The program also uses the Trilinos linear solvers (Aztec/Amesos) and
an automatic differentiation package, Sacado, which is also part of Trilinos.
-<h2>Euler flow</h2>
+<h3>Euler flow</h3>
The equations for a compressible, inviscid gas (the Euler equations) are
a basic system of conservation laws, in spatial dimension $d$,
problem. A general overview of how to deal with these problems in
deal.II can be found in the @ref vector_valued module.
-<h2>Discretization</h2>
+<h3>Discretization</h3>
Discretization happens in the usual way, taking into account that this
is a hyperbolic problem in the same style as the simple one discussed
until $|R(\mathbf{W}^k)|$ (the residual) is sufficiently small.
-<h2> Auto-Differentiation </h2>
+<h3> Auto-Differentiation </h3>
Since computing the Jacobian $\partial R$ is a terrible beast, we use an automatic differentiation package,
Sacado, to do this. Sacado is a package within the <a
used in this program. However, understanding the example above is
enough to understand the use of Sacado in this Euler flow program.
-<h2> Trilinos solvers </h2>
+<h3> Trilinos solvers </h3>
The program uses either the Aztec iterative solvers, or the Amesos
sparse direct solver, both provided by
the Trilinos package. This package is inherently designed to be used in a parallel program, however,
on these vectors. Please view the commented code for more details on how these solvers are used
within the example.
-<h2> Adaptivity </h2>
+<h3> Adaptivity </h3>
The example uses an ad-hoc refinement indicator that shows some usefulness 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 computing the numerical flux across cells that are of differing
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>
+<h3> Input Deck </h3>
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