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

收稿日期:2023-07-03 修回日期:2023-09-18 接受日期:2023-10-07 发布日期:2024-12-20
通讯作者: 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
基金资助:
The work of M.L.and Y.Y.was partially funded by the Gutenberg Research College and by Chinesisch-Deutschen Zentrum für Wissenschaftsförderung —Sino-German Project No.GZ1465.M.L.is grateful to the Mainz Institute of Multiscale Modelling and SPP 2410 Hyperbolic Balance Laws in Fluid Mechanics:Complexity,Scales,Randomness (CoScaRa) for supporting her research.

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.

摘要/Abstract

摘要： 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.

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

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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Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

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