On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes

Michael Dumbser; Claus-dieter Munz

International Journal of Applied Mathematics and Computer Science (2007)

  • Volume: 17, Issue: 3, page 297-310
  • ISSN: 1641-876X

Abstract

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This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.

How to cite

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Dumbser, Michael, and Munz, Claus-dieter. "On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes." International Journal of Applied Mathematics and Computer Science 17.3 (2007): 297-310. <http://eudml.org/doc/207837>.

@article{Dumbser2007,
abstract = {This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.},
author = {Dumbser, Michael, Munz, Claus-dieter},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {ADER approach; boundary conditions; unstructured meshes; source terms; discontinuous Galerkin schemes; nonlinear hyperbolic systems},
language = {eng},
number = {3},
pages = {297-310},
title = {On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes},
url = {http://eudml.org/doc/207837},
volume = {17},
year = {2007},
}

TY - JOUR
AU - Dumbser, Michael
AU - Munz, Claus-dieter
TI - On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 3
SP - 297
EP - 310
AB - This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.
LA - eng
KW - ADER approach; boundary conditions; unstructured meshes; source terms; discontinuous Galerkin schemes; nonlinear hyperbolic systems
UR - http://eudml.org/doc/207837
ER -

References

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