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

doi:10.1007/s42967-020-00108-z

Abstract

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

CLC Number:

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.

TrendMD

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)

[1]	Hongjuan Zhang, Boying Wu, Xiong Meng. A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 84-107.
[2]	Haijin Wang, Qiang Zhang. The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 271-292.
[3]	Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 319-352.
[4]	Jie Du, Eric Chung, Yang Yang. Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 353-379.
[5]	Junhong Tian, Hengfei Ding. A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 571-584.
[6]	Zheng Sun, Chi-Wang Shu. Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 671-700.
[7]	Giacomo Albi, Lorenzo Pareschi. High Order Semi-implicit Multistep Methods for Time-Dependent Partial Diferential Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 701-718.
[8]	Rathan Samala, Biswarup Biswas. Arc Length-Based WENO Scheme for Hamilton-Jacobi Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 481-496.
[9]	Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven. A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations [J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 689-709.
[10]	Junying Cao, Ziqiang Wang, Chuanju Xu. A High-Order Scheme for Fractional Ordinary Diferential Equations with the Caputo-Fabrizio Derivative [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 179-199.
[11]	Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong. An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients [J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 215-239.
[12]	Yuxin Zhang, Hengfei Ding. Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 57-72.
[13]	Can Li, Shuming Liu. Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 73-91.
[14]	Yanyong Wang, Yubin Yan, Ye Hu. Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 505-523.
[15]	Kamran Kazmi, Abdul Khaliq. A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 525-544.