity
+                                   // to discover this.
+  int cur_row = 0;
+  int cur_col = 0;
+  int max_df = -1;
+  for (SparsityPattern::iterator s_i = sparsity_pattern.begin(); 
+       s_i != sparsity_pattern.end(); s_i++) {
+    if (s_i->row() != cur_row) {
+      cur_col = 0;
+      cur_row = s_i->row();
+    }
+    cur_col++;
+   if (cur_col >= max_df) max_df = cur_col;
+  }
+
+  if (cur_col >= max_df) max_df = cur_col;
+  std::cout << "max_df:" << max_df << std::endl;
+
+                                  // Now we build the matrix, using the constructor
+                                  // that optimizes with the <code> max_df </code> variable.
+  if (Matrix) delete Matrix;
+  Matrix = new Epetra_CrsMatrix(Copy, *Map, max_df+1, true);
+
+                                 // We add the sparsity pattern to the matrix by
+                                 // inserting zeros.
+  std::vector<double> vals(max_df, 0);
+  std::vector<int> row_indices(max_df);
+ 
+  cur_row = 0;
+  cur_col = 0;
+  for (SparsityPattern::iterator s_i = sparsity_pattern.begin(); 
+     s_i != sparsity_pattern.end(); s_i++) {
+    if (s_i->row() != cur_row) {
+      Matrix->InsertGlobalValues(cur_row, cur_col, &vals[0], &row_indices[0]);
+      cur_col = 0;
+      cur_row = s_i->row();
+    }
+  row_indices[cur_col++] = s_i->column();
+  }
+                                 // The last row.
+  Matrix->InsertGlobalValues(cur_row, cur_col, &vals[0], &row_indices[0]);
+
+                                 // Epetra requires this function after building or
+                                 // filling a matrix.  It typically does some parallel
+                                 // bookeeping; perhaps more.
+  Matrix->FillComplete();
+
+}
+
+                                 // @sect3{Solving the linear system}
+                                 // Actually solve the linear system, using either
+                                 // Aztec of Amesos.
+template <int dim>
+void ConsLaw<dim>::solve (Vector<double> &dsolution, int &niter, double &lin_residual) 
+{
+
+                                 // We must hand the solvers Epetra vectors.
+                                 // Luckily, they support the concept of a 
+                                 // 'view', so we just send in a pointer to our
+                                 // dealii vectors.
+    Epetra_Vector x(View, *Map, dsolution.begin());
+    Epetra_Vector b(View, *Map, right_hand_side.begin());
+
+                                 // The Direct option selects the Amesos solver.
+  if (solver_params.SOLVER == solver_params_type::DIRECT) {
+   
+                                 // Setup for solving with
+                                 // Amesos.
+     Epetra_LinearProblem prob;
+     prob.SetOperator(Matrix);
+     Amesos_BaseSolver *solver;
+     Amesos Factory;
+
+                                 // Other solvers are available
+                                 // and may be selected by changing this
+                                 // string.
+     char *stype = "Amesos_Klu";
+
+     solver = Factory.Create(stype, prob);
+
+     Assert (solver != NULL, ExcInternalError());
+
+                                 // There are two parts to the direct solve.
+                                 // As I understand, the symbolic part figures
+                                 // out the sparsity patterns, and then the
+                                 // numerical part actually performs Gaussian
+                                 // elimination or whatever the approach is.
+     if (solver_params.OUTPUT == solver_params_type::VERBOSE)
+       std::cout << "Starting Symbolic fact\n" << std::flush;
+
+     solver->SymbolicFactorization();
+
+     if (solver_params.OUTPUT == solver_params_type::VERBOSE)
+         std::cout << "Starting Numeric fact\n" << std::flush;
+
+     solver->NumericFactorization();
+
+    
+                                 // Define the linear problem by setting the
+                                 // right hand and left hand sides.
+     prob.SetRHS(&b);
+     prob.SetLHS(&x);
+                                 // And finally solve the problem.
