TY - JOUR
T1 - A note on stability of pseudospectral methods for wave propagation
AU - Jackiewicz, Zdzislaw
AU - Renaut, Rosemary
N1 - Funding Information: The authors wish to express their gratitude to Professor L.N. Trefethen for useful comments and for providing us with his program for plotting pseudospectra. This work was supported by the National Science Foundation under grants DMS-9971164 (Z. Jackiewicz) and DMS-9402943 (R.A. Renaut).
PY - 2002/6/1
Y1 - 2002/6/1
N2 - In this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation. We suppose that spatial derivatives are approximated by Chebyshev pseudospectral discretizations. Through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time. However, the resulting stability, and hence efficiency, properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated. Specifically, mathematically equivalent well-posed approaches are not equivalent numerically. An analysis of the spectrum of the resultant system operator predicts these properties.
AB - In this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation. We suppose that spatial derivatives are approximated by Chebyshev pseudospectral discretizations. Through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time. However, the resulting stability, and hence efficiency, properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated. Specifically, mathematically equivalent well-posed approaches are not equivalent numerically. An analysis of the spectrum of the resultant system operator predicts these properties.
KW - Absorbing boundary conditions
KW - Eigenvalue stability
KW - Hyperbolic systems
KW - Pseudospectral Chebyshev method
KW - Runge-Kutta methods
KW - Wave equation
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U2 - 10.1016/S0377-0427(01)00495-2
DO - 10.1016/S0377-0427(01)00495-2
M3 - Article
SN - 0377-0427
VL - 143
SP - 127
EP - 139
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 1
ER -