An Optimal Compact Finite Difference Scheme For The 3D Helmholtz Equation

The Helmholtz equation governs wave propagations and scattering phenomena,and has important applications in many areas,for example,in acoustics,optics,electromagnetics,geophysics.Therefore,obtaining an efficient and more accurate numerical solution for the Helmoltz equation has always been a hot topic.For large wavenumbers,the quality of the numerical results usually deteriorates as the wave number increases.Meanwhile,the numerical method usually requires a finer mesh to ensure the accuracy with the increasing wavenumber.Therefore,for the Helmholtz equation with large wavenumbers,the resulting matrix is very large and ill-conditioned.Usually,direct methods do not perform well.In this thesis,to solve the Helmholtz equation efficiently,we pay attention to two issues: one is the numerical accuracy,while the other is the solver cost.This thesis is divided into five parts.In chapter 1,we briefly introduce some backgrounds of the Helmholtz equation and the developments of the numerical methods for solving the Helmholtz equation.In addition,we summarize the main work of this paper.In chapter 2,we develop an optimal compact finite difference scheme for the3 D Helmholtz equation.Firstly,we propose a compact finite difference scheme for the 3D Helmholtz equation with constant wavenumbers.A convergence analysis is also provided to show that the scheme enjoys the forth-order accuracy.Secondly,we provide an error analysis between the numerical wavenumber and the exact wavenumber.Based on minimizing the numerical dispersion,a refined strategy is presented to choose optimal parameters for the scheme.In the end,we propose a compact finite difference scheme for the 3D Helmholtz equation with variable wavenumbers.In chapter 3,we provide the preconditioned Bi-CGSTAB solver to solve the resulting linear system.In this chapter,we present the shifted-Laplacian preconditioning technique for the 3D Helmholtz equation,and perform spectral analysis for the preconditioned linear system.From the perspective of linear fractal mapping,we give some theoretical results for the spectrum being enclosed by a certain circle.Based on the shifted-Laplacian preconditioner and the matrix-based multigrid,the implementation of the preconditioned Bi-CGSTAB solver is then provided.In chapter 4,numerical experiments,which include the Helmholtz problem with constant and variable wavenumbers under different boundary conditions,are presented to demonstrate the efficiency of the new compact fourth-order difference schemes and the preconditioned iterative solver.In the last chapter,we give some conclusions of this thesis and propose a prospect of the future work.