Volume 45, pp. 283-304, 2016.

A matrix-free Legendre spectral method for initial-boundary value problems

Bernd Brumm and Emil Kieri

Abstract

We present a Legendre spectral method for initial-boundary value problems with variable coefficients and of arbitrary dimensionality, where the computational work in each time step scales linearly with the number of unknowns. Boundary conditions are enforced weakly, allowing for stable solutions of many classes of problems. Working in coefficient space, derivatives can be evaluated recursively in linear time. We show how also the action of variable coefficients can be implemented without transforming back to coordinate space using a recursive, linearly scaling matrix-free algorithm, under the assumption that the coefficients vary on a much longer scale than the solution. We also prove that spectral accuracy is preserved for smooth solutions. Numerical results for the wave equation in two and three dimensions corroborate the theoretical predictions.

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Key words

spectral methods, matrix-free methods, method of lines, stability, computational wave propagation, boundary conditions

AMS subject classifications

65M12, 65M15, 65M20, 65M70

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