Communications on Applied Mathematics and Computation >

2022 , Vol. 4 >Issue 1: 250 - 270

DOI: https://doi.org/10.1007/s42967-020-00108-z

Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations

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  • 1 Beijing Computational Science Research Center, Beijing 100193, China;
    2 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
    3 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

Received date: 2020-08-31

  Revised date: 2020-11-10

  Online published: 2022-03-01

Supported by

Q. Tao:Research supported by the fellowship of China Postdoctoral Science Foundation, no:2020TQ0030. Y. Xu:Research supported by National Numerical Windtunnel Project NNW2019ZT4-B08, Science Challenge Project TZZT2019-A2.3, NSFC Grants 11722112, 12071455. X. Li:Research supported by NSFC Grant 11801062.

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Abstract

In this paper, we present the negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method solving nonlinear hyperbolic equations with smooth solutions. The smoothness-increasing accuracy-conserving (SIAC) flter is a postprocessing technique to enhance the accuracy of the discontinuous Galerkin (DG) solutions. This work is the essential step to extend the SIAC flter to the moving mesh for nonlinear problems. By the post-processing theory, the negative norm estimates are vital to get the superconvergence error estimates of the solutions after post-processing in the L2 norm. Although the SIAC flter has been extended to nonuniform mesh, the analysis of fltered solutions on the nonuniform mesh is complicated. We prove superconvergence error estimates in the negative norm for the ALE-DG method on moving meshes. The main diffculties of the analysis are the terms in the ALE-DG scheme brought by the grid velocity feld, and the time-dependent function space. The mapping from time-dependent cells to reference cells is very crucial in the proof. The numerical results also confrm the theoretical proof.

Key words： Arbitrary Lagrangian-Eulerian discontinuous Galerkin method; Nonlinear hyperbolic equations; Negative norm estimates; Smoothness-increasing accuracyconserving flter; Post-processing

Cite this article

Qi Tao, Yan Xu, Xiaozhou Li . Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations[J]. Communications on Applied Mathematics and Computation, 2022 , 4(1) : 250 -270 . DOI: 10.1007/s42967-020-00108-z

References

1. Bramble, J.H., Schatz, A.H.:Higher order local accuracy by averaging in the fnite element method. Math. Comput. 31(137), 94-111 (1997)
2. Brenner, S.:The Mathematical Theory of Finite Element Methods. Springer, New York (2002)
3. Ciarlet, P.G.:The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia (2002)
4. Cockburn, B., Karniadakis, G.E., Shu, C.-W.:The development of discontinuous Galerkin methods. In:Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods:Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, Part I:Overview, vol. 11, pp. 3-50. Springer, Berlin (2000)
5. Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.:Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39(1), 264-285 (2001)
6. Cockburn, B., Shu, C.-W.:The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. Journal of Computational Physics 141(2), 199-224 (1998)
7. Cockburn, B., Luskin, M., Shu, C.-W., Süli, E.:Enhanced accuracy by post-processing for fnite element methods for hyperbolic equations. Math. Comput. 72(242), 577-606 (2003)
8. Farhat, C., Geuzaine, P., Grandmont, C.:The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of fow problems on moving grids. J. Comput. Phys. 174(2), 669-694 (2001)
9. Fu, P., Gero, S., Xia, Y.:Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes. Math. Comput. 88(319), 2221-2255 (2019)
10. Ji, L., Xu, Y., Ryan, J.K.:Accuracy-enhancement of discontinuous Galerkin solutions for convectiondifusion equations in multiple-dimensions. Math. Comput. 81(280), 1929-1950 (2012)
11. Ji, L., Xu, Y., Ryan, J.K.:Negative-order norm estimates for nonlinear hyperbolic conservation laws. J. Sci. Comput. 54(2/3), 531-548 (2013)
12. Hong, X., Xia, Y.:Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for hyperbolic equations involving δ-singularities. SIAM J. Num. Anal. 58(1), 125-152 (2020)
13. Klingenberg, C., Schnücke, G., Xia, Y.:Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws:analysis and application in one dimension. Math. Comput. 86(305), 1203-1232 (2017)
14. Klingenberg, C., Schnücke, G., Xia, Y.:An Arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for Hamilton-Jacobi equations. J. Sci. Comput. 73(2/3), 906-942 (2017)
15. Li, X., Ryan, J.K., Kirby, R.M., Vuik, C.:Smoothness-increasing accuracy-conserving (SIAC) flters for derivative approximations of discontinuous Galerkin (DG) solutions over nonuniform meshes and near boundaries. J. Comput. Appl. Math. 294, 275-296 (2016)
16. Li, X., Ryan, J.K., Kirby, R.M., Vuil, C.:Smoothness-increasing accuracy-conserving (SIAC) fltering for discontinuous Galerkin solutions over nonuniform meshes:superconvergence and optimal accuracy. J. Sci. Comput. 81(3), 1150-1180 (2019)
17. Meng, X., Ryan, J.K.:Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws:divided diference estimates and accuracy enhancement. Numerische Mathematik 136(1), 27-73 (2017)
18. Mirzaee, H., Ji, L., Ryan, J.K., Kirby, R.M.:Smoothness-increasing accuracy-conserving (SIAC) postprocessing for discontinuous Galerkin solutions over structured triangular meshes. SIAM J. Num. Anal. 49(5), 1899-1920 (2011)
19. Mirzaee, H., King, J., Ryan, J.K., Kirby, R.M.:Smoothness-increasing accuracy-conserving flters for discontinuous Galerkin solutions over unstructured triangular meshes. SIAM J. Num. Anal. 35(1), A212-A230 (2013)
20. Reed, W., Hill, T.:Triangular Mesh Nethods for the Neutron Transport Equation. Los Alamos Scientifc Laboratory, New Mexico (1973)
21. Ryan, J.K., Shu, C.-W.:On a one-sided post-processing technique for the discontinuous Galerkin methods. Methods and Applications of Analysis 10(2), 295-308 (2003)
22. Ryan, J.K., Shu, C.-W., Atkins, H.:Extension of a post processing technique for the discontinuous Galerkin method for hyperbolic equations with application to an aeroacoustic problem. SIAM J. Sci. Comput. 26(3), 821-843 (2005)
23. Stefen, M., Kirby, S., Kirby, R.M., Ryan, J.K.:Investigation of smoothness-increasing accuracyconserving flters for improving streamline integration through discontinuous felds. IEEE TVCG 14, 680-692 (2008)
24. Tao, Q., Xia, Y.:Error estimates and post-processing of local discontinuous Galerkin method for Schrödinger equations. J. Comput. Appl. Math. 356, 198-218 (2019)
25. Thomée, V.:High order local approximations to derivatives in the fnite element method. Math. Comput. 31(139), 652-660 (1997)
26. van Slingerland, P., Ryan, J.K., Vuik, C.:Position-dependent smoothness-increasing accuracy-conserving (SIAC) fltering for improving discontinuous Galerkin solutions. SIAM J. Sci. Comput. 33(2), 802-825 (2011)
27. Zhang, Q., Shu, C.-W.:Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Num. Anal. 42(2), 641-666 (2004)
28. Zhang, Q., Shu, C.-W.:Stability analysis and a priori error estimates of the third order explicit RungeKutta discontinuous Galerkin method for scalar conservation law. SIAM J. Num. Anal. 48(3), 1038- 1063 (2010)
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