High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

doi:10.1007/s42967-021-00183-w

Abstract

Abstract: A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory (WENO) scheme. The exact C-property is investigated, and comparison with the standard finite difference WENO scheme is made. Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems. The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems, indicating smaller errors compared with the Lax-Friedrichs solver. In addition, we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.

Key words: Shallow water equation, Well-balanced, High order accuracy, WENO scheme, Curvilinear meshes, Positivity-preserving limiter

CLC Number:

65M06

Zepeng Liu, Yan Jiang, Mengping Zhang, Qingyuan Liu. High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes[J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 485-528.

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URL: https://www.camc.shu.edu.cn/EN/10.1007/s42967-021-00183-w

https://www.camc.shu.edu.cn/EN/Y2023/V5/I1/485

1. Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000)
2. Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluid 23(8), 1049–1071 (1994)
3. Bryson, S., Epshteyn, Y., Kurganov, A., Petrova, G.: Well-balanced positivity preserving centralupwind scheme on triangular grids for the Saint-Venant system. ESAIM: Math. Modell. Numer. Anal. 45(3), 423–446 (2011)
4. Cai, X., Ladeinde, F.: Performance of WENO scheme in generalized curvilinear coordinate systems. In: 46th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2008-36, Reno, Nevada (2008)
5. Christlieb, A.J., Feng, X., Jiang, Y., Tang, Q.: A high-order finite difference WENO scheme for ideal magnetohydrodynamics on curvilinear meshes. SIAM J. Sci. Comput. 40(4), A2631–A2666 (2018)
6. Gao, Z., Hu, G.: High order well-balanced weighted compact nonlinear schemes for shallow water equations. Commun. Comput. Phys. 22(4), 1049–1068 (2017)
7. Hubbard, M.: On the accuracy of one-dimensional models of steady converging/diverging open channel flows. Int. J. Numer. Meth. Fluids 35(7), 785–808 (2001)
8. Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)
9. Jiang, Y., Shu, C.-W., Zhang, M.: An alternative formulation of finite difference weighted ENO schemes with Lax-Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35(2), A1137–A1160 (2013)
10. Jiang, Y., Shu, C.-W., Zhang, M.: Free-stream preserving finite difference schemes on curvilinear meshes. Methods Appl. Anal. 21(1), 1–30 (2014)
11. LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146(1), 346–365 (1998)
12. Li, P., Don, W.S., Gao, Z.: High order well-balanced finite difference WENO interpolation-based schemes for shallow water equations. Comput. Fluid 201, 104476 (2020)
13. Liu, H., Qiu, J.: Finite difference Hermite WENO schemes for conservation laws, II: an alternative approach. J. Sci. Comput. 66(2), 598–624 (2016)
14. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)
15. Navas-Montilla, A., Murillo, J.: Improved Riemann solvers for an accurate resolution of 1D and 2D shock profiles with application to hydraulic jumps. J. Comput. Phys. 378, 445–476 (2019)
16. Nonomura, T., Iizuka, N., Fujii, K.: Freestream and vortex preservation properties of high-order WENO and WCNs on curvilinear grids. Comput. Fluid 39(2), 197–214 (2010)
17. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)
18. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)
19. Thomas, P.D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17(10), 1030–1037 (1979)
20. Visbal, M.R., Gaitonde, D.V.: On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys. 181(1), 155–185 (2002)
21. Xing, Y., Shu, C.-W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208(1), 206–227 (2005)
22. Xing, Y., Shu, C.-W.: High-order finite volume WENO schemes for the shallow water equations with dry states. Adv. Water Resour. 34(8), 1026–1038 (2011)
23. Xing, Y., Zhang, X.: Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes. J. Sci. Comput. 57(1), 19–41 (2013)
24. Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476–1493 (2010)
25. Xiong, T., Qiu, J.M., Xu, Z.: A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows. J. Comput. Phys. 252, 310–331 (2013)
26. Xiong, T., Qiu, J.M., Xu, Z.: Parametrized positivity preserving flux limiters for the high order finite difference WENO scheme solving compressible Euler equations. J. Sci. Comput. 67(3), 1066–1088 (2016)
27. Xu, Z.: Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem. Math. Comput. 83(289), 2213–2238 (2014)
28. Yu, Y., Jiang, Y., Zhang, M.: Free-stream preserving finite difference schemes for ideal magnetohydrodynamics on curvilinear meshes. J. Sci. Comput. 82(1), 1–26 (2020)

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High Order Finite Difference WENO Methods for Shallow Water Equations on Curvilinear Meshes

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