Regularization for accurate numerical wave propagation in discontinuous media.

*(English)*Zbl 1136.65084Summary: Structured computational grids are the basis for highly efficient numerical approximations of wave propagation. When there are discontinuous material coefficients the accuracy is typically reduced and there may also be stability problems. In a sequence of recent papers B. Gustafsson and E. Mossberg [SIAM J. Sci. Comput. 26, No. 1, 259–271 (2004; Zbl 1075.65112), B. Gustafsson and P. Wahlund, ibid. 26, No. 1, 272–293 (2004; Zbl 1077.65092); Time compact high order difference methods for wave propagation, 2-D, J. Sci. Comput. 25, 195–211 (2005)] proved stability of the Yee scheme [cf. K. S. Yee, IEEE Trans. Antennas Propag. 14, 302–307 (1966)] and a higher order difference approximation based on a similar staggered structure, for the wave equation with general coefficients.

In this paper, the Yee discretization is improved from first to second order by modifying the material coefficients close to the material interface. This is proven in the \(L^2\) norm. The modified higher order discretization yields a second order error component originating from the discontinuities, and a fourth order error from the smooth regions. The efficiency of each original method is retained since there is no special structure in the difference stencil at the interface. The main focus of this paper is on one spatial dimension, with the derivation of a second order algorithm for a two dimensional example given in the last section.

In this paper, the Yee discretization is improved from first to second order by modifying the material coefficients close to the material interface. This is proven in the \(L^2\) norm. The modified higher order discretization yields a second order error component originating from the discontinuities, and a fourth order error from the smooth regions. The efficiency of each original method is retained since there is no special structure in the difference stencil at the interface. The main focus of this paper is on one spatial dimension, with the derivation of a second order algorithm for a two dimensional example given in the last section.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35L05 | Wave equation |

35R05 | PDEs with low regular coefficients and/or low regular data |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |