A level set moving mesh method in static form for one dimensional PDEs

Document Type : Research Article

Author

Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.

Abstract

In this paper, we propose an adaptive mesh approach for time dependent parial differential equations, based on a so-called moving mesh PDE(MMPDE) and level set method. It means that the velocity of mesh nodes is calculated by MMPDE and is employed as veocity in the level set equation. Then, at each time level, the mesh points are considered as the level contours of the level set function. Finally the method is merged with local time step technique.

Keywords

References

1. Aslam, T., A level set algorithm for tracking discontinuities in hyperbolic conservation laws,I:Scalar equations, J. Comput. Phys. 167, (2001), 413-438.
2. Ghorbani, M. and Soheili, A.R.,Moving element free Petrov Galerkian viscous method,special issue on meshless methodes, Journal of the Chinese Institue of engineers, 27, (2004), 473-479.
3. Huang, W., Ren, Y. and Russell, R., Moving mesh partial differential equations(MMPDEs)based on the equiditribution principle, SIAM J.Number. Anal. 31, (1994), 709-730.
4. Huang, W., Ren, Y. and Russell, R., Moving mesh methods based on moving mesh partial differential equations, J. Compute. Phys., 113, (1994), 279-290.
5. Jin, S. and Osher, S.,A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations, SIAM .Numer. Analys. 28, (1991), 907-921.
6. Liao, G., Liu, F., Pena, G.D., peng, D. and Osher, S., Level-set-based deformation methods for adaptive grids, J.Compute. Phys., 159, (2000), 103-122.
 7. Mazhukin, A.V., Dynamic adaptation in convection-diffusion equations, Comput. method. Appl. Math, 8, (2008), 171-186.
8. Mazhukin, A.V. and Chuiko, M.M., Solution of the multi-interface Stefan problem by the method of dynamic adaptation, Compute. method. Appl. Math, 2, (2002), 283-294.
9. Osher, S. and Fedkiw, R., Level set methods and dynamic implicit surfaces, Springer Verlag, NY, (2002)
10. Osher, S. and Sanders, R., Numerical approxiamtion to nonlinear conservation laws with locally varying time and spaces grid , Math. Compute. 41, (1983), 321-336.
11. Osher, S. and Sethian, J., Front propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Compute. Phys. 79, 12 (1988).
12. Sethian, J.A.Level set methods and fast marching methods, New York: Cambridge University Press,(1999).
13. Soheili, A.R. and Ameri, M.A., Adaptive grid based on geometric conservation law level set method for time dependent PDE , Numerical methods for PDEs, 25, (2009), 582-597.
14. Soheili, A.R. and Salahshour, S., Moving mesh methods with local time step refinement for blow-up problems, Appl.Math.Copmpute. 195, (2008), 76-85.
15. Soheili, A.R. and Stockie, J.M.,A moving mesh method with variable mesh relaxation time, Appl.Number.Math. 58, (2008), 249-263.
16. Stockie, J.M., Mackenzie, J.A. and Russell, R., A moving mesh method for one dimensional hyperbolic conservation laws, SIAM. J.S.Compute. 22, (2001), 1791-1831.
17. Tan, H.Z. and Tang, T., Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM. J.Numer. Anal. 41, (2003), 487-515.
18. Tan, Z., Zhang, Z., Huang, Y. and Tang, T.,Moving mesh methods with locally varying time step, J. Comp. Phys. 200, (2004), 347-367.
19. Tasi, Y.H.R. and Giga, Y. and Osher, S.,A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations, J. Math. Compute. 72, (2002), 159-181.
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