Finite Difference Scheme Of Helmholtz Euquation With Neumann Boundary Problem

The Helmholtz equations are very important in engineering and scientific applications. Such as in electromagnetic wave diffraction problem and cycle power systems, both of them can be modeled with the Helmholtz equations with different boundary conditions do not have exact solutions, so we try our best to minimize the error to get the approximate solution.We firstly introduce the Helmholtz equation with constant coefficients and the way to construct six order finite difference schemes. In this way, we use HODIE methods to get difference of right thing. We also introduced Neumann boundary problem of the Helmholtz equation with fourth order accuracy difference schemes. Based on above information, we use two different methods to treat Neumann boundary condition, and the accuracy of the discrete boundary condition can reach up to fifth order. The first method we use the adjacent points of the Neumann boundary to approximate the derivatives of Neumann boundary for the Poisson equation. Then we apply such scheme to the Helmholtz equation, with the analysis of the differences between these two equations make up the difference to get the part. The second approach is taken by using more Neumann boundary points, Using the first derivatives at these points, we then get a relationship between these points and finally come to a proper finite difference scheme. Both of these methods can improve the accuracy of the Helmholtz equation with Neumann boundary problem and get its difference schemes. At last, several numerical examples are illustrated, which prove numerically that both of the schemes are fifth order accurate.