+     if (solver_params.OUTPUT == solver_params_type::VERBOSE)
+       std::cout << "Starting solve\n" << std::flush;
+     solver->Solve();
+     niter = 0;
+     lin_residual = 0;
+
+                                 // We must free the solver that was created
+                                 // for us.
+     delete solver;
+
+  } else if (solver_params.SOLVER == solver_params_type::GMRES) {
+
+                                 // For the iterative solvers, we use Aztec.
+    AztecOO Solver;
+
+                                 // Select the appropriate level of verbosity.
+    if (solver_params.OUTPUT == solver_params_type::QUIET)
+      Solver.SetAztecOption(AZ_output, AZ_none);
+
+    if (solver_params.OUTPUT == solver_params_type::VERBOSE)
+      Solver.SetAztecOption(AZ_output, AZ_all);
+
+                                 // Select gmres.  Other solvers are available.
+    Solver.SetAztecOption(AZ_solver, AZ_gmres);
+    Solver.SetRHS(&b);
+    Solver.SetLHS(&x);
+
+                                 // Set up the ILUT preconditioner.  I do not know
+                                 // why, but we must pretend like we are in parallel
+                                 // using domain decomposition or the preconditioner
+                                 // refuses to activate.
+    Solver.SetAztecOption(AZ_precond, AZ_dom_decomp);
+    Solver.SetAztecOption(AZ_subdomain_solve, AZ_ilut);
+    Solver.SetAztecOption(AZ_overlap, 0);
+    Solver.SetAztecOption(AZ_reorder, 0);
+
+                                 // ILUT parameters as described above.
+    Solver.SetAztecParam(AZ_drop, solver_params.ILUT_DROP);
+    Solver.SetAztecParam(AZ_ilut_fill, solver_params.ILUT_FILL);
+    Solver.SetAztecParam(AZ_athresh, solver_params.ILUT_ATOL);
+    Solver.SetAztecParam(AZ_rthresh, solver_params.ILUT_RTOL);
+    Solver.SetUserMatrix(Matrix);
+
+                                 // Run the solver iteration.  Collect the number
+                                 // of iterations and the residual.
+    Solver.Iterate(solver_params.MAX_ITERS, solver_params.RES);
+    niter = Solver.NumIters();
+    lin_residual = Solver.TrueResidual();
+  }
+}
+
+                                 // @sect3{Postprocessing and Output}
+                                 // Recover the physical variables from the conservative
+                                 // variables so that output will be (perhaps) more
+                                 // meaningfull.
+template <int dim>
+void ConsLaw<dim>::postprocess() {
+  const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;
+  std::vector<unsigned int> dofs (dofs_per_cell);
+  UpdateFlags update_flags = update_values
+			     | update_gradients
+			     | update_q_points
+			     | update_JxW_values;
+  UpdateFlags update_flags1 = update_values
+			     | update_gradients
+			     | update_q_points
+			     | update_JxW_values;
+
+ QGauss<dim>  quadrature_formula(4);
+
+ const std::vector<Point<dim> > &us = fe_ptr->base_element(0).get_unit_support_points();
+
+
+ Quadrature<dim>  unit_support(us);
+
+ int n_q_points = quadrature_formula.n_quadrature_points;
+ int n_uq_points = unit_support.n_quadrature_points;
+
+  FEValues<dim> fe_v (
+    mapping, *fe_ptr, quadrature_formula, update_flags);
+
+  FEValues<dim> fe_v_unit (
+    mapping, *fe_ptr, unit_support, update_flags1);
+
+  std::vector<Vector<double> > U(n_uq_points,
+                                 Vector<double>(get_n_components()));
+  std::vector<Vector<double> > UU(n_q_points,
+                                 Vector<double>(get_n_components()));
+  std::vector<std::vector<Tensor<1,dim> > > dU(n_uq_points,
+                                            std::vector<Tensor<1,dim> >(get_n_components()));
+  
