Compact optimal quadratic spline collocation methods for the Helmholtz equation

  • Authors:

  • Graeme Fairweather;Andreas Karageorghis;Jon Maack

  • Affiliations:

  • Mathematical Reviews, American Mathematical Society, 416 Fourth Street, Ann Arbor, Michigan 48103, USA;Department of Mathematics and Statistics, University of Cyprus, Nicosia CY-1678, Cyprus;3255 W. Chenango Ave., Englewood, Colorado 80110, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2011

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Abstract

Quadratic spline collocation methods are formulated for the numerical solution of the Helmholtz equation in the unit square subject to non-homogeneous Dirichlet, Neumann and mixed boundary conditions, and also periodic boundary conditions. The methods are constructed so that they are: (a) of optimal accuracy, and (b) compact; that is, the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems. Using fast Fourier transforms, the computational cost of such an algorithm is O(N^2logN) on an NxN uniform partition of the unit square. The results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena. In particular, it is shown that the methods are fourth-order accurate at the nodes of the partition.