Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

doi:10.1007/s42967-023-00338-x

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (4): 2215-2238.doi: 10.1007/s42967-023-00338-x

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Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

Mária Lukáčová-Medvid'ová¹, Yuhuan Yuan²

1 Institute of Mathematics, Johannes Gutenberg-University Mainz, Staudingerweg 9, 55 128 Mainz, Germany;
2 School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, Jiangsu, China

Received:2023-07-03 Revised:2023-09-18 Accepted:2023-10-07 Published:2024-12-20
Contact: Yuhuan Yuan,E-mail:yuhuanyuan@nuaa.edu.cn;Mária Lukáčová-Medvid’ová,E-mail:lukacova@uni-mainz.de E-mail:yuhuanyuan@nuaa.edu.cn;lukacova@uni-mainz.de
Supported by:
DSB acknowledges support via NSF grants NSF-19-04774, NSF-AST-2009776, NASA-2020-1241, and NASA grant 80NSSC22K0628. DSB and HK acknowledge support from a Vajra award, VJR/2018/00129 and also a travel grant from Notre Dame International. CWS acknowledges support via AFOSR grant FA9550-20-1-0055 and NSF grant DMS-2010107.

Abstract

Abstract: In this paper, we study the convergence of a second-order finite volume approximation of the scalar conservation law. This scheme is based on the generalized Riemann problem (GRP) solver. We first investigate the stability of the GRP scheme and find that it might be entropy-unstable when the shock wave is generated. By adding an artificial viscosity, we propose a new stabilized GRP scheme. Under the assumption that numerical solutions are uniformly bounded, we prove the consistency and convergence of this new GRP method.

Key words: Scalar conservation law, Finite volume method, Generalized Riemann problem(GRP)solver, Entropy stability, Consistency, Convergence

Mária Lukáčová-Medvid'ová, Yuhuan Yuan. Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation[J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2215-2238.

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URL: https://www.camc.shu.edu.cn/EN/10.1007/s42967-023-00338-x

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Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

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