A Noncompact Optimal Fourth-order Finite Difference Method For The Helmholtz Equation

The Helmholtz equation has great significance in theory and application,which is frequently used to model the physical phenomena,such as wave propagation and inverse scattering in the scientific fields of acoustics,optics and electromagnetics.The Helmholtz equation finds wide applications in the field of aerospace,marine technology and oil-gas exploration.Currently,it is an important research content to establish an effective method for solving the Helmholtz equation numerically in many areas of science and technology.Especially for the circumstances of large wavenumbers,the accuracy of the numerical solutions usually deteriorates with the increasing wavenumbers,due to the terrible oscillation of the Helmholtz equation.Therefore,it is still a challenging task to solve the Helmholtz equation with high wavenumbers in the field of computational science.In this paper,we aim to deveop an efficient high-order finite difference method for solving the Helmholtz equation based on the Taylor's formula and the mechanism of minimizing the numerical dispersion.The organization of this paper is as follows.Chapter 1 is mainly introduces the background of Helmholtz equation,and points out the difficulties encountered in the numerical solution of the Helmholtz equation.In Chapter 2,we propose a non-compact optimal fourth-order 25-point finite difference method for solving the Helmholtz equation.In Section 1,three sets of fourth-order difference schemes are constructed to discretize the higher-order derivative term(35)u.In Section 2,another three sets of fourth-order difference schemes are desgined to approximate the zero-order derivative termsk ~2u.Then,we obtain a noncompact optimal 25-point fourth-order finite difference method by combing linearly the above schemes.Finally in Section 3,the convergence analysis are presented.Chapter 3 is devoted to the dispersion analysis of the non-compact optimal fourth-order 25-point finite difference method method.Section 1 performs the numerical dispersion analysis,based on which the dispersion error and dispersion relation is given.Then,in Section 2,the weight parameters are determined by minimizing the dispersion error.To illustrate the efficiency of the new schem in suppressing the numerical dispersion,Section 3 gives the normalized phase velocity curve diagram.In Chapter 4,an extended MIB(Matched Interface and Boundary)method is developed to deal with the discretization on the points which is near the boundary.An introduction to the existing boundary-processing method is given in Section 1.Then,we reviews the classical MIB method for the central FD methods in Section 2.Finally,we proposes an extended MIB method for the non-compact optimal fourth-order 25-point finite difference method in Section 3.In Chapter 5,various numerical examples are given to demonstrate the efficiency of the new scheme.We also compare it with other existing difference schemes.And some conclusions and prospects are presented.