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Consistency of finite volume approximations to nonlinear hyperbolic balance laws

Authors: Matania Ben-Artzi and Jiequan Li
Journal: Math. Comp. 90 (2021), 141-169
MSC (2010): Primary 65M12; Secondary 35L65, 65M08
Published electronically: October 6, 2020
MathSciNet review: 4166456
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Abstract: This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.

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Additional Information

Matania Ben-Artzi
Affiliation: Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel
MR Author ID: 34290
ORCID: 0000-0002-6782-4085

Jiequan Li
Affiliation: Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, People’s Republic of China; Center for Applied Physics and Technology, Peking University, People’s Republic of China; and State Key Laboratory for Turbulence Research and Complex System, Peking University, People’s Republic of China

Keywords: Balance laws, conservation laws, consistency, convergence, discontinuous solutions, Lax-Wendroff theorem, finite volume schemes, high-order schemes, numerical flux, Riemann problem, generalized Riemann problem
Received by editor(s): September 23, 2019
Received by editor(s) in revised form: March 16, 2020, May 9, 2020, and May 30, 2020
Published electronically: October 6, 2020
Additional Notes: The first author thanks the Institute of Applied Physics and Computational Mathematics, Beijing, for the hospitality and support.
The second author was supported by NSFC (Nos. 11771054, 91852207), the Sino-German Research Group Project (No. GZ1465) and Foundation of LCP. \thanks{The second author is the corresponding author.}
Article copyright: © Copyright 2020 American Mathematical Society