二维Helmholtz方程的改进六阶紧致差分法

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

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。

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.