A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

Steven Britt*, Eli Turkel, Semyon Tsynkov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.

Original languageEnglish
Pages (from-to)777-811
Number of pages35
JournalJournal of Scientific Computing
Volume76
Issue number2
DOIs
StatePublished - 1 Aug 2018

Keywords

  • Compact finite differences
  • High order accuracy
  • Variable coefficients
  • Wave equation

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