+  typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+
+                                    // Loop the cells
+  for (unsigned int cell_no=0; cell!=endc; ++cell, ++cell_no) {
+    cell->get_dof_indices (dofs);
+    fe_v_unit.reinit(cell);
+    fe_v.reinit(cell);
+
+    fe_v_unit.get_function_values(solution, U);
+    fe_v_unit.get_function_grads(solution, dU);
+    fe_v.get_function_values(solution, UU);
+
+    const std::vector<double> &JxW = fe_v.get_JxW_values ();
+
+    for (int q = 0; q < fe_v.get_fe().base_element(0).n_dofs_per_cell(); q++) {
+      unsigned int didx = fe_v.get_fe().component_to_system_index(DENS_IDX, q);
+      unsigned int eidx = fe_v.get_fe().component_to_system_index(ENERGY_IDX, q);
+      double rho_normVsqr = 0;
+      for (int d = 0; d < dim; d++) {
+        unsigned int vidx = fe_v.get_fe().component_to_system_index(d, q);
+        ppsolution(dofs[vidx]) = solution(dofs[vidx])/solution(dofs[didx]);
+        rho_normVsqr += solution(dofs[vidx])*solution(dofs[vidx]);
+      }
+      rho_normVsqr /= solution(dofs[didx]);
+                                 // Pressure
+      ppsolution(dofs[eidx]) = (GAMMA-1.0)*(solution(dofs[eidx]) - 0.5*rho_normVsqr);
+
+                                 // Either output density or gradient squared of density,
+                                 // depending on what the user wants.
+      if (!schlieren_plot) {
+        ppsolution(dofs[didx]) = solution(dofs[didx]);
+      } else {
+        double ng = 0;
+        for (int i = 0; i < dim; i++) ng += dU[q][DENS_IDX][i]*dU[q][DENS_IDX][i];
+        ng = std::sqrt(ng);
+        ppsolution(dofs[didx]) = ng;
+      }
+    }
+
+  } // cell
+
+}
+
+                            // Loop and assign a value for refinement.  We
+                            // simply use the density squared, which selects
+                            // shocks with some success.
+template <int dim>
+void ConsLaw<dim>::estimate() {
+  
+  const unsigned int dofs_per_cell = dof_handler.get_fe().dofs_per_cell;
+  std::vector<unsigned int> dofs (dofs_per_cell);
+  UpdateFlags update_flags = update_values
+			     | update_gradients
+			     | update_q_points
+			     | update_JxW_values;
+
+ QGauss<dim>  quadrature_formula(1);
+ int n_q_points = quadrature_formula.n_quadrature_points;
+
+
+  FEValues<dim> fe_v (
+    mapping, *fe_ptr, quadrature_formula, update_flags);
+
+  std::vector<Vector<double> > U(n_q_points,
+                                 Vector<double>(get_n_components()));
+  std::vector<std::vector<Tensor<1,dim> > > dU(n_q_points,
+                                            std::vector<Tensor<1,dim> >(get_n_components()));
+  
+  typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+  for (unsigned int cell_no=0; cell!=endc; ++cell, ++cell_no) {
+    fe_v.reinit(cell);
+
+    fe_v.get_function_values(predictor, U);
+    fe_v.get_function_grads(predictor, dU);
+
+    indicator(cell_no) = 0;
+    for (int q = 0; q < n_q_points; q++) {
+      double ng = 0;
+      for (int d = 0; d < dim; d++) ng += dU[q][DENS_IDX][d]*dU[q][DENS_IDX][d];
+
+      indicator(cell_no) += std::log(1+std::sqrt(ng));
+      
+    } 
+    indicator(cell_no) /= n_q_points;
+
+  } 
+}
+
+template <int dim>
+void ConsLaw<dim>::refine_grid ()
+{
+
+  SolutionTransfer<dim, double> soltrans(dof_handler);
+
+  typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+
+                                  // Loop cells.  If the indicator
+                                  // for the cell matches the refinement criterion,
+                                  // refine, else unrefine.  The unrefinement has
+                                  // a slight hysterisis to avoid 'flashing' from refined
+                                  // to unrefined.
