High‐order implicit staggered‐grid finite differences methods for the acoustic wave equation

Abstract

Motivated by the idea that staggered‐grid methods give a greater stability and give energy conservation, this article presents a new family of high‐order implicit staggered‐grid finite difference methods with any order of accuracy to approximate partial differential equations involving second‐order derivatives. In particular, we numerically analyze our new methods for the solution of the one‐dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ( 2 M + 2 ) ‐point operator: the explicit formula is ( 2 M ) th‐order and the implicit formula is ( 2 M + 2 ) th‐order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.

A hybrid explicit implicit staggered grid finite-difference scheme for the first-order acoustic wave equation modeling

“…One can use a larger time step with the coefficient c M +1 . At the same time, one can use a short operator length with the coefficient b 27 – 30 . We propose to use the simplest explicit second-order SGFD operator for the spatial derivatives in Eqs.…”