Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

THE NUMERICAL SOLUTION OF SHALLOW WATER EQUATION BY MOVING MESH METHODS

  • Shin, Suyeon (Department of Mathematics Korea University) ;
  • Hwang, Woonjae (Department of Information and Mathematics Korea University)
  • Published : 2012.08.15
⟨ Previous Next ⟩

Abstract

This paper presents a moving mesh method for solving the hyperbolic conservation laws. Moving mesh method consists of two independent parts: PDE evolution and mesh- redistribution. We compute numerical solution of shallow water equation by using moving mesh methods. In comparison with computations on a fixed grid, the moving mesh method appears more accurate resolution of discontinuities.

Keywords

References

  1. B. N. Azarenok, S. A. Ivanenko, and T. Tang, Adaptive mesh redistribution method based on Godunov's scheme, Communications in Mathematical Sciences 1 (2003), 152-179. https://doi.org/10.4310/CMS.2003.v1.n1.a10
  2. B. N. Azarenok and T. Tang, Second-order Godunov-type scheme for reactive flow calculations on moving meshes, Journal of Computational Physics 206 (2005), 48-80. https://doi.org/10.1016/j.jcp.2004.12.002
  3. Y. Di, R. Li, T. Tang and P.W. Zhang, Moving mesh finite element methods for the incompressible Navier-Stokes equations, AMJ. Sci. Comput. 26 (2005), 1036-1056.
  4. A. S. Dvinsky, Adaptive grid generation from harmonic maps on Riemannian manifolds, Journal of Computational Physics, 95 (1991), 450-476. https://doi.org/10.1016/0021-9991(91)90285-S
  5. J. Felcman and L. Kadrnka, On a Moving Mesh Method Applied to the Shallow Water Equations, Numerical Analysis and Applied Mathematics, International Conference 1 (2010), 195-198.
  6. A. Harten, J. M. Hyman, Self-adjusting grid methods for one-dimensional hy- perbolic conservation laws, Journal of Computational Physics 50 (1983), 235- 269. https://doi.org/10.1016/0021-9991(83)90066-9
  7. W. Huang, Y. Ren and R. D. Russell, Moving mesh methods based on moving mesh partial differential equations, Journal of Computational Physics 113 (1994), 279-290. https://doi.org/10.1006/jcph.1994.1135
  8. R. J. Leveque, Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge, 2002.
  9. R. Li, T. Tang and P. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, Journal of Computational Physics 170 (2001), 562- 588. https://doi.org/10.1006/jcph.2001.6749
  10. R. Li, T. Tang and P. Zhang, A moving mesh finite element algorithm for singular problems in two and three space dimensions, Journal of Computational Physics 177 (2002), 365-393. https://doi.org/10.1006/jcph.2002.7002
  11. S. Li, L. Petzold, Moving mesh methods with upwinding schemes for time-dependent PDEs, Journal of Computational Physics 131 (1997), 368-377. https://doi.org/10.1006/jcph.1996.5611
  12. Y. Ren, R. Ren and Russell, Moving mesh partial differential equations(MMPDES) based on the equidistribution principle, SIAM Journal on Numerical Analysis 31 (1994), 709-730. https://doi.org/10.1137/0731038
  13. K. Saleri, S. Steinberg, Flux-corrected transport in a moving grid, Journal of Computational Physics 111 (1994), 24-32. https://doi.org/10.1006/jcph.1994.1040
  14. S. Shin, Adaptive mesh method for conservation laws, Master thesis, Korea University, Seoul, 2006.
  15. S. Shin and W. Hwang, Numerical solution of burgers' equation by moving mesh methods, Proceedings of the National Institute for Mathematical Sciences 1 (2006), 100-103.
  16. J. M. Stockie, J. A. Mackenzie, and R. D. Russell, A moving mesh method for one-dimensional hyperbolic conservation laws, SIAM Journal on Scientific Computing 22 (2001), 1791-1813. https://doi.org/10.1137/S1064827599364428
  17. Z. Tan, Z. Zhang, Y. Huang, and T. Tang, Moving mesh methods with locally varying time steps, Journal of Computational Physics 200 (2004), 347-367. https://doi.org/10.1016/j.jcp.2004.04.007
  18. H. Tang, Solution of the shallow-water equations using an adaptive moving mesh method, International Journal for Numerical Methods in Fluids 44 (2004), 789-810. https://doi.org/10.1002/fld.681
  19. H. Tang and T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM Journal on Numerical Analysis 41 (2003), 487-515. https://doi.org/10.1137/S003614290138437X
  20. H. Z. Tang and T. Tang, Multi-dimensional moving mesh methods for shock computations, Inter. Conf. on Sci. Comput. and PDEs,on the Occasion of S. Osher's 60th birthday, Dec. 12-15, 2002, Hongkong. Contemporary Mathematics 330 (2003), 169-183.
  21. H. Z. Tang, T. Tang, P. Zhang, An adaptive mesh redistribution method for nonlinear Hamiton-Jacobi equations in two- and three-dimensions, Journal of Computational Physics 188 (2003), 543-572. https://doi.org/10.1016/S0021-9991(03)00192-X
  22. T. Tang, Moving mesh methods for computational fluid dynamics, Contemporary Mathematics 383 (2005), 141-173. https://doi.org/10.1090/conm/383/07162
  23. P. A. Zegeling, W. D. de Boer, and H. Z. Tang, Robust and efficient adaptive moving mesh solution of the 2-D Euler equations, Contemporary Mathematics 383 (2005), 375-386. https://doi.org/10.1090/conm/383/07179
  24. Z. Zhang and T. Tang, An adaptive mesh redistribution algorithm for convection-dominated problems, Communications on pure and Applied Analysis 1 (2002), 341-357. https://doi.org/10.3934/cpaa.2002.1.341
  25. A. Winslow, Numerical solution of the quasi-linear Poisson equation in a nonuniform triangle mesh, Journal of Computational Physics 1 (1967), 149-172.