+  for (unsigned int cell_no=0; cell!=endc; ++cell, ++cell_no) {
+    cell->clear_coarsen_flag();
+    cell->clear_refine_flag();
+    if (cell->level() < refinement_params.shock_levels &&
+        std::fabs(indicator(cell_no)) > refinement_params.shock_val ) {
+      cell->set_refine_flag();
+    } else {
+      if (cell->level() > 0 &&
+         std::fabs(indicator(cell_no)) < 0.75*refinement_params.shock_val)
+           cell->set_coarsen_flag();
+    }
+  }
+
+                                  // The following code prolongs the solution
+                                  // to the new grid and carries out the refinement.
+  std::vector<Vector<double> > interp_in;
+  std::vector<Vector<double> > interp_out;
+
+  interp_in.push_back(solution);
+  interp_in.push_back(predictor);
+
+  triangulation.prepare_coarsening_and_refinement();
+  soltrans.prepare_for_coarsening_and_refinement(interp_in);
+
+  triangulation.execute_coarsening_and_refinement ();
+
+  dof_handler.clear();
+  dof_handler.distribute_dofs (*fe_ptr);
+
+  {
+  Vector<double> new_solution(1);
+  Vector<double> new_predictor(1);
+
+  interp_out.push_back(new_solution);
+  interp_out.push_back(new_predictor);
+  interp_out[0].reinit(dof_handler.n_dofs());
+  interp_out[1].reinit(dof_handler.n_dofs());
+  }
+
+  soltrans.interpolate(interp_in, interp_out);
+  
+                               // Let the vector delete a very small vector
+  solution.reinit(1);
+  predictor.reinit(1);
+  solution.swap(interp_out[0]);
+  predictor.swap(interp_out[1]);
+
+                               // resize these vectors for the new grid.
+  nlsolution.reinit(dof_handler.n_dofs());
+  ppsolution.reinit(dof_handler.n_dofs());
+  nlsolution = solution;
+  dsolution.reinit (dof_handler.n_dofs());
+  right_hand_side.reinit (dof_handler.n_dofs());
+
+  indicator.reinit(triangulation.n_active_cells());
+
+}
+
+template <int dim>
+void ConsLaw<dim>::output_results (const unsigned int cycle) const
+{
+  char filename[512];
+  std::sprintf(filename, "solution-%03d.vtk", cycle);
+  std::ofstream output (filename);
+
+  DataOut<dim> data_out;
+  data_out.attach_dof_handler (dof_handler);
+  std::vector<std::string> solution_names;
+
+                         // Rename the output with the physical variable
+                         // names.  Send the post-processed values.
+  solution_names.clear();
+  for (int i = 0; i < dim; i++) {
+    char buf[512];
+    std::sprintf(buf, "v_%d", i);
+    solution_names.push_back (buf);	    
+  }
+  solution_names.push_back("density");
+  solution_names.push_back("pressure");
+  data_out.add_data_vector (ppsolution, solution_names);
+
+  data_out.add_data_vector (indicator, "error");
+  data_out.build_patches ();
+  data_out.write_vtk (output);
+
+  output.close();
+}
+
+                                   // @sect3{Parsing the Input Deck}
+                                   // Declare the parameters for the
+                                   // input deck.  We assume a certain
+                                   // maximum number of boundaries and process
+                                   // any boundary the user supplies up to
+                                   // that maximum number.  We
+                                   // leave a detailed explanation of these 
+                                   // parameters to our description of the input
+                                   // sample file.
