Finite Difference Method For Helmholtz Equation With PML Boundary Condition

The computational area of wave propagation in the medium is unbounded,When numerical methods such as finite element,finite difference,and finite volume are used for numerical solution,they must be limited by computer storage and time.To regulate the contradiction between the infinite wave propagation and the finiteness of the calculation area,the most direct way is to artificially cut off the wave propagation area and turn it into a bounded area.The wave reflection phenomenon occurs at the boundary,When domain is truncated directly,the reflected wave and the incident wave will be superimposed in the calcu-lation region,which will have a greater impact on the solution in the calculation region.Therefore,how to use effective methods to eliminate or reduce the reflection of waves at the boundary is a key issue in numerical simulation of wave propagation.On the basis of the literature,Firstly the Helmholtz equation is truncated at the boundary with Per-fectly Matched Layer,which transforms the Helmholtz equation into a variable coefficient equation.By analyzing the exact solution error and truncation error,the parameters in the PML boundary conditions are selected,including thickness,attenuation function,etc.Secondly The Helmholtz equation with vari-able coefficients is discretized by the finite difference method,When solving the equations,the coefficient matrix is subjected to product polynomial preconditoned.Finally the numerical example is used to verify the feasibility of PML in the truncated wave propagation and the efficiency of preprocessing for solving algebra equations.