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 • Previous Articles Next Articles

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.

TrendMD

References

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]	Matania Ben-Artzi, Jiequan Li. Hyperbolic Conservation Laws, Integral Balance Laws and Regularity of Fluxes [J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2048-2063.
[2]	Wasilij Barsukow, Jonas P. Berberich. A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying [J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2385-2430.
[3]	Yifan Chen, Thomas Y. Hou, Yixuan Wang. Exponentially Convergent Multiscale Finite Element Method [J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 862-878.
[4]	Wes Whiting, Bao Wang, Jack Xin. Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent [J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 1175-1188.
[5]	Xuechun Liu, Haijin Wang, Jue Yan, Xinghui Zhong. Superconvergence of Direct Discontinuous Galerkin Methods: Eigen-structure Analysis Based on Fourier Approach [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 257-278.
[6]	Changpin Li, Dongxia Li, Zhen Wang. L1/LDG Method for the Generalized Time-Fractional Burgers Equation in Two Spatial Dimensions [J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1299-1322.
[7]	R. Abgrall, J. Nordström, P. Öffner, S. Tokareva. Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 573-595.
[8]	Johannes Markert, Gregor Gassner, Stefanie Walch. A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 679-721.
[9]	J. C. González-Aguirre, J. A. González-Vázquez, J. Alavez-Ramírez, R. Silva, M. E. Vázquez-Cendón. Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 885-922.
[10]	Xucheng Meng, Yaguang Gu, Guanghui Hu. A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 315-342.
[11]	Liang Li, Jun Zhu, Chi-Wang Shu, Yong-Tao Zhang. A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 403-427.
[12]	Xiaoying Han, Habib N. Najm. Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1457-1493.
[13]	Xiaozhou Li. How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 759-782.
[14]	Mohammed Homod Hashim, Akil J. Harfash. Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part I: Space Convergence [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 1011-1056.
[15]	Mohammed Homod Hashim, Akil J. Harfash. Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part II: Time Convergence, Error Analysis and Numerical Results [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 1057-1104.

Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 15

Recommended Articles

Metrics

Comments