+const UInt MAX_BD = 10;
+template <int dim>
+void ConsLaw<dim>::declare_parameters() {
+
+                                   // Global scope parameters/
+  prm.declare_entry("mesh", "grid.inp",
+                    Patterns::Anything(),
+                    "intput file");
+
+  prm.declare_entry("diffusion power", "2.0",
+                     Patterns::Double(),
+                     "power of mesh size for diffusion");
+
+  prm.declare_entry("gravity", "0.0",
+                     Patterns::Double(),
+                     "gravity forcing");
+
+                                   // Time stepping block
+  prm.enter_subsection("time stepping");
+    prm.declare_entry("time step", "0.1",
+                     Patterns::Double(),
+                     "simulation time step");
+    prm.declare_entry("final time", "10.0",
+                     Patterns::Double(),
+                     "simulation end time");
+  prm.leave_subsection();
+
+
+                                  // Declare the boundary parameters
+  for (int b = 0; b < MAX_BD; b++) {
+    char bd[512];
+    std::sprintf(bd, "boundary_%d", b);
+    prm.enter_subsection(bd);
+    prm.declare_entry("no penetration", "false",
+                       Patterns::Selection("true|false"),
+                       "<true|false>");
+                                  // declare a slot for each of the conservative
+                                  // variables.
+    for (int di = 0; di < N_COMP; di++) {
+      char var[512];
+      std::sprintf(var, "w_%d", di);
+      prm.declare_entry(var, "outflow",
+                     Patterns::Selection(
+        "inflow|outflow|pressure"),
+        "<inflow|outflow|pressure>");
+      
+                                   // for dirichlet, a function in x,y,z
+      std::sprintf(var, "w_%d value", di);
+      prm.declare_entry(var, "0.0",
+                     Patterns::Anything(),
+                 "expression in x,y,z");
+    }
+
+    prm.leave_subsection();
+  }
+
+                                // Initial condition block.
+  prm.enter_subsection("initial condition");
+    for (int di = 0; di < N_COMP; di++) {
+      char var[512];
+      std::sprintf(var, "w_%d", di);
+      
+      // for dirichlet, a function in x,y,z
+      std::sprintf(var, "w_%d value", di);
+      prm.declare_entry(var, "0.0",
+                     Patterns::Anything(),
+                 "expression in x,y,z");
+    }
+  prm.leave_subsection();
+
+                              // The linear solver block.
+  prm.enter_subsection("linear solver");
+    prm.declare_entry("output", "quiet",
+                     Patterns::Selection(
+                     "quiet|verbose"),
+                      "<quiet|verbose>");
+    prm.declare_entry("method", "gmres",
+                     Patterns::Selection(
+                     "gmres|direct"),
+                      "<gmres|direct>");
+    prm.declare_entry("residual", "1e-10",
+                     Patterns::Double(),
+                     "linear solver residual");
+    prm.declare_entry("max iters", "300",
+                     Patterns::Double(),
+                     "maximum solver iterations");
+    prm.declare_entry("ilut fill", "2",
+                     Patterns::Double(),
+                     "ilut preconditioner fill");
+    prm.declare_entry("ilut absolute tolerance", "1e-9",
+                     Patterns::Double(),
+                     "ilut preconditioner tolerance");
+    prm.declare_entry("ilut relative tolerance", "1.1",
+                     Patterns::Double(),
+                     "rel tol");
+    prm.declare_entry("ilut drop tolerance", "1e-10",
+                     Patterns::Double(),
+                     "ilut drop tol");
+  prm.leave_subsection();
+
+
+                           // A refinement controller block.
+  prm.enter_subsection("refinement");
+    prm.declare_entry("refinement", "none",
+                     Patterns::Selection(
+                     "none|fixed number|shock"),
+                      "<on|off>");
+    prm.declare_entry("refinement fraction", "0.1",
+                     Patterns::Double(),
+                     "Fraction of high refinement");
+    prm.declare_entry("unrefinement fraction", "0.1",
+                     Patterns::Double(),
+                     "Fraction of low unrefinement");
+    prm.declare_entry("max elements", "1000000",
+                     Patterns::Double(),
+                     "maximum number of elements");
+    prm.declare_entry("shock value", "4.0",
+                     Patterns::Double(),
+                     "value for shock indicator");
+    prm.declare_entry("shock levels", "3.0",
+                     Patterns::Double(),
+                     "number of shock refinement levels");
+  prm.leave_subsection();
+
+                      // Output control.
