Numerical analysis of a wave equation for lossy media obeying a frequency power law

TL;DR

This paper analyzes a wave equation with fractional damping modeling frequency-dependent acoustic attenuation, proving solution existence, developing a numerical scheme, and providing error estimates supported by numerical experiments.

Contribution

It introduces a novel finite element-based numerical scheme for a fractional damping wave equation and provides rigorous error analysis for both smooth and singular solutions.

Findings

Proved existence and uniqueness of solutions.

Developed an explicit time-stepping numerical scheme.

Validated error estimates through numerical experiments.

Abstract

We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency dependence power law. First we prove existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time stepping scheme based on the finite element method in space and a combination of convolution quadrature and second order central differences in time. We conduct a full error analysis of the mixed time discretization and in turn the fully space time discretized scheme. Error estimates are given for both smooth solutions and solutions with a singularity at $t = 0$ of a type that is typical for equations involving fractional time-derivatives. A number of numerical results are presented to support the error analysis.