(untitled) (Communications in Computational Physics, 2018)
This paper presents new finite difference schemes for solving the Helmholtz equation in the polar and spherical coordinates. The most important result presented in this study is that the developed difference schemes are pollution free, and their convergence orders are independent of the wave number k. Let h denote the step size, it will be demonstrated that when solving the Helmholtz equation at large wave numbers and considering kh is fixed, the errors of the proposed new schemes decrease as h decreases even when k is increasing and kh>1.