Compact optimal quadratic spline collocation methods for the Helmholtz equation

• Published in: Journal of computational physics Vol. 230; no. 8; pp. 2880 - 2895
• Main Authors: Fairweather, Graeme, Karageorghis, Andreas, Maack, Jon
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.04.2011; Elsevier
• Subjects: Algorithms; Collocation methods; Computational techniques; Dirichlet problem; Exact sciences and technology; Fast Fourier transforms; Helmholtz equation; Helmholtz equations; Mathematical analysis; Mathematical methods in physics; Mathematical models; Matrix decomposition algorithms; Optimal global convergence rates; Optimization; Partitions; Physics; Quadratic spline collocation; Splines; Superconvergence; Tridiagonal linear systems; Fast Fourier transforms; Optimal global convergence rates; Helmholtz equation; Quadratic spline collocation; Superconvergence; Tridiagonal linear systems; Matrix decomposition algorithms; Linear systems; Helmholtz equations; Digital simulation; Boundary conditions; Matrix decomposition; Spline; Calculation methods; Algorithms; Numerical solution; Convergence rate; Calculation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.12.041

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 2 log N) on an N × N 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.