Study On A High-Order Finite Difference Method For The Helmholtz Equation With Discontinuous Coefficients

There are a broad range of physical background and many application fields described by Helmholtz equation, such as the acoustic scattering problem, the vibration of the structure and electromagnetic scattering problems, etc. In actual application, the analytical solutions of Helmholtz equation are usually harder to obtain, hence, we often use the finite difference method, finite element method, finite volume method and other numerical methods to find out the higher precision numerical solution. Due to the characteristics of the problem itself and the complexity of mathematical model for numerical calculation which have brought many challenges, especially for the unbounded and large wave number problems, there are still lots of problems unsolved. Effective numerical calculation method, in particular, has yet to be further study.In general, through interface, the solution of the interface problem of partial differential equation is discontinuous. When the finite difference method is used to solve the Helmholtz equation with interface problem, and cannot guarantee the corresponding order as the same as the expected order, we can use the immersed boundary method, the harmonic average method and immersed interface method to achieve the expected results.The thesis mainly employs the immersed interface method used for dealing with Helmholtz equation with discontinuous coefficient and the singular source term. First of all, the fourth order and sixth order compact difference schemes are utilized for 1D and 2D Helmholtz equation with continuous wave number which are proposed within literatures, then combined with the immersed interface method, and established fourth order and sixth order compact difference schemes for 1D Helmholtz equation with discontinuous wave number.The corresponding order Neumann boundary conditions are proposed. Secondly, the proposed schemes are extended to solve 2D Helmholtz equation, fourth order compact difference scheme is established for Helmholtz equation with discontinuous wave number. In addition, when the sixth order difference scheme is constructed for solving 2D Helmholtz equation with piecewise wave number, sixth order difference scheme is used far away the interface, while on the interface, fourth order difference scheme is adopted on the interface in order to keep the compactness of the finite difference schemes, and the global accuracy can achieve fourth order. The fourth order difference scheme is adopted for approximating Neumann boundary condition for 2D problems. In the end, the numerical experiments are conducted to verify the accuracy and efficiency of the proposed schemes.