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

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 t...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 76; no. 2; pp. 777 - 811
Main Authors: Britt, Steven, Turkel, Eli, Tsynkov, Semyon
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2018
Springer Nature B.V
Subjects:
Accuracy
Acoustics
Algorithms
Approximation
Boundary conditions
Boundary value problems
Computational Mathematics and Numerical Analysis
Conjugate gradient method
Convergence
Efficiency
Eigenvalues
Finite difference method
Helmholtz equations
Iterative methods
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Multigrid methods
Theoretical
Time dependence
Time marching
Upper bounds
Wave equations
Compact finite differences
Wave equation
Variable coefficients
High order accuracy
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary: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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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-017-0639-9