Iteration Method For Solving Helmholtz Equation With Variable Coefficients

Many physical phenomena in life can be modeled by Helmholtz equation,such as Time-Harmonic Wave propagation,underwater acoustics,aeroacoustics,electromagnetic scattering,etc.In recent years,scholars at home and abroad have done a lot of research on solving Helmholtz equation by using finite element method,finite volume method and finite difference method.In real life,there are some problem that need to be modeled by Helmholtz equation with variable coefficients due to some changes in physical properties,or time-harmonic waves propagation in inhomogeneous medium.First of all,a simple transformation for the 2D variable coefficient Helmholtz equations is made.The center difference is used to get the second order compact finite difference schemes.And then,the Taylor expansion is used to analysis the truncation errors for the finite differences and to eliminate the error terms.And the fourth order compact finite difference schemes for the 2D variable coefficient Helmholtz equations is developed,at the same time the fourth order compact finite difference schemes for Local Sommerfeld-type boundary condition is considered.The similar dteps can be used for 3D,and got the fourth order compact finite difference schemes for the 3D variable coefficient Helmholtz equations.Finally the GMRES(m)iteration method is applied to solve linear system.The numerical results show effectiveness and feasibility.The results show that the solution of the system of linear equations,especially on the three dimensional problem solving,GMRES(m)algorithm has great advantage.