二维Helmholtz方程的改进六阶紧致差分法

为得到求解二维Helmholtz方程的高精度差分法, 构造了一种改进六阶紧致差分格式: 首先, 给出一种带优化参数的六阶紧致差分格式的截断误差; 然后, 对此截断误差的部分项进行二阶紧致逼近, 得到一种改进紧致差分格式; 其次, 对该格式进行了收敛性分析, 证明其为六阶收敛的; 最后, 基于极小化数值频散的思想, 给出该格式优化参数的加细选取策略。与带优化参数的六阶紧致差分格式相比, 数值实验说明改进六阶紧致差分格式的数值精度有了显著提高, 且其误差对波数k的依赖性更低。

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.