Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

Fernández, Ernesto Guerrero; Escalante, Cipriano; Díáz, Manuel

doi:10.3390/math10010015

Cited by 7 publications

(2 citation statements)

References 94 publications

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“…Note that the Legendre polynomials basis has largely been used in similar contexts, although one can find other different choices (see [52,40,22,23,44,53,43,41,12,7,33,42]). It is worth mentioning that, while the regular integrals are numerically approximated in other works (see for instance [24,37,26]), here we are computing those integrals exactly.…”

Section: A General Vertical Decomposition Of Euler Equationsmentioning

confidence: 99%

A general vertical decomposition of Euler equations: Multilayer-moment models

Garres-Díaz

Escalante

Luna

et al. 2023

Applied Numerical Mathematics

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Section: A General Vertical Decomposition Of Euler Equationsmentioning

confidence: 99%

A general vertical decomposition of Euler equations: Multilayer-moment models

Garres-Díaz

Escalante

Luna

et al. 2023

Applied Numerical Mathematics

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“…Well-balanced methods were originally introduced for the shallow water equations in [14,120,91,28,6,37,142,38,132,138,139] and then they have been used in many other context: as an illustration, we cite [36,5,39,145,1] and for the extension to discontinuous Galerkin schemes we mention [96,129,171,75]. In particular, nowadays well-balancing is of interest in the framework of astrophysical applications.…”

Section: Introductionmentioning

confidence: 99%

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

Gaburro

Boscheri

Chiocchetti

et al. 2020

Journal of Computational Physics

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We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPOtype approach [1], which rapidly rebuilds a new high quality mesh exploiting the previous one, but rearranging the element shapes and neighbors in order to guarantee that the mesh evolution is robust even for vortex flows and for very long computational times. The old and new Voronoi elements associated to the same generator point are connected in space-time to construct closed space-time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also need to incorporate some degenerate space-time sliver elements, which are needed in order to fill the space-time holes that arise because of the topology changes in the mesh between time t n and time t n+1 . The final ALE FV-DG scheme is obtained by a novel redesign of the high order accurate fully discrete direct ALE schemes of Boscheri and Dumbser [2,3], which have been extended here to general moving Voronoi meshes and space-time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed spacetime control volumes combined with a fully-discrete space-time conservation formulation of the governing hyperbolic PDE system. In this way the discrete solution is conservative and satisfies the geometric conservation law (GCL) by construction. Numerical convergence studies as well as a large set of benchmark problems for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes that allow topology changes in each time step lead to substantial improvements over the existing state of the art in direct ALE methods.Keywords: Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes, arbitrary high order in space and time, moving Voronoi tessellations with topology change, a posteriori sub-cell finite volume limiter, fully-discrete one-step ADER approach for hyperbolic PDE, compressible Euler and MHD equations directly integrated by means of a high order fully discrete one-step ADER method. To the best knowledge of the authors, this is the first time that arbitrary high order accurate direct ALE FV and DG schemes are developed with an embedded mesh generator that builds a new mesh with a different topology at each time step. State of the artLagrangian algorithms [4,5,6,7,8,9,10,11,12] are characterized by a moving computational mesh displaced with a velocity chosen as close as possible to the local fluid velocity. In the Lagrangian description of the fluid, the ...

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Nonstaggered Central Scheme Under Steady-State Discretization for Solving the Ripa Model

Li,

2024

J Sci Comput

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Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

Cited by 7 publications

References 94 publications

A general vertical decomposition of Euler equations: Multilayer-moment models

A general vertical decomposition of Euler equations: Multilayer-moment models

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

Nonstaggered Central Scheme Under Steady-State Discretization for Solving the Ripa Model

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