High Order Compact Schemes For The Helmholtz Equation In Polar Coordinates

Helmholtz equation is a kind of important physical equation, which is often encountered in engineer-ing field, such as wave guide problems in electromagnetic field, the control of noise, thin film vibration problem. In recent years, many researchers have done a large amount of work for Helmholtz equation us-ing finite volume method, finite element method and finite difference method in the Cartesian coordinates when wave number is continuous. The accuracy of those numerical methods can be reached the fourth order, sixth order or even higher. When the wave number is piecewise constant, the solving accuracy with finite difference method for solving Helmholtz equation can not be reached the expected one, even the method does not work. For the case, the immersed interface method can effectively deal with It. Based on this method, the point on one side of the interface can be expressed to the form about the other side by utilizing the jump condition, then the desired accuracy will be got.In this thesis, we focus on solving the Helmholtz equation with discontinuous wave number in po-lar coordinate by finite difference method and immersed interface method. First of all, one dimension-al Helmholtz equation with continuous wave number is solved by the fourth order compact difference scheme developed in literature. Under the combination of the immersed interface method and Taylor se-ries expansion, one dimensional Helmholtz equation with piecewise wave number is solved. Then, we added correction terms on the scheme for the case of continuous wave number, and applied jump con-ditions to express the grid points near the interface to the other side of it. Consequently, the coefficient of correction term is determined, and accordingly, the fourth order compact difference scheme of one dimensional Helmholtz equation with piecewise wave number is finally established.Secondly, for solving two-dimensional Helmholtz equation with continuous wave number, we changed the equation into two Laplacian operators, discretized them respectively and combined the fourth-order compact difference scheme of two operators together. When wave number is discontinuous, similar to one dimensional case, we determined the coefficient of correction terms by applying jump conditions to expressing the grid point near the interface to the other side of it. In this way, the two-dimensional fourth-order compact finite difference format of Helmholtz equation in polar coordinates system when wave number is discontinuous can be developed. Finally, the accuracy and efficiency of the proposed method are validated through the numerical experiment of examples.