Stability of finite difference schemes for the problem of elastic wave propagation in a quarter plane

Abstract

For the problem of elastic waves propagating in a quarter plane, the stability of the finite difference method is critically dependent on the approximation of the boundary conditions and the treatment of the 90° corner. The stability of two classical and two composed approximations to the boundary conditions is studied using analysis of the local propagating matrix and by computer experiments. Mistreatment of certain grid points near the corner is found to be the cause of the unstable solution reported by Alterman and Rotenberg (1969). Correction of this results in a stable scheme in which the range of stability of the different approximations of boundary conditions for the quarter plane is the same as for the half plane. The two classical approximations, which use fictitious lines of grid points, are reliable for quarter planes only when the ratio of shear velocity to compressional velocity β α > 0.5 . For β α < 0.5 the results they give are contaminated, by an oscillation about the true solution in the case of the central difference approximation and also by a phase delay in the one-sided approximation. The first composed approximation ( A. Ilan et al., Geophys. J. R. Astron. Soc. 34 (1975) , 727–742) is stable only for β α > 0.575 . However, the new composed approximation ( A. Ilan and D. Loewenthal, Geophys. Prospect. 24 (1976) , 431–453) is shown to be stable even for small values of β α .