Abstract
Objectives: The study reviews the iterative solvers for solving Helmholtz equation. It also surveys the preconditioners for Krylov subspace methods for discrete Helmholtz equation. Numerical experiments based conclusions are drawn, after a thorough comparison of various preconditioners. Methods/Statistical Analysis: Five point finite difference scheme for two-dimensional Helmholtz equation is employed to obtain discrete analogue of Helmholtz equation. Krylov subspace method for is considered for iterations. Matrix based preconditioner ILU with variety of fill-in and operator based preconditioner complex shifted Laplace preconditioner (CSLP) are tested. Different shifts are proposed and experimented in CSLP. Optimization of shifts is recommended on basis of results. Spectral analysis is performed to validate results. Findings: For small wave number problem, ILU type preconditioned performs better and is fairly comparable with CSLP. However, increasing wave number makes unaffordable in terms of computational cost. The pure imaginary shift (0,1) and complex shift (1, / 4 π ) costsless time and number of iterations compared to other choices of shift. These choices are recommended when preconditioner is inverted directly. Approximate inversion may affect iterations, but will save solve time for large problems. Extensive spectral analysis, validating results, highlights this work. Application/Improvements: Dependency upon wave number is decreased with this recommended choice of shifts in CSLP. The number of iterations increases at very low rate when wave number is increased at rate of 2. Keywords: Helmholtz Equation, Iterative Methods, Krylov Subspace Methods, Preconditioners, Spectral Analysis
Highlights
Helmholtz equation often arises in the study of physical problems involving partial differential equation (PDE) in both space and time; in this paper the time-independent case considered.Many problems related to steady state oscillations are modeled by the Helmholtz equation1
The pure imaginary shift 0,1 ( ) and complex shift 1,π / 4 costsless time and number of iterations compared to other choices of shift. These choices are recommended when preconditioner is inverted directly
The linear system arising from a discretization of the Helmholtz equation in two dimensions (2D) or three dimensional (3D), domain, converges typically characterized by indefiniteness of the Eigen values of the corresponding coefficient matrix
Summary
Helmholtz equation often arises in the study of physical problems involving partial differential equation (PDE) in both space and time; in this paper the time-independent case considered.Many problems related to steady state oscillations (mechanical, acoustical, and thermal) are modeled by the Helmholtz equation. The linear system arising from a discretization of the Helmholtz equation in two dimensions (2D) or three dimensional (3D), domain, converges typically characterized by indefiniteness of the Eigen values of the corresponding coefficient matrix. With such a property, an iterative method either basic or advanced, encounters convergence problems. The method usually converges very slowly or divergences, for high frequency problems, as the linear system becomes extremely large and an indefinite. This makes the problem even harder to solve
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