%0 Journal Article
%T Preconditioned Krylov Subspace Methods for Sixth Order Compact Approximations of the Helmholtz Equation
%A Yury Gryazin
%J ISRN Computational Mathematics
%D 2014
%R 10.1155/2014/745849
%X We consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann, and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and lower order preconditioned Krylov subspace methodology. The resulting systems of finite-difference equations are solved by different preconditioned Krylov subspace-based methods. In the analysis of the lower order preconditioning developed here, we introduce the term ”°kth order preconditioned matrix”± in addition to the commonly used ”°an optimal preconditioner.”± The necessity of the new criterion is justified by the fact that the condition number of the preconditioned matrix in some of our test problems improves with the decrease of the grid step size. In a simple 1D case, we are able to prove this analytically. This new parameter could serve as a guide in the construction of new preconditioners. The lower order direct preconditioner used in our algorithms is based on a combination of the separation of variables technique and fast Fourier transform (FFT) type methods. The resulting numerical methods allow efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative approach. 1. Introduction In recent years, the problem of increasing the resolution of existing numerical solvers has become an urgent task in many areas of science and engineering. Most of the existing efficient solvers for structured matrices were developed for lower-order approximations of partial differential equations. The need for improved accuracy of underlying algorithms leads to modified discretized systems and as a result to the modification of the numerical solvers (see, e.g., [1]). The use of a lower-order preconditioner for efficient implementation of high-order finite-difference and finite-element schemes has been under consideration for a long time (see, e.g., [2, 3]). In this paper, a compact sixth order approximation finite-difference scheme is developed, and a lower-order approximation direct solver as a preconditioner for an efficient implementation of this compact scheme for the Helmholtz equation in the Krylov subspace method framework is considered. This approach allows us to utilize the existing lower-order approximation solvers which significantly simplifies the implementation process of the higher resolution numerical methods. The model problem considered in the paper is the numerical solution of the Helmholtz equation with the Dirichlet, Neumann, and/or Sommerfeld-like (radiation) boundary
%U http://www.hindawi.com/journals/isrn.computational.mathematics/2014/745849/