High-order Difference Methods For The Helmholtz Equation

The Helmholtz equation is widely used in wave propagation,scattering theory and other scientific fields.Therefore,numerically solving the Helmholtz equation has been a hot topic.The main difficulty for numerically solving the Helmholtz equation is the problem with high wavenumbers.Because the solution of the Helmholtz equation with high wavenumbers oscillates violently,the corresponding numerical solution's accuracy decreases while the wavenumber increases.Therefore,it is crucial to solve the problem with high wavenumbers.This thesis is divided into five chapters.The first chapter is the introduction.In this chapter,the background of the Helmholtz equation is presented,and the classical numerical methods for the Helmholtz equation is recalled.Furthermore,the main work of this thesis is summarized in this chapter.In Chapter 2,we develop an optimal compact sixth-order difference scheme for the 3D Helmholtz equation.Firstly,we construct a compact sixth-order difference scheme with constant wavenumbers,and a convergence analysis is provided to show that the scheme is sixth-order in accuracy.Secondly,based on minimizing the numerical dispersion,we propose a refined choice strategy for choosing optimal parameters of the finite difference scheme.In the end,we establish a compact sixthorder difference scheme for the Helmholtz equation with variable wavenumbers.In Chapter 3,we construct an improved compact sixth-order difference scheme for the 3D Helmholtz equation.The truncation error of the compact sixth-order difference scheme proposed in chapter 2 is further dealt with,and then an improved compact sixth-order difference scheme with higher accuracy is proposed.The accuracy of this scheme is less dependent on the wavenumber.We begin with establishing an improved sixth-order difference scheme with constant wavenumbers,and analyze the convergence of the scheme.Then,based on the idea of minimizing the numerical dispersion,we propose a refined strategy for choosing the optimal parameters of the finite difference scheme.Finally,an improved compact sixth-order difference scheme is presented for the Helmholtz equation with variable wavenumbers.In Chapter 4,numerical experiments are given to demonstrate that the proposed schemes not only suppress the numerical dispersion,but also improve the accuracy.In Chapter 5,the conclusion of this thesis is presented,and the research work in the future is also discussed.