High-order Leapfrog and Rapid Expansion Time Integrations on Staggered Finite Difference Wave Simulations

Abstract

Summary This work is an exploratory study of coupling high-order time integrations to a finite-difference (FD) spatial discretization of the 1-D wave equation that combines eigth-order differencing at grid interior, with lateral formulas of order sixth and fourthat boundary neighborhood. This reduction of spatial accuracy at the grid vecinity of free surfaces is a known stability limitation of FD methods, when coupled to the two-step Leap-frog (LF) time stepping, which is widely used on seismic modeling. We first implement LF time integrations with an arbitrary accuracy order, as given from a standard Lax-Wendroff procedure, and compare results from the fourth-, sixth-, and twelfth order schemes, against the popular second-order LF. Our emphirical analyses establish the CFL stability constraints for propagation on an homogeneous medium, as first results, and then consider velocity heterogeneities when assessing dispersion and dissipation anomalies. Finally, we use a rapid expansion method (REM) to approximate the exponential of the semidiscrete FD discretization operator by a truncated Chebyshev matrix expansion. Althougth, REM has been previouslly applied to peudospectral (PS) wave simulations, this REM-FD scheme is the first reported in the technical literature according to our knowledge.