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

doi:10.1007/s42967-021-00175-w

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (1): 370-402.doi: 10.1007/s42967-021-00175-w

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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

R. Abgrall

Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH 8057 Zurich, Switzerland

Received:2020-11-24 Revised:2021-10-23 Online:2023-03-20 Published:2023-03-08
Contact: R. Abgrall,E-mail:remi.abgrall@math.uzh.ch E-mail:remi.abgrall@math.uzh.ch
Supported by:
This work was done while the author was partially funded by the SNF project 200020_ 175784. The support of Inria via the International Chair of the author at Inria Bordeaux-Sud Ouest is also acknowledged. Discussions with Dr. Wasilij Barsukow are acknowledged, as well as the encouragements of Anne Burbeau (CEA DAM, France). Last, I would like to thank, warmly, the two anonymous referees: their critical comments have led to big improvements.

Abstract

Abstract: 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.

Key words: Hyperbolic problems, high order, Active flux, MOOD, Residual distribution methods

CLC Number:

R. Abgrall. 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[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 370-402.

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References

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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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