A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

doi:10.1007/s42967-021-00179-6

Abstract

Abstract: Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show highorder accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

Key words: Fixed-point fast sweeping methods, Multi-resolution WENO schemes, Steady state, ILW procedure, Convergence

CLC Number:

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.

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References

1. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)
2. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)
3. Chen, S.: Fixed-point fast sweeping WENO methods for steady state solution of scalar hyperbolic conservation laws. Int. J. Numer. Anal. Mod. 11(1), 117–130 (2014)
4. Fomel, S., Luo, S., Zhao, H.: Fast sweeping method for the factored Eikonal equation. J. Comput. Phys. 228(17), 6440–6455 (2009)
5. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)
6. Huang, L., Shu, C.-W., Zhang, M.: Numerical boundary conditions for the fast sweeping high order WENO methods for solving the Eikonal equation. J. Comput. Math. 26(3), 336–346 (2008)
7. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)
8. Krivodonova, L., Berger, M.: High-order accurate implementation of solid wall boundary conditions in curved geometries. J. Comput. Phys. 211(2), 492–512 (2006)
9. Li, F., Shu, C.-W., Zhang, Y.-T., Zhao, H.-K.: A second order discontinuous Galerkin fast sweeping method for Eikonal equations. J. Comput. Phys. 227(17), 8191–8208 (2008)
10. Li, L., Zhu, J., Zhang, Y.-T.: Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws. J. Comput. Phys. 443, 110516 (2021)
11. 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)
12. Luo, H., Baum, J.D., Löhner, R.: A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys. 225(1), 686–713 (2007)
13. Qian, J., Zhang, Y.-T., Zhao, H.-K.: Fast sweeping methods for Eikonal equations on triangular meshes. SIAM J. Numer. Anal. 45(1), 83–107 (2007)
14. Qian, J., Zhang, Y.-T., Zhao, H.-K.: A fast sweeping method for static convex Hamilton-Jacobi equations. J. Sci. Comput. 31(1), 237–271 (2007)
15. Shi, J., Zhang, Y.-T., Shu, C.-W.: Resolution of high order WENO schemes for complicated flow structures. J. Comput. Phys. 186(2), 690–696 (2003)
16. Shida, Y., Kuwahara, K., Ono, K., Takami, H.: Computation of dynamic stall of a NACA-0012 airfoil. AIAA J. 25(3), 408–413 (1987)
17. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E. (eds) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998)
18. Shu, C.-W.: High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)
19. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)
20. Sjögreen, B., Petersson, N.A.: A Cartesian embedded boundary method for hyperbolic conservation laws. Commun. Comput. Phys. 2(6), 1199–1219 (2007)
21. Tan, S., Shu, C.-W.: Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws. J. Comput. Phys. 229(21), 8144–8166 (2010)
22. Tan, S., Wang, C., Shu, C.-W., Ning, J.: Efficient implementation of high order inverse Lax-Wendroff boundary treatment for conservation laws. J. Comput. Phys. 231(6), 2510–2527 (2012)
23. Wu, L., Zhang, Y.-T.: A third order fast sweeping method with linear computational complexity for Eikonal equations. J. Sci. Comput. 62(1), 198–229 (2015)
24. Wu, L., Zhang, Y.-T., Zhang, S., Shu, C.-W.: High order fixed-point sweeping WENO methods for steady state of hyperbolic conservation laws and its convergence study. Commun. Comput. Phys. 20(4), 835–869 (2016)
25. Xiong, T., Zhang, M., Zhang, Y.-T., Shu, C.-W.: Fast sweeping fifth order WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment. J. Sci. Comput. 45(1), 514–536 (2010)
26. Zhang, S., Jiang, S., Shu, C.-W.: Improvement of convergence to steady state solutions of Euler equations with the WENO schemes. J. Sci. Comput. 47(2), 216–238 (2011)
27. Zhang, S., Shu, C.-W.: A new smoothness indicator for the WENO schemes and its effect on the convergence to steady state solutions. J. Sci. Comput. 31(1), 273–305 (2007)
28. Zhang, Y.-T., Chen, S., Li, F., Zhao, H., Shu, C.-W.: Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations. SIAM J. Sci. Comput. 33(4), 1873–1896 (2011)
29. Zhang, Y.-T., Zhao, H.-K., Chen, S.: Fixed-point iterative sweeping methods for static Hamilton-Jacobi equations. Meth. Appl. Anal. 13(3), 299–320 (2006)
30. Zhang, Y.-T., Zhao, H.-K., Qian, J.: High order fast sweeping methods for static Hamilton-Jacobi equations. J. Sci. Comput. 29(1), 25–56 (2006)
31. Zhao, H.-K.: A fast sweeping method for Eikonal equations. Math. Comput. 74(250), 603–627 (2005)
32. Zhu, J., Qiu, J.: A new type of finite volume WENO schemes for hyperbolic conservation laws. J. Sci. Comput. 73(5), 1338–1359 (2017)
33. Zhu, J., Shu, C.-W.: A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J. Comput. Phys. 375(3), 659–683 (2018)
34. Zhu, J., Shu, C.-W.: Numerical study on the convergence to steady-state solutions of a new class of finite volume WENO schemes: triangular meshes. Shock Wav. 29(1), 3–25 (2019)
35. Zhu, J., Shu, C.-W.: Convergence to steady-state solutions of the new type of high-order multi-resolution WENO schemes: a numerical study. Commun. Appl. Math. Comput. 2(6), 429–460 (2020)

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