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A Combination of residual distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem:Application to the 1D Euler Equations

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Communications on Applied Mathematics and Computation 2023年第1期5卷 370-402页

作者： R.Abgrall Institute of Mathematics University of ZürichWinterthurerstrasse 190CH 8057 ZurichSwitzerland

We show how to combine in a natural way(i.e.,without any test nor switch)the conservative and non-conservative formulations of an hyperbolic system that has a conservative form.This is inspired from two different classes of schemes:the residual distribution one(Abgrall in Commun Appl Math Comput 2(3):341–368,2020),and the active flux formulations(Eyman and Roe in 49th AIAA Aerospace Science Meeting,2011;Eyman in active flux.PhD thesis,University of Michigan,2013;Helzel et al.in J Sci Comput 80(3):35–61,2019;Barsukow in J Sci Comput 86(1):paper No.3,34,2021;Roe in J Sci Comput 73:1094–1114,2017).The solution is globally continuous,and as in the active flux method,described by a combination of point values and average values.Unlike the“classical”active flux methods,the meaning of the point-wise and cell average degrees of freedom is different,and hence follow different forms of PDEs;it is a conservative version of the cell average,and a possibly non-conservative one for the points.This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem.We also develop a method to perform nonlinear stability.We illustrate the behaviour on several benchmarks,some quite challenging.

关键词： Hyperbolic problems high order Active flux MOOD residual distribution methods

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A Review of residual distribution Schemes for Hyperbolic and Parabolic Problems:The July 2010 State of the Art

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Communications in Computational Physics 2012年第4期11卷 1043-1080页

作者： Remi Abgrall Team Bacchus Institut de Mathematiques de BordeauxINRIA and University of Bordeaux33405 Talence cedexFrance

We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil.We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes,show that their are really non oscillatory.We also discuss the extension to these methods to parabolic problems.We also draw some research perspectives.

关键词： Hyperbolic problems Navier Stokes equations unstructuredmeshes residual distribution methods

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