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Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation. (English) Zbl 0544.65071
Summary: Two Chebyshev solvers are presented for the linear Helmholtz equation. The first algorithm is a 3-D direct spectral solver based on a diagonalization technique, whilst the second performs an iterative pseudospectral 2-D calculation with finite difference preconditioning. Both techniques handle general nonhomogeneous boundary conditions. Computing times and accuracies of the two methods are compared.

##### MSC:
 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65L15 Numerical solution of eigenvalue problems involving ordinary differential equations 34L99 Ordinary differential operators 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35P15 Estimates of eigenvalues in context of PDEs
FISHPAK
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##### References:
 [1] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, SIAM monograph no. 26, (1977), Philadelphia, Penn. · Zbl 0412.65058 [2] Haidvogel, D.B.; Zang, T., The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials, J. comput. phys., 30, 167-180, (1979) · Zbl 0397.65077 [3] Gottlieb, D.; Lustman, L., The spectrum of the Chebyshev collocation operator for the heat equation, () · Zbl 0537.65085 [4] Sweet, R.A.; Boisvert, R.F., A survey of mathematical software for elliptic boundary value problems, (), 449-451 [5] Orszag, S.A., Spectral methods for problems in complex geometries, () · Zbl 0217.25803 [6] Adams, J.; Swarztrauber, P.; Sweet, R., FISHBACK, a package of Fortran subprograms for the solution of separable elliptic partial differential equations, (1979), NCAR Boulder, Colo [7] Swarztrauber, P., A direct method for the discrete solution of separable elliptic equations, SIAM J. numer. anal., 11, 1136-1150, (1974) · Zbl 0292.65054 [8] Rivlin, T.J., The Chebyshev polynomials, (1974), Wiley New York · Zbl 0291.33012 [9] Deville, M.; Labrosse, G., An algorithm for the evaluation of multidimensional (direct and inverse) discrete Chebyshev transforms, J. comput. appl. math, 8, 4, 293-304, (1982) · Zbl 0494.65003
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