+  prm.enter_subsection("output");
+    prm.declare_entry("density", "standard",
+                     Patterns::Selection(
+                     "standard|schlieren"),
+                      "<standard|schlieren>");
+    prm.declare_entry("step", "-1",
+                     Patterns::Double(),
+                     "output once per this period");
+  prm.leave_subsection();
+
+                      // Flux control
+  prm.enter_subsection("flux");
+    prm.declare_entry("stab", "alpha",
+                     Patterns::Selection(
+                     "alpha|constant|mesh"),
+                      "<alpha|constant|mesh>");
+    prm.declare_entry("stab value", "1",
+                     Patterns::Double(),
+                     "alpha stabilization");
+  prm.leave_subsection();
+
+
+}
+
+                     // Code to actually parse an input file.  This function
+                     // matches the declarations above.
+template <int dim>
+void ConsLaw<dim>::load_parameters(const char *infile){
+
+  prm.read_input(infile);
+
+                     // The global parameters.
+  mesh = prm.get("mesh");
+
+  diffusion_power = prm.get_double("diffusion power");
+
+  gravity = prm.get_double("gravity");
+
+                    // The time stepping.
+  prm.enter_subsection("time stepping");
+  dT = prm.get_double("time step");
+  std::cout << "dT=" << dT << std::endl;
+  if (dT == 0) {
+    is_stationary = true;
+    dT = 1.0;
+    TF = 1.0;
+    std::cout << "Stationary mode" << std::endl;
+  }
+  TF = prm.get_double("final time");
+  std::cout << "TF=" << TF << std::endl;
+  prm.leave_subsection();
+
+                   // The boundary info
+  for (int b = 0; b < MAX_BD; b++) {
+    std::vector<bc_type> flags(N_COMP, OUTFLOW_BC);
+
+                   // Define a parser for every boundary, though it may be
+                   // unused.
+    SideCondition<dim> *sd = new SideCondition<dim>(N_COMP);
+    char bd[512];
+    std::sprintf(bd, "boundary_%d", b);
+    prm.enter_subsection(bd);
+
+    const std::string &nopen = prm.get("no penetration");
+
+                    // Determine how each component is handled.
+    for (int di = 0; di < N_COMP; di++) {
+      char var[512];
+      std::sprintf(var, "w_%d", di);
+      std::string btype = prm.get(var);
+      std::sprintf(var, "w_%d value", di);
+      std::string var_value = prm.get(var);
+
+      if (di < dim && nopen == "true") {
+        flags[di] = NO_PENETRATION_BC;
+      } else if (btype == "inflow") {
+        flags[di] = INFLOW_BC;
+        sd->set_coeff_row(di, var_value);  
+      } else if (btype == "pressure") {
+        flags[di] = PRESSURE_BC;
+        sd->set_coeff_row(di, var_value);  
+      }
+    } 
+    prm.leave_subsection();
+
+                     // Add the boundary condition to the law.
+    sd->Init();
+    add_boundary(b, flags, sd);
+   }
+
+                     // Initial conditions.
+   prm.enter_subsection("initial condition");
+    for (int di = 0; di < N_COMP; di++) {
+      char var[512];
+
+      std::sprintf(var, "w_%d value", di);
+      std::string var_value = prm.get(var);
+      ic.set_ic(di, var_value);  
+    }
+    ic.Init();
+   prm.leave_subsection();
+
+                    // The linear solver.
