On accuracy of the finite-difference and finite-element schemes with respect to P-wave to S-wave speed ratio

Abstract

Numerical modelling of seismic motion in sedimentary basins often has to account for P-wave to S-wave speed ratios as large as five and even larger, mainly in sediments below groundwater level. Therefore, we analyse seven schemes for their behaviour with a varying P-wave to S-wave speed ratio. Four finite-difference (FD) schemes include (1) displacement conventional-grid, (2) displacement-stress partly-staggered-grid, (3) displacement-stress staggered-grid and (4) velocity-stress staggered-grid schemes. Three displacement finite-element schemes differ in integration: (1) Lobatto four-point, (2) Gauss four-point and (3) Gauss one-point. To compare schemes at the most fundamental level, and identify basic aspects responsible for their behaviours with the varying speed ratio, we analyse 2-D second-order schemes assuming an elastic homogeneous isotropic medium and a uniform grid. We compare structures of the schemes and applied FD approximations. We define (full) local errors in amplitude and polarization in one time step, and normalize them for a unit time. We present results of extensive numerical calculations for wide ranges of values of the speed ratio and a spatial sampling ratio, and the entire range of directions of propagation with respect to the spatial grid. The application of some schemes to real sedimentary basins in general requires considerably finer spatial sampling than usually applied. Consistency in approximating first spatial derivatives appears to be the key factor for the behaviour of a scheme with respect to the P-wave to S-wave speed ratio. © 2010 The Authors Journal compilation © 2010 RAS.