Sixth-Order Finite Difference Scheme For The Helmholtz Equation With Inhomogeneous Robin Boundary Condition

The Helmholtz equation is a partial differential equation which describes wave propagation in the frequency domain.This equation has a wide range of applications in acoustics,electromagnetic,seismology and other related research areas.The processes such as acoustic wave propagation in time harmonics,electromagnetic wave scattering in specific cases,and seismic wave diffusion can be simplified to this model under certain conditions.Therefore,efficient numerical methods for the Helmholtz equation has been a research central issue.However,due to the special properties of this model,many numerical schemes have poor approximation effects when simulating it.The main reason is that the model is often built on an unbounded area,and when the propagation frequency is high,the solution will violently oscillate.For the former,the existing research generally converts an unbounded region into a bounded computing region by giving an appropriate radiation boundary condition(such as Robin boundary condition).For the latter,designing high-order high-precision schemes is a common processing technique.So studying the high-order scheme for the Helmholtz equation with Robin boundary conditions is a frequently used approach to overcome the difficulties encountered in the calculation of this problem.This article is based on this aspect.The first chapter of this paper describes the research background,significance and current situation of this model problem.In second Chapter,in view of the inhomogeneous Robin boundary conditions in two-dimensional space,this paper first derived a special fourth-order finite difference scheme,and used different approximations to the high-order derivatives to ensure the successful processing of the ghost points;Further,based on a special fourth-order scheme,the paper have studied the general derivation of the inhomogeneous Robin boundary condition sixth-order finite difference scheme in two-dimensional space.Because the order of the derivative to be processed is higher at this time,the process is more complicated.This is also an important difficulties in the processing process.In third Chapter,we carried out numerical experiments by using the above scheme through several calculation examples,showing the calculation errors and convergence order results of different difference schemes when simulating the Helmholtz equation,and verified the correctness of the scheme derivation process and effectiveness of the scheme.Finally,this paper summarizes the work done and looks forward to possible future work.