Analysis and Hermite spectral approximation of diffusive-viscous wave equations in unbounded domains arising in geophysics

TL;DR

This paper develops a Hermite spectral Galerkin method for the diffusive-viscous wave equation in unbounded domains, avoiding artificial reflections and achieving spectral convergence, with theoretical analysis and numerical validation.

Contribution

It introduces a spectral Galerkin scheme for the DVWE in unbounded domains, providing error estimates and demonstrating effectiveness without artificial boundary reflections.

Findings

Spectral convergence rate achieved for smooth solutions.

Method effectively avoids artificial boundary reflections.

Numerical experiments confirm theoretical error estimates.

Abstract

The diffusive-viscous wave equation (DVWE) is widely used in seismic exploration since it can explain frequency-dependent seismic reflections in a reservoir with hydrocarbons. Most of the existing numerical approximations for the DVWE are based on domain truncation with ad hoc boundary conditions. However, this would generate artificial reflections as well as truncation errors. To this end, we directly consider the DVWE in unbounded domains. We first show the existence, uniqueness, and regularity of the solution of the DVWE. We then develop a Hermite spectral Galerkin scheme and derive the corresponding error estimate showing that the Hermite spectral Galerkin approximation delivers a spectral rate of convergence provided sufficiently smooth solutions. Several numerical experiments with constant and discontinuous coefficients are provided to verify the theoretical result and to demonstrate the effectiveness of the proposed method. In particular, We verify the error estimate for both smooth and non-smooth source terms and initial conditions. In view of the error estimate and the regularity result, we show the sharpness of the convergence rate in terms of the regularity of the source term. We also show that the artificial reflection does not occur by using the present method.