Helmholtz方程有着广泛的物理背景和众多的应用领域,如电磁波传输问题，水下声波的传播、散射等。由于问题本身的特点和数学模型的复杂性为数值计算带来了挑战，尤其是无界域和大波数问题的数值处理，许多实际问题在极坐标或球坐标下处理会带来降维的优势，但因此需要处理界面问题，因此建立求解带有界面的Helmholtz方程的高效、稳定、易于实现的数值计算方法具有非常重要的理论和实际应用价值。本项目拟采用有限差分方法针对带有界面的二维、三维Helmholtz方程在极坐标系、球坐标系下建立高阶紧致差分格式，结合浸入界面方法来处理界面问题；在无界域上带有界面Helmholtz方程分别采用完美匹配层，人工边界条件及项目组设计的边界条件进行处理，对所构造方法进行理论分析；建立非均网格上的高阶紧致差分格式，对离散后的方程组开发相应的预条件子，构造具有并行性质的迭代方法如CARP-CG对离散后的方程组进行求解。

A variety of physical phenomena, including acoustic and electromagnetic waves transmission and scattering are described by the Helmholtz equation, it is a big challenge for researcher to numerical solving Helmholtz equations on unbounded domain and with high wave number. We have to deal with interface problem when the advantage of lowing dimension is taken to solve Helmholtz equations in polar coordinates or spherical coordinates. Iit is very important to develop efficient numerical methods for solving Helmholtz equations with interface. The project aim to establish high order compact finite difference method for 2D and 3D Helmholtz equations in polar coordinates or spherical coordinates, solve interface problem with immersed interface methods; solve Helmholtz equation with interface problem by matching different boundary condition skill, such as perfectly matched layer, artificial boundary condition and ours, discuss the relationship between wave number and space step size , develop the finite difference method to apply on nonuniform grids ,design ideal preconditioner for linear system and solve it by iteration methods with paralleling quality like CARP-CG method.