An Improved Compact Sixth-order Finite Difference Scheme for the 2D Helmholtz Equation

To obtain a finite difference scheme with high accuracy for solving the 2D Helmholtz equation, an improved compact sixth-order difference scheme is constructed. Firstly, the truncation error of a compact sixth-order difference scheme with optimal parameters is presented. Then, some terms of the truncation error are approximated with second-order compact formulas to obtain an improved compact difference scheme. Next, the convergence analysis of the improved compact difference scheme is given, and it is proved that the proposed scheme enjoys sixth-order convergence. Based on minimizing the numerical dispersion, a refined choice strategy is proposed for choosing weight parameters. Compared with the compact sixth-order difference scheme with optimal parameters, numerical experiments show that, the numerical accuracy of the improved compact sixth-order difference scheme has been significantly improved, and the scheme's error is less dependent on the wavenumber k.