×

zbMATH — the first resource for mathematics

Convergence results for pseudospectral approximations of hyperbolic systems by a penalty-type boundary treatment. (English) Zbl 0736.65074
A new method of imposing boundary conditions in the pseudospectral approximation of a scalar hyperbolic equation is presented. The idea is to collocate the equation at the boundary points as well as in the inner grid points, using the boundary conditions as penalty terms. This technique is extended to general constant coefficient hyperbolic systems of equations. Error estimates for the pseudospectral Legendre method are given.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. · Zbl 0658.76001
[2] Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. · Zbl 0537.65020
[3] D. Funaro and D. Gottlieb, A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations, Math. Comp. 51 (1988), no. 184, 599 – 613. · Zbl 0699.65079
[4] David Gottlieb, Liviu Lustman, and Eitan Tadmor, Stability analysis of spectral methods for hyperbolic initial-boundary value systems, SIAM J. Numer. Anal. 24 (1987), no. 2, 241 – 256. · Zbl 0628.65109 · doi:10.1137/0724020 · doi.org
[5] David Gottlieb, Liviu Lustman, and Eitan Tadmor, Convergence of spectral methods for hyperbolic initial-boundary value systems, SIAM J. Numer. Anal. 24 (1987), no. 3, 532 – 537. · Zbl 0622.65110 · doi:10.1137/0724038 · doi.org
[6] Bertil Gustafsson, Heinz-Otto Kreiss, and Arne Sundström, Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972), 649 – 686. · Zbl 0293.65076
[7] Stanley Osher, Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I, Math. Comp. 23 (1969), 335 – 340. · Zbl 0177.20403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.