Discontinuous Galerkin

CompHydroTutorial.tex

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higher-order/higher-order-euler.tex

@@ -98,3 +98,116 @@ \subsection{Extensions}
 by maximising over the characteristic speed within the stencil at that point, is
 less diffusive but slightly more expensive. Roe-style flux splittings are
 possible but not always stable.
+
+
+\section{Discontinuous Galerkin methods for the Euler equations}
+\label{sec:dg_euler}
+
+The Discontinuous Galerkin methods introducted in section~\ref{sec:dg} extend to
+nonlinear systems directly. For a nonlinear scalar conservation law,
+%
+\begin{equation}
+  q_t + f(q)_x = 0,
+\end{equation}
+%
+the Discontinuous Galerkin method becomes
+%
+\begin{equation}
+  \label{eq:dg_nodal_scalar_nonlinear}
+  M \ddt{\bm{\hat{q}}} + S^T f (\bm{\hat{q}}) = -\left[ \bm{\psi F} \right]_{x_{j-1/2}}^{x_{j+1/2}}.
+\end{equation}
+%
+This should be compared with equation~\eqref{eq:dg_nodal_final}
+%
+\begin{equation}
+  \tag{\ref{eq:dg_nodal_final}}
+  M \ddt{\bm{\hat{a}}} + S^T u \bm{\hat{a}} = -\left[ \bm{\psi F} \right]_{x_{j-1/2}}^{x_{j+1/2}}
+\end{equation}
+%
+for the advection equation with advection speed $u$. The crucial change is that
+the stiffness matrix $S$ is now multiplying the nonlinear flux term, and the
+boundary flux on the right hand side now needs a nonlinear Riemann solver or an
+appropriate approximation. The Lax-Friedrichs flux is often used here. Extending
+to a system of nonlinear conservation laws means applying the above scheme
+component-by-component, either directly to the conserved variables, or by
+transforming to the characteristic variables.
+
+\begin{figure}[t]
+\centering
+% figure generated by hydro_examples/compressible/dg_euler.py
+\includegraphics[width=0.8\linewidth]{dg_sod_32}
+\caption[Discontinuous Galerkin methods applied to the Sod problem, low resolution]
+{\label{fig:dg-euler-sod32} Various Discontinuous Galerkin methods, using moment
+limiting, applied to the Sod problem with $N=32$. We see that the method
+struggles near discontinuities, and that increasing $m$ alone does not improve
+the results.\\
+\hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/dg_euler.py}{dg\_euler.py}}}
+\end{figure}
+%
+
+\begin{figure}[t]
+\centering
+% figure generated by hydro_examples/compressible/dg_euler.py
+\includegraphics[width=0.8\linewidth]{dg_sod_128}
+\caption[Discontinuous Galerkin methods applied to the Sod problem, high resolution]
+{\label{fig:dg-euler-sod128} Various Discontinuous Galerkin methods, using moment
+limiting, applied to the Sod problem with $N=128$. We see that the method
+now has enough resolution to produce good results, but that increasing $m$ alone
+still does not improve the results.\\
+\hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/dg_euler.py}{dg\_euler.py}}}
+\end{figure}
+%
+
+The strength of the Discontinuous Galerkin method is its high order accuracy and
+minimal coupling to neighbouring cells. Its weakness is its behaviour near
+discontinuities. As many tests of the Euler equations seen above focus on
+results involving shocks, we will not see the DG methods at their best. For
+example, in Figures~\ref{fig:dg-euler-sod32} and \ref{fig:dg-euler-sod128} the
+Sod problem is solved. The DG method and limiting is applied directly to the
+conserved variables. At low resolution the method struggles. There are so few
+cells covering the shock and contact that it has difficulty clearly resolving
+the jumps, although the oscillations are not large enough to be problematic. As
+the resolution is increased the results are much improved, but in both cases it
+is the increased resolution through the number of grid cells $N$ rather than the
+number of modes $m$ that is important. The is in complete contrast with the
+smooth solutions seen for the advection equation, where the larger gains were
+seen by increasing $m$.
+
+\begin{figure}[t]
+\centering
+% figure generated by hydro_examples/compressible/dg_euler.py
+\includegraphics[width=0.8\linewidth]{dg_shock_entropy_m3_N128}
+\caption[Discontinuous Galerkin methods applied to the shock-entropy problem, low resolution]
+{\label{fig:dg-euler-shockentropy-m3-N128} The shock-entropy interaction problem,
+solve using Discontinuous Galerkin methods with moment limiting, with $m=3$ and
+$N=128$. We see that the method is excellent at resolving the smooth features to
+the right, which have not interacted with the shock. The finer features just
+after the shock are not well captured. Modifying $m$ has little effect here. The
+reference solution uses a WENO method with $r=4$ and $N=1024$.\\
+\hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/dg_euler.py}{dg\_euler.py}}}
+\end{figure}
+%
+
+\begin{figure}[t]
+\centering
+% figure generated by hydro_examples/compressible/dg_euler.py
+\includegraphics[width=0.8\linewidth]{dg_shock_entropy_m3_N256}
+\caption[Discontinuous Galerkin methods applied to the shock-entropy problem, high resolution]
+{\label{fig:dg-euler-shockentropy-m3-N256} The shock-entropy interaction problem,
+solve using Discontinuous Galerkin methods with moment limiting, with $m=3$ and
+$N=256$. The results are similar to Figure~\ref{fig:dg-euler-shockentropy-m3-N128}
+but there is now sufficient resolution to capture the finer features after the
+shock.\\
+\hydroexdoit{\href{https://github.com/zingale/hydro_examples/blob/master/compressible/dg_euler.py}{dg\_euler.py}}}
+\end{figure}
+%
+
+A test that better captures the advantages as well as the disadvantages of these
+methods is the shock-entropy interaction test. A right-going shock interacts
+with an entropy wave, with the final results showing both smooth features (that
+should be captured well by high order schemes) and shocks (which are more
+problematic). Results are shown in Figures~\ref{fig:dg-euler-shockentropy-m3-N128}
+and \ref{fig:dg-euler-shockentropy-m3-N256}. The smooth features are mostly
+extremely well captured even at these low resolutions, \emph{except} immediately
+after the shock. Here again varying $m$ has limited impact, and the crucial
+thing is to increase $N$ until the features are resolvable.

refs.bib

@@ -1154,3 +1154,34 @@ @techreport{Shu1997
 url = {https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19980007543.pdf},
 year = {1997}
 }
+
+@book{hesthaven2018numerical,
+  title={Numerical methods for conservation laws: From analysis to algorithms},
+  author={Hesthaven, Jan S},
+  volume={18},
+  year={2018},
+  publisher={SIAM},
+  url={https://books.google.co.uk/books?id=uyZKDwAAQBAJ&lpg=PR2&ots=udFqVc-kcT}
+}
+
+@article{burbeau2001problem,
+  title={{A problem-independent limiter for high-order Runge--Kutta discontinuous Galerkin methods}},
+  author={Burbeau, Anne and Sagaut, Pierre and Bruneau, Ch-H},
+  journal={Journal of Computational Physics},
+  volume={169},
+  number={1},
+  pages={111--150},
+  year={2001},
+  publisher={Elsevier}
+}
+
+@article{biswas1994parallel,
+  title={Parallel, adaptive finite element methods for conservation laws},
+  author={Biswas, Rupak and Devine, Karen D and Flaherty, Joseph E},
+  journal={Applied Numerical Mathematics},
+  volume={14},
+  number={1-3},
+  pages={255--283},
+  year={1994},
+  publisher={Elsevier}
+}