Accurate Finite-Difference Methods for Helmholtz and Wave Equations

Accurate finite difference methods (FDMs) for the numerical solution of two-dimensional Helmholtz and one-dimensional wave equations are proposed. The accurate finite difference equations and the boundary conditions are formulated as algebraic and algebraic-eigenvalue problems. Calculation schemes for solving these problems using conventional numerical methods and Newton’s method or its continuous analogue, and the corresponding test exactly solvable examples are presented and analyzed. A new exact FDM for solving the two-dimensional Helmholtz equation is also presented. This method is implemented by solving two one-dimensional problems under the separation of variables restriction. The main feature of this method is that it can be applied for solving the two-dimensional Helmholtz equation with any wavenumber without using a fine mesh. The method accuracy is analyzed theoretically. The effectiveness and performance of the proposed FDMs are demonstrated by comparing known methods with several examples.