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

Katherine Baker, Lehel Banjai

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Abstract

We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency-dependent power law. First we prove the 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.
Original languageEnglish
Pages (from-to)2083–2117
Number of pages35
JournalIMA Journal of Numerical Analysis
Volume42
Issue number3
Early online date10 May 2021
DOIs
Publication statusPublished - Jul 2022

Keywords

  • Convergence
  • Existence and uniqueness
  • Fractional calculus
  • Wave equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

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