Exact Finite Difference Schemes for Solving Helmholtz Equation at Any Wavenumber

Abstract. In this study, we consider new finite difference schemes for solving the Helmholtz equation. Novel difference schemes which do not introduce truncation error are presented, consequently the exact solution for the Helmholtz equation can be computed numerically. The most important features of the new schemes are that while the resulting linear system has the same simple structure as those derived from the standard central difference method, the technique is capable of solving Helmholtz equation at any wavenumber without using a fine mesh. The proof of the uniqueness for the discretized Helmholtz equation is reported. The power of this technique is illustrated by comparing numerical solutions for solving oneand two-dimensional Helmholtz equations using the standard second-order central finite difference and the novel finite difference schemes.