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

• ORIGINAL PAPERS • 上一篇下一篇

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.

1. Ball, J.M.: A version of the fundamental theorem for young measures. In: PDEs and Continuum Models of Phase Transitions, pp. 207-215. Springer, Berlin (1989)
2. Ben-Artzi, M., Falcovitz, J., Li, J.: The convergence of the GRP scheme. Discrete Contin. Dyn. Syst. 23, 1-27 (2009)
3. Ben-Artzi, M., Li, J.: Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem. Numer. Math. 106(3), 369-425 (2007)
4. Ben-Artzi, M., Li, J.: Consistency of finite volume approximations to nonlinear hyperbolic balance laws. Math. Comput. 90(327), 141-169 (2021)
5. Ben-Artzi, M., Li, J., Warnecke, G.: A direct Eulerian GRP scheme for compressible fluid flows. J. Comput. Phys. 218(1), 19-43 (2006)
6. Bressan, A., Crasta, G., Piccoli, B.: Well-Posedness of the Cauchy Problem for n×n Systems of Conservation Laws. Memoirs of the American Mathematical Society, London (2000)
7. Bressan, A., Lewicka, M.: A uniqueness condition for hyperbolic systems of conservation laws. Discrete Contin. Dyn. Syst. 6(3), 673-682 (2000)
8. Cancès, C., Mathis, H., Seguin, N.: Error estimate for time-explicit finite volume approximation of strong solutions to systems of conservation laws. SIAM J. Numer. Anal. 54(2), 1263-1287 (2016)
9. Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Math. 68(7), 1157-1190 (2015)
10. Chiodaroli, E., Kreml, O., Mácha, V., Schwarzacher, S.: Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data. Trans. Am. Math. Soc. 374(4), 2269- 2295 (2021)
11. De Lellis, C., Székelyhidi, L., Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225-260 (2010)
12. DiPerna, R.J.: Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88(3), 223- 270 (1985)
13. Feireisl, E., Klingenberg, C., Kreml, O., Markfelder, S.: On oscillatory solutions to the complete Euler system. J. Differential Equations 269(2), 1521-1543 (2020)
14. Feireisl, E., Lukáčová-Medvid’ová, M., Mizerová, H.: Convergence of finite volume schemes for the Euler equations via dissipative measure-valued solutions. Found. Comput. Math. 20(4), 923-966 (2020)
15. Feireisl, E., Lukáčová-Medvid’ová, M., Mizerová, H.: A finite volume scheme for the Euler system inspired by the two velocities approach. Numer. Math. 144(1), 89-132 (2020)
16. Feireisl, E., Lukáčová-Medvid’ová, M., Mizerová, H., She, B.: Numerical Analysis of Compressible Flows. MS & A Series, vol. 20. Springer, Lodnon (2021)
17. Fjordholm, U.K., Käppeli, R., Mishra, S., Tadmor, E.: Construction of approximate entropy measurevalued solutions for hyperbolic systems of conservation laws. Found. Comput. Math. 17(3), 763-827 (2017)
18. Fjordholm, U.S., Mishra, S., Tadmor, E.: ENO reconstruction and ENO interpolation are stable. Found. Comput. Math. 13, 139-159 (2013)
19. Fjordholm, U.S., Mishra, S., Tadmor, E.: On the computation of measure-valued solutions. Acta Numer. 25, 567-679 (2016)
20. Jovanović, V., Rohde, Ch.: Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws. SIAM J. Numer. Anal. 43(6), 2423-2449 (2006)
21. Kröner, D., Noelle, S., Rokyta, M.: Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math. 71, 527- 560 (1995)
22. Kröner, D., Rokyta, M.: Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal. 31(2), 324-343 (1994)
23. Kružkov, S.: First order quasilinear equations in several independent variables. USSR Math. Sbornik 10(2), 217-243 (1970)
24. Li, J., Du, Z.F.: A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers I. Hyperbolic conservation laws. SIAM J. Sci. Comput. 38(5), A3046-A3069 (2016)
25. Lukáčová-Medvid’ová, M., Yuan, Y.: Convergence of first-order finite volume method based on exact Riemann solver for the complete compressible Euler equations. Numer. Methods Partial Differential Equations 39(5), 3777-3810 (2023)
26. Lukáčová-Medvid’ová, M., She, B., Yuan, Y.: Error estimates of the Godunov method for the multidimensional compressible Euler system. J. Sci. Comput. 91(3), Paper No. 71 (2022)

[1]	Fei Li, Sha Li, Liuqiang Zhong, Wanwan Zhu. Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 831-850.
[2]	Hongying Man, Shangyou Zhang. On the Superconvergence of a Conforming Mixed Finite Element for Linear Elasticity on Uniform n-Square Grids[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 851-867.
[3]	Reetika Chawla, Devendra Kumar, Haitao Qi. A Numerical Technique to Solve Time-Fractional Delay Diffusion Wave Equation via Trigonometric Collocation Approach[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 909-929.
[4]	B. Yu. Irgashev. Robin Type Problem for a Degenerate Equation of Even Order with Caputo Fractional Derivative[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 930-952.
[5]	Zhengge Huang. New Gradient-Based Iterative-Like Algorithms for Solving a Class of Sylvester Tensor Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1005-1049.
[6]	Bei-Bei Li, Jing-Jing Cui, Zheng-Ge Huang, Xiao-Feng Xie. Two Quasi-combining Real and Imaginary Parts Iteration Methods for Solving Complex Symmetric System of Linear Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 427-455.
[7]	Lisha Chen, Zhibo Wang. A Second-Order Scheme with Nonuniform Time Grids for the Two-Dimensional Time-Fractional Zakharov-Kuznetsov Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 456-471.
[8]	Yongjun Chen, Liping Zhang. Linear Convergence of the Collatz Method for Computing the Perron Eigenpair of a Primitive Dual Number Matrix[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 232-247.
[9]	Chunxiu Liu, Junying Cao, Tong Lyu, Xingyang Ye. Numerical Analysis of a High-Order Scheme with Nonuniform Time Grids for Caputo-Hadamard Fractional Reaction Sub-diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 338-365.
[10]	Qin-Qin Shen, Geng-Chen Yang, Chen-Can Zhou. Convergence Analysis of the Projected SOR Iteration Method for Horizontal Linear Complementarity Problems[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1617-1638.
[11]	Xu Li, Jian-Sheng Feng. Parameterized QHSS Iteration Method and Its Variants for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1665-1683.
[12]	Jin-Feng Mao, Fang Chen. On Multi-step Partially Randomized Extended Kaczmarz Method for Solving Large Sparse Inconsistent Linear Systems[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1724-1743.
[13]	Lu-Xin Wang, Yang Cao, Qin-Qin Shen. Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1769-1790.
[14]	Yan-Xia Dai, Ren-Yi Yan, Ai-Li Yang. Minimum Residual BAS Iteration Method for Solving the System of Absolute Value Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1815-1825.
[15]	Rong Ma, Yu-Jiang Wu, Lun-Ji Song. Modified Newton-PAGSOR Method for Solving Nonlinear Systems with Complex Symmetric Jacobian Matrices[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 1880-1906.

Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

PDF

可视化

摘要/Abstract

引用本文

使用本文

相关文章 15

编辑推荐

Metrics

本文评价