The Numerical Simulation Of The Two-dimensional Helmholtz Equation On Unbounded Domains Based On The Compact Finite Volume Method

The Helmholtz equation describes a kind of wave propagation phenomenon,including sound wave,electromagnetic wave,light radiation,etc,which plays an im-portant role in engineering practice and science and technology,and it has important research significance.In this paper,we consider the numerical solution of two-dimensional Helmholtz equation as follows:?u + k~2u? f(x,y),where ?u =(a2)/(ax~2)+a~2/(ay~2)represent two-dimensional Laplace operator,k =2?f/? is the wave number,f and v represent frequency and speed,respectively.The unboundedness of the region brings great difficulty to solving the the nu-merical solution of the above problems.At present,the artificial boundary method is one of the effective methods to solve this difficulty.After the artificial boundary is introduced,the unbounded region is divided into two parts:bounded computing re-gions and unbounded exterior regions,and artificial boundary conditions are added to the artificial boundary,so that the boundary value problem with artificial bound-ary conditions in the finite computing area is a good approximation to the original problem.In this paper,we consider the artificial boundary with positive thick-ness:an artificial layer that can enclose the computation area,named the perfectly matched layer(PML),which can absorb most of the external waves and internal reflections.Therefore,the two-dimensional Helmholtz equation in the unbound-ed region is reduced to the initial boundary value problem in bounded computing regions,which can solve the problem.The main content of this article includes the following parts:part one,we studies the physics background of Helmholtz equation,and analyzes the perfect matching layer and domestic research status.Part two,we study the numerical format of the compact finite volume method of the two-dimensional Helmholtz equation in the bounded domain,so as to achieve the fourth-order precision and the validity of the numerical example analysis method.The third part,the research on two-dimensional unbounded region Helmholtz equation of compact finite volume method of numerical format to achieve fourth order accuracy in the inner region and second order accuracy on the PMLlayer,and numerical examples are used to analyze the effectiveness of the method.Part four,we will make a summary of completed work and outlook on unfinished work.