Helmholtz方程基于变限积分法的数值求解

The Variable Limit Integral Method for Helmholtz Equation

Helmholtz方程是一类描述电磁波的椭圆型偏微分方程,在力学、声学和电磁学等领域应用广泛。为了消除因高波数引起的污染效应,数值求解Helmholtz方程的传统方法是对网格进行加密,网格加密不仅增加了时间复杂度,且离散后的矩阵通常是病态的。因此,寻求对任意波数都有效的方法是必要的。在有限体积法的基础上,引入变限因子,将微分方程完全转换成积分方程,利用一元三点和二元九点Lagrange插值公式,构造含三对角矩阵的离散格式,分别对一维和二维Helmholtz方程进行变限积分法的数值求解。该方法适用于任意波数,求解过程物理意义明确,数值格式简单。对于一维Helmholtz方程研究了变限因子对误差的影响,利用Taylor展式及Lagrange插值余项公式进行误差估计,证明离散格式的截断误差达到二阶。数值实例表明该离散格式的变限因子和步长相等时,误差阶较低。对二维Helmholtz方程,探究不同波数对数值解的影响,证明离散格式的截断误差达到三阶。数值实例表明,对于不同的波数,数值格式都有较好的精度,高波数没有引起污染效应。

Helmholtz equation is a class of elliptic partial differential equations that describe electromagnetic waves,and are widely used in mechanics,acoustics,electromagnetism,and other fields.In order to eliminate the pollution effect of high wavenumbers,the traditional method for numerically solving Helmholtz equation is to refine the grid,which not only increases the time complexity,but also usually makes the discrete matrix ill conditioned.Therefore,it is necessary to find an efficient numerical method for any wavenumbers.Based on the finite volume method,variable limit factors are introduced to completely convert the differential equations into integral equations.A discrete scheme containing a tridiagonal matrix is constructed using the univariable three-point and bivariable nine-point Lagrange interpolation formulas to perform numerical solutions of one-dimensional and two-dimensional Helmholtz equations using the variable limit integration method,respectively.The proposed method is suitable for arbitrary wave numbers,and the physical meaning of the solution process is clear.For the one-dimensional Helmholtz equation,the influence of the variable limit factor on the error is studied.The error estimation of the numerical solution is performed using Taylor expansion and Lagrange interpolation residual formula,and it is proved that the truncation error of the discrete scheme reaches second order.Numerical examples show that when the variable limit factor and step size of the discrete scheme are equal,the error order is lower.For the two-dimensional Helmholtz equation,the influence of different wave numbers on the numerical solution is investigated.It is proved that the truncation error of the discrete scheme reaches third order.Numerical examples indicate that the numerical scheme has good accuracy for different wave numbers,and high wavenumbers do not cause the pollution effect.