Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case

Abstract

Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model — that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable.