Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Method for 1D Conservation Laws

doi:10.1007/s42967-024-00391-0

Abstract

Abstract: In this paper, we propose a new class of discontinuous Galerkin (DG) methods for solving 1D conservation laws on unfitted meshes. The standard DG method is used in the interior cells. For the small cut elements around the boundaries, we directly design approximation polynomials based on inverse Lax-Wendroff (ILW) principles for the inflow boundary conditions and introduce the post-processing to preserve the local conservation properties of the DG method. The theoretical analysis shows that our proposed methods have the same stability and numerical accuracy as the standard DG method in the inner region. An additional nonlinear limiter is designed to prevent spurious oscillations if a shock is near the boundary. Numerical results indicate that our methods achieve optimal numerical accuracy for smooth problems and do not introduce additional oscillations in discontinuous problems.

Key words: Discontinuous Galerkin (DG) method, Hyperbolic conservation laws, Numerical boundary conditions, Inverse Lax-Wendroff (ILW) method, High-order accuracy, Stability analysis

CLC Number:

65M60
65M12

Lei Yang, Shun Li, Yan Jiang, Chi-Wang Shu, Mengping Zhang. Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Method for 1D Conservation Laws[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 796-826.

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References

1. Bastian, P., Engwer, C., Fahlke, J., Ippisch, O.: An unfitted discontinuous Galerkin method for pore-scale simulations of solute transport. Math. Comput. Simul. 81(10), 2051–2061 (2011)
2. Burman, Erik, Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62(4), 328–341 (2012)
3. Carpenter, M.H., Gottlieb, D., Abarbanel, S., Don, W.-S.: The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error. SIAM J. Sci. Comput. 16(6), 1241–1252 (1995)
4. Cheng, Z., Liu, S., Jiang, Y., Lu, J., Zhang, M., Zhang, S.: A high order boundary scheme to simulate complex moving rigid body under impingement of shock wave. Appl. Math. Mech. 42(6), 841–854 (2021)
5. Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990)
6. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)
7. Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)
8. Cockburn, B., Shu, C.-W.: The Runge-Kutta local projection-discontinuous-Galerkin finite element method for scalar conservation laws. ESAIM Math. Model. Numer. Anal. 25(3), 337–361 (1991)
9. Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)
10. Ding, S., Shu, C.-W., Zhang, M.: On the conservation of finite difference WENO schemes in nonrectangular domains using the inverse Lax-Wendroff boundary treatments. J. Comput. Phys. 415, 109516 (2020)
11. Fu, P., Frachon, T., Kreiss, G., Zahedi, S.: High order discontinuous cut finite element methods for linear hyperbolic conservation laws with an interface. J. Sci. Comput. 90(3), 84 (2022)
12. Fu, P., Kreiss, G.: High order cut discontinuous Galerkin methods for hyperbolic conservation laws in one space dimension. SIAM J. Sci. Comput. 43(4), A2404–A2424 (2021)
13. Giuliani, A.: A two-dimensional stabilized discontinuous Galerkin method on curvilinear embedded boundary grids. SIAM J. Sci. Comput. 44(1), A389–A415 (2022)
14. Gurkan, C., Sticko, S., Massing, A.: Stabilized cut discontinuous Galerkin methods for advection-reaction problems. SIAM J. Sci. Comput. 42(5), A2620–A2654 (2020)
15. Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014)
16. Li, T., Lu, J., Shu, C.-W.: Stability analysis of inverse Lax-Wendroff boundary treatment of high order compact difference schemes for parabolic equations. J. Comput. Appl. Math. 400, 113711 (2022)
17. Li, T., Lu, J., Wang, P.: Stability analysis of inverse Lax-Wendroff procedure for a high order compact finite difference schemes. Commun. Appl. Math. Comput. 6(1), 142–189 (2024)
18. Li, T., Shu, C.-W., Zhang, M.: Stability analysis of the inverse Lax-Wendroff boundary treatment for high order upwind-biased finite difference schemes. J. Comput. Appl. Math. 299, 140–158 (2016)
19. Li, T., Shu, C.-W., Zhang, M.: Stability analysis of the inverse Lax-Wendroff boundary treatment for high order central difference schemes for diffusion equations. J. Sci. Comput. 70, 576–607 (2017)
20. Liu, S., Cheng, Z., Jiang, Y., Lu, J., Zhang, M., Zhang, S.: Numerical simulation of a complex moving rigid body under the impingement of a shock wave in 3D. Adv. Aerodyn. 4(1), 1–29 (2022)
21. Liu, S., Jiang, Y., Shu, C.-W., Zhang, M., Zhang, S.: A high order moving boundary treatment for convection-diffusion equations. J. Comput. Phys. 473, 111752 (2023)
22. Lu, J., Fang, J., Tan, S., Shu, C.-W., Zhang, M.: Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations. J. Comput. Phys. 317, 276–300 (2016)
23. Lu, J., Shu, C.-W., Tan, S., Zhang, M.: An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary. J. Comput. Phys. 426, 109940 (2021)
24. Müller, B., Krämer-Eis, S., Kummer, F., Oberlack, M.: A high-order discontinuous Galerkin method for compressible flows with immersed boundaries. Int. J. Numer. Methods Eng. 110(1), 3–30 (2017)
25. Qin, R., Krivodonova, L.: A discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. J. Comput. Sci. 4(1/2), 24–35 (2013)
26. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical report, Los Alamos Scientific Lab., N. Mex. (USA) (1973)
27. Schoeder, S., Sticko, S., Kreiss, G., Kronbichler, M.: High-order cut discontinuous Galerkin methods with local time stepping for acoustics. Int. J. Numer. Methods Eng. 121(13), 2979–3003 (2020)
28. Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. In: Bertoluzza, S., Falletta, S., Russo, G., Shu, C.-W. (eds.) Numerical Solutions of Partial Differential Equations. Advanced Courses in Mathematics-CRM Barcelona, pp. 149–201. Birkhäuser, Basel (2009)
29. Song, T., Main, A., Scovazzi, G., Ricchiuto, M.: The shifted boundary method for hyperbolic systems: embedded domain computations of linear waves and shallow water flows. J. Comput. Phys. 369, 45–79 (2018)
30. Sticko, S., Kreiss, G.: A stabilized Nitsche cut element method for the wave equation. Comput. Methods Appl. Mech. Eng. 309, 364–387 (2016)
31. Sticko, S., Kreiss, G.: Higher order cut finite elements for the wave equation. J. Sci. Comput. 80, 1867– 1887 (2019)
32. Tan, S., Shu, C.-W.: Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229(21), 8144–8166 (2010)
33. Tan, S., Shu, C.-W.: A high order moving boundary treatment for compressible inviscid flows. J. Comput. Phys. 230(15), 6023–6036 (2011)
34. Tan, S., Shu, C.-W.: Inverse Lax-Wendroff procedure for numerical boundary conditions of hyperbolic equations: survey and new developments. In: Melnik, R., Kotsireas, I. (eds.) Advances in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol. 66, pp. 41–63. Springer, New York (2013)
35. Vilar, F., Shu, C.-W.: Development and stability analysis of the inverse Lax-Wendroff boundary treatment for central compact schemes. ESAIM Math. Model. Numer. Anal. 49(1), c115 (2014)
36. Zhao, W., Huang, J., Ruuth, S.J.: Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws. J. Comput. Phys. 421, 109697 (2020)

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