Summary
The Sinc-Galerkin method is applied to the canonical forms of second order partial differential equations in multiple space dimensions. For time dependent problems the scheme is fully Galerkin; i.e., the domain of the basis elements includes time. Hence the approximate solution for each equation is specified by solving the associated linear system. Notation is developed which facilitates description of both the linear systems and the algorithms used to solve them. Alternative algorithms are offered dependent on a scalar versus vector computing environment. Numerical results for the test problems presented sustain the exponential convergence rate of the method even in the presence of singularities.
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References
Bowers, K.L., Lund, J.: Numerical solution of singular Poisson problems via the Sinc-Galerkin Method. SIAM J. Numer. Anal.24, 36–51 (1987)
Doyle, R.: A fully Sinc Galerkin method for nonlinear parabolic problems. Ph.D. Thesis: Montana State University 1990
Fairweather, G., Saylor, A.V.: On the application of extrapolation, deferred correction and defect correction to discrete-time Galerkin methods for parabolic problems. IMA J. Numer. Anal.3, 173–192 (1983)
Gear, C.W.: Numerical initial value problems in ordinary differential equations, 1st Ed. Englewood Cliffs: Prentice-Hall 1971
Gottlieb, D., Orszag, S.A.: Numerical analysis of spectral methods: Theory and applications. Philadelphia: SIAM 1977
Kaufman, L., Warner, D.D.: High-order, fast-direct methods for separable elliptic equations. SIAM J. Numer. Anal.21, 672–694 (1984)
Lewis, D.L., Lund, J., Bowers, K.L.: The space-time Sinc-Galerkin method for parabolic problems. Int. J. Numer. Methods Eng.24, 1629–1644 (1987)
Lund, J.: Symmetrization of the Sinc-Galerkin method for boundary value problems. Math. Comput.47, 571–588 (1986)
Lund, J., Bowers, K.L., McArthur, K.M.: Symmetrization of the Sinc-Galerkin method with block techniques for elliptic equations. IMA J. Numer. Anal.9, 29–46 (1989)
McArthur, K.M.: Sinc-Galerkin solution of second-order hyperbolic problems in multiple space dimensions. Ph.D. Thesis: Montana State University 1987
McArthur, K.M., Bowers, K.L., Lund, J.: Numerical implementation of the Sinc-Galerkin method for second order hyperbolic equations. Numer. Methods Partial Diff. Equations3, 169–185 (1987)
Stenger, F.: A Sinc-Galerkin method of solution of boundary value problems. Math. Comput.33, 85–109 (1979)
Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev.23, 165–224 (1981)
Stenger, F.: Sinc methods of approximate solution of partial differential equations. IMACS Proc. Adv. Comput. Methods Partial Diff. Equations 244–251 (1984)
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The work of the authors was supported in part by NSF-MONTS grant No. ISP-8011449 and NSF grant No. ECS-8515083
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McArthur, K.M., Bowers, K.L. & Lund, J. The Sinc method in multiple space dimensions: Model problems. Numer. Math. 56, 789–816 (1989). https://doi.org/10.1007/BF01405289
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DOI: https://doi.org/10.1007/BF01405289