Stable and high-order accurate finite difference methods for the diffusive viscous wave equation

TL;DR

This paper introduces a stable, high-order finite difference method for the diffusive viscous wave equation, effectively modeling wave propagation in media with elastic and viscous effects, with proven stability and accuracy.

Contribution

The paper develops a novel finite difference scheme with summation-by-parts property for the diffusive viscous wave equation, ensuring stability and high-order accuracy.

Findings

Method is stable and high-order accurate

Error estimates are derived for constant and variable coefficients

Numerical examples confirm theoretical properties

Abstract

The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the dissipative effects due to internal friction and viscosity; acoustic waves propagating through biological tissues, where both elastic and viscous effects play a significant role. We propose a stable and high-order finite difference method for solving the governing equations. By designing the spatial discretization with the summation-by-parts property, we prove stability by deriving a discrete energy estimate. In addition, we derive error estimates for problems with constant coefficients using the normal mode analysis and for problems with variable coefficients using the energy method. Numerical examples are presented to demonstrate the stability and accuracy properties of the developed method.