Difference Approximations for the Second Order Wave Equation

Difference approximations are derived for the second order wave equation in one and two space dimensions, without first writing it as a first order system. Both the Dirichlet and the Neumann problems are treated for the one-dimensional case. Relations between the boundary error and the interior phas...

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Bibliographic Details
Published inSIAM journal on numerical analysis Vol. 40; no. 5; pp. 1940 - 1967
Main Authors Kreiss, Heinz-Otto, Petersson, N. Anders, Jacob Yström
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2003
Subjects
Approximation
Boundary conditions
Boundary points
Cartesianism
Dirichlet problem
Error rates
Exact sciences and technology
Interpolation
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Phase error
Sciences and techniques of general use
Traveling waves
Wave equations
Wave equation
Second order equation
Neumann problem
Convergence of numerical methods
Difference scheme
Error estimation
Differential equation
Partial differential equation
Numerical approximation
Accuracy
Dirichlet problem
Embedded boundary
Numerical stability
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Summary:Difference approximations are derived for the second order wave equation in one and two space dimensions, without first writing it as a first order system. Both the Dirichlet and the Neumann problems are treated for the one-dimensional case. Relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary. General two-dimensional domains are considered for the Dirichlet problem where the domain is embedded in a Cartesian grid and the boundary conditions are approximated by interpolation. A stable conservative scheme is derived where the time step is determined only by the interior discretization formula. Discretization cells cut by the boundary are treated implicitly, but the resulting scheme becomes explicit because the implicit dependence only is pointwise. Numerical examples are provided to verify the stability and accuracy of the proposed method.
ISSN:0036-1429
1095-7170