SBP-SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids

TL;DR

This paper introduces a flexible finite difference method for simulating acoustic wave equations on non-uniform, block-wise grids, improving efficiency and accuracy in seismic modeling.

Contribution

It develops SBP-SAT finite difference operators on nonconforming grids with rational ratios, enabling more efficient seismic simulations with energy conservation.

Findings

Energy-conserving discretization verified on test cases

Accurate and stable simulation results achieved

Handles nonconforming grid interfaces effectively

Abstract

We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid restriction on finite difference methods and allow the grids to be block-wise uniform with nonconforming interfaces. In doing so, variations in the wave speeds of the subterranean media can be accounted for more efficiently. Staggered grid finite difference operators satisfying the summation-by-parts (SBP) property are devised to approximate the spatial derivatives appearing in the acoustic wave equation. These operators are applied within each block independently. The coupling between blocks is achieved through simultaneous approximation terms (SATs), which impose the interface condition weakly, i.e., by penalty. Ratio of the grid spacing of neighboring blocks is allowed to be rational number, for which specially designed interpolation formulas are presented. These interpolation formulas constitute key pieces of the simultaneous approximation terms. The overall discretization is shown to be energy-conserving and examined on test cases of both theoretical and practical interests, delivering accurate and stable simulation results.