+ prm.enter_subsection("linear solver");
+    const std::string &op = prm.get("output");
+    if (op == "verbose") solver_params.OUTPUT = solver_params_type::VERBOSE;
+    if (op == "quiet") solver_params.OUTPUT = solver_params_type::QUIET;
+    const std::string &sv = prm.get("method");
+    if (sv == "direct") {
+      solver_params.SOLVER = solver_params_type::DIRECT;
+    } else if (sv == "gmres") {
+      solver_params.SOLVER = solver_params_type::GMRES;
+    } 
+
+    solver_params.RES = prm.get_double("residual");
+    solver_params.MAX_ITERS = (int) prm.get_double("max iters");
+    solver_params.ILUT_FILL = prm.get_double("ilut fill");
+    solver_params.ILUT_ATOL = prm.get_double("ilut absolute tolerance");
+    solver_params.ILUT_RTOL = prm.get_double("ilut relative tolerance");
+    solver_params.ILUT_DROP = prm.get_double("ilut drop tolerance");
+    solver_params.RES = prm.get_double("residual");
+  prm.leave_subsection();
+
+
+                       // And refiement.
+  prm.enter_subsection("refinement");
+    const std::string &ref = prm.get("refinement");
+    if (ref == "none") {
+      refinement_params.refine = refinement_params_type::NONE;
+    } else if (ref == "fixed number") {
+      refinement_params.refine = refinement_params_type::FIXED_NUMBER;
+    } else if (ref == "shock") {
+      refinement_params.refine = refinement_params_type::SHOCK;
+    } else
+    refinement_params.high_frac = prm.get_double("refinement fraction");
+    refinement_params.high_frac_sav = refinement_params.high_frac;
+    refinement_params.low_frac = prm.get_double("unrefinement fraction");
+    refinement_params.max_cells = prm.get_double("max elements");
+    refinement_params.shock_val = prm.get_double("shock value");
+    refinement_params.shock_levels = prm.get_double("shock levels");
+  prm.leave_subsection();
+    
+                           // Output control.
+  prm.enter_subsection("output");
+    const std::string &dens = prm.get("density");
+    schlieren_plot = dens == "schlieren" ? true : false;
+    output_step = prm.get_double("step");
+  prm.leave_subsection();
+
+                           // Flux control.
+  prm.enter_subsection("flux");
+    const std::string &stab = prm.get("stab");
+    if (stab == "constant") {
+      flux_params.LF_stab = flux_params_type::CONSTANT;
+    } else if (stab == "mesh ") {
+      flux_params.LF_stab = flux_params_type::MESH;
+    }
+    flux_params.LF_stab_value = prm.get_double("stab value");
+  prm.leave_subsection();
+
+
+}
+
+template<int dim>
+void ConsLaw<dim>::zero_matrix() {
+  Matrix->PutScalar(0); Matrix->FillComplete();
+}
+
+                          // We use a predictor to try and make adaptivity
+                          // work better.  The idea is to try and refine ahead
+                          // of a front, rather than stepping into a coarse
+                          // set of elements and smearing the solution.  This
+                          // simple time extrapolator does the job.
+template<int dim>
+void ConsLaw<dim>::compute_predictor() {
+  predictor = nlsolution;
+  predictor.sadd(3/2.0, -1/2.0, solution);
+}
+
+                          // @sect3{Run the simulation}  Contains the initialization,
+                          // the time loop, and the inner Newton iteration.
+template <int dim>
+void ConsLaw<dim>::run () 
+{
+
+                          // Open and load the mesh.
+  GridIn<dim> grid_in;
+  grid_in.attach_triangulation(triangulation);
+  std::cout << "Opening mesh <" << mesh << ">" << std::endl;
+  std::ifstream input_file(mesh.c_str(), std::ios::in);
+
+  Assert (infile,
+	  ExcFileNotOpen());
+
+  grid_in.read_ucd(input_file);   
+  input_file.close();
+  
+  build_fe();
+
+  unsigned int nstep = 0;
+  
+                           // Initialize fields and matrices.
+  initialize_system (); 
+  setup_system();
+  initialize(); 
+  predictor = solution;
+
+                          // Initial refinement.  We apply the ic,
+                          // estimate, refine, and repeat until
+                          // happy.
+  if (refinement_params.refine != refinement_params_type::NONE)
+  for (int i = 0; i < refinement_params.shock_levels; i++) {
+    estimate();
+    refine_grid();
+    setup_system();
+    initialize(); 
+    predictor = solution;
+  }
+  postprocess();
+  output_results (nstep);
+
+                           // Determine when we will output next.
