High order immersed interface method for acoustic wave equation with discontinuous coeffcients

Iranian Journal of Numerical Analysis and Optimization

مقاله 2، دوره 4، شماره 1 - شماره پیاپی 5، مرداد 2014، صفحه 1-24   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v4i1.26174

نویسندگان

J. Farzi^* ؛ S. M. Hosseini

Department of Mathematics, Tarbiat Modares University, P.O. Box 14115-175, Tehran, Iran.

چکیده

This paper concerns the numerical solution of the acoustic wave equation that contains interfaces in the solution domain. To solve the interface problems with high accuracy, more attention should be paid to the interfaces. In fact, any direct application of a high order finite diffierence method to these problems leads to inaccurate proximate solutions with high oscillations at the interfaces. There is however, the possibility of deriving some high order methods to resolve this phenomenon at the interfaces. In this paper, a sixth order immersed interface method for acoustic wave equation is presented. The order of accuracy is also maintained at the discontinuity using the jump conditions. Some numerical experiments are included which confirm the order of accuracy and numerical stability of the presented method.

کلیدواژه‌ها

Interface methods؛ High order methods؛ Lax-Wendroff method؛ Discontinuous coeffcients؛ Jump conditions

مراجع

1. Driscoll, T. A. and Fornberg, B. A block pseudospectral method for maxwells equations I. one-dimensional case, Journal of Comput. Phys. 140 (1998) 47-65.
2. Farzi, J. and Hosseini, S. M. A high order method for the solution of a one-way wave equation in heterogeneous media, Far East J. Appl. Math. 36 (3) (2009) 317-330.
3. Gustafsson, B. and Wahlund, P. Time compact high order difference methods for wave propagation in discontinuous media, SIAM J. Sci. Comput. 26 (2004) 272-293.
4. Gustafsson, B. and Mossberg, E. High order difference methods for wave propagation, SIAM J. Sci. Comput. 26 (2004) 259-271.
5. Gustafsson, B., Kreiss, H. O. and Oliger, J. Time dependent problems and difference methods, first ed., John Wiley and Sons, (1996).
6. Jiang, G. S. and Shu, C. W. Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996) 202-228.
7. Larsson, J. and Gustafsson, B. Stability criteria for hybrid difference methods, J. comp. phys. 227 (2008) 2886-2898.

8. LeVeque, R. J. Wave propagation algorithms for multidimensional hyper bolic systems, J. comp. phys. 131 (1997) 327-353.
9. LeVeque, R. J. Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2004.
10. Li, Z. and Ito, K. The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains, SIAM, 2006.
11. Peskin, C. S. Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977) 220-252.
12. Piraux, J. and Lombard, B. A new interface method for hyperbolic problems with discontinuous coefficients. one-dimensional acoustic example, J. Comput. Phys., 168 (2001) 227-248.
13. Qiu, J. and Shu, C. W. Finite difference WENO schemes with Lax Wendroff type time discretizations, SIAM J. Sci. Comput. 24 (2003) 2185-2198.
14. Sei, A. and Symes, W. W. Error analysis of numerical schemes for the wave equation in heterogeneous media, Appl. Numer. Math. 15 (1994) 465-480.
15. Symes, W. W. and Vdovina, T. Interface error analysis for numerical wave propagation, Comput. Geosci. 13 (2009) 363-371.
16. Trefethen, L. N. Group velocity in finite difference schemes, SIAM. Rev. 24 (1982) 113-136.
17. Wiegmann, A. and Bube, K. P. The explicit-jump immersed interface method: finite difference methods for PDE with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000) 827-862.
18. Zhang, C. and LeVeque, R. J. Immersed interface methods for wave equations with discontinuous coefficients, Wave Motion, 25 (1997) 237-263.
19. Zhao, S. and Wei, G. W. High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces, J. Comput. Phys., 200 (2004) 60-103.