Journal
JOURNAL OF SCIENTIFIC COMPUTING
Volume 76, Issue 2, Pages 777-811Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-017-0639-9
Keywords
Compact finite differences; Wave equation; Variable coefficients; High order accuracy
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Funding
- US-Israel Binational Science Foundation (BSF) [2014048]
- US Army Research Office (ARO) [W911NF-16-1-0115]
- Raymond and Beverly Sackler Post-Doctoral Scholarship at Tel Aviv University
- Fulbright Postdoctoral Scholarship - US-Israel Educational Foundation
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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.
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