In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension

Pimentel-García, Ernesto; Castro, Manuel J.; Chalons, Christophe; Luna, Tomás Morales de; Parés, Carlos

doi:10.1016/j.jcp.2022.111152

Cited by 17 publications

(2 citation statements)

References 26 publications

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“…Instead, one can introduce Borel measure solutions [19,30,31]. This concept of weak solutions was utilized to develop path-conservative finite-volume (FV) methods; see, e.g., [1,[6][7][8][9]35,36,38] and reference therein. Third, a novel scheme should be well-balanced (WB) in the sense that it should be capable of exactly preserving some of the physically relevant steady-state solutions satisfying…”

Section: Introductionmentioning

confidence: 99%

Local characteristic decomposition based central-upwind scheme

Chertock

Chu²,

Herty³

et al. 2023

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 99%

Local characteristic decomposition based central-upwind scheme

Chertock

Chu²,

Herty³

et al. 2023

Journal of Computational Physics

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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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High‐order in‐cell discontinuous reconstruction path‐conservative methods for nonconservative hyperbolic systems–DR.MOOD method

Pimentel‐García,

Castro,

Chalons

et al. 2024

Numerical Methods Partial

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In this work, we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in‐cell discontinuous reconstruction operator are the key points to develop a new family of high‐order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two‐Layer Shallow Water system.

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In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension

Cited by 17 publications

References 26 publications

Local characteristic decomposition based central-upwind scheme

Local characteristic decomposition based central-upwind scheme

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

High‐order in‐cell discontinuous reconstruction path‐conservative methods for nonconservative hyperbolic systems–DR.MOOD method

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