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SUMMARYThis paper contains the formulation of a space-time Sinc-Galerkin method for the numerical solution of the parabolic partial differential equation in one space dimension. The space^^ time adjective means that the Galerkin technique is employed simultaneously in time and space. Salient features of the method include: exponential rate of convergence, ease of assembly of the discrete system, a global approximation and the ability to handle singular problems. Two methods of solution for the discrete system are offered and numerical results for test problems, selected from the literature, are included.

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A fully Galerkin method in both space and time is developed for the second-order, linear hyperbolic problem. Sinc basis functions are used and error bounds are given which show the exponential convergence rate of the method. The matrices necessary for the formulation of the discrete system are easily assembled. They require no numerical integrations (merely point evaluations) to be filled. The discrete problem is formulated in two different ways and solution techniques for each are described. Consideration of the two formulations is motivated by the computational architecture available. Each has advantages for the appropriate hardware. Numerical results reported show that if 2N + 1 basis functions are used then the exponential convergence rate q e x p ( -~f i ) J , K > 0, is attained for both analytic and singular problems.

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a b s t r a c tA new Alternating-Direction Sinc-Galerkin (ADSG) method is developed and contrasted with classical Sinc-Galerkin methods. It is derived from an iterative scheme for solving the Lyapunov equation that arises when a symmetric Sinc-Galerkin method is used to approximate the solution of elliptic partial differential equations. We include parameter choices (derived from numerical experiments) that simplify existing alternating-direction algorithms. We compare the new scheme to a standard method employing Gaussian elimination on a system produced using the Kronecker product and Kronecker sum, as well as to a more efficient algorithm employing matrix diagonalization. We note that the ADSG method easily outperforms Gaussian elimination on the Kronecker sum and, while competitive with matrix diagonalization, does not require the computation of eigenvalues and eigenvectors.