+  double next_output = T + output_step;
+
+                           // @sect4{Main time stepping loop}
+  predictor = solution;
+  while(T < TF)
+    {
+      std::cout << "T=" << T << ", ";
+
+
+      std::cout << "   Number of active cells:       "
+		<< triangulation.n_active_cells()
+		<< std::endl;
+
+
+      std::cout << "   Number of degrees of freedom: "
+		<< dof_handler.n_dofs()
+		<< std::endl;
+
+      bool nonlin_done = false;
+      double res_norm;
+      int lin_iter;
+
+                              // Print some relevant information during the
+                              // Newton iteration.
+      std::cout << "NonLin Res:       Lin Iter     Lin Res" << std::endl;
+      std::cout << "______________________________________" << std::endl;
+
+      int max_nonlin = 7;
+      int nonlin_iter = 0;
+      double lin_res;
+
+                             // @sect5{Newton iteration}
+      nlsolution = predictor;
+      while (!nonlin_done) {
+        lin_iter = 0;
+        zero_matrix();
+        right_hand_side = 0;
+        assemble_system (res_norm);
+                            // Flash a star to the screen so one can
+                            // know when the assembly has stopped and the linear
+                            // solution is starting.
+        std::cout << "* " << std::flush;
+
+                            // Test against a (hardcoded) nonlinear tolderance.
+                            // Do not solve the linear system at the last step 
+                            // (since it would be a waste).
+                      
+        if (fabs(res_norm) < 1e-10) {
+          nonlin_done = true;
+        } else {
+                            // Solve the linear system and update with the
+                            // delta.
+           dsolution = 0;
+           solve (dsolution, lin_iter, lin_res);
+           nlsolution.add(1.0, dsolution);
+        }
+
+                            // Print the residuals.
+        std::printf("%-16.3e %04d        %-5.2e\n",
+              res_norm, lin_iter, lin_res);
+
+        nonlin_iter++;
+      } 
+
+                           // Various post convergence tasks.
+      compute_predictor();
+
+      solution = nlsolution;
+
+
+      estimate();
+
+      postprocess();
+
+      T += dT;
+
+                          // Output if it is time.
+      if (output_step < 0) {
+        output_results (++nstep);
+      } else if (T >= next_output) {
+        output_results (++nstep);
+        next_output += output_step;
+      }
+
+                          // Refine, if refinement is selected.
+      if (refinement_params.refine != refinement_params_type::NONE) {
+        refine_grid();
+        setup_system();
+      }
+    }
+}
+
+				 // The following ``main'' function is
+				 // similar to previous examples and
+				 // need not to be commented on.
+int main (int argc, char *argv[]) 
+{
+
+  MPI_Init(&argc, &argv);
+  Comm = new Epetra_MpiComm(MPI_COMM_WORLD);
+
+  if (argc != 2) {
+    std::cout << "Usage:" << argv[0] << " infile" << std::endl;
+    std::exit(1);
+  }
+  try
+    {
+      ConsLaw<DIMENSION> cons;
+      cons.declare_parameters();
+      cons.load_parameters(argv[1]);
+      cons.run ();
+    }
+  catch (std::exception &exc)
+    {
+      std::cerr << std::endl << std::endl
+		<< "----------------------------------------------------"
+		<< std::endl;
+      std::cerr << "Exception on processing: " << std::endl
+		<< exc.what() << std::endl
+		<< "Aborting!" << std::endl
+		<< "----------------------------------------------------"
+		<< std::endl;
+      return 1;
+    }
+  catch (...) 
+    {
+      std::cerr << std::endl << std::endl
+		<< "----------------------------------------------------"
+		<< std::endl;
+      std::cerr << "Unknown exception!" << std::endl
+		<< "Aborting!" << std::endl
+		<< "----------------------------------------------------"
+		<< std::endl;
+      return 1;
+    };
+  
+  return 0;
+}
+