求解Helmholtz方程的无网格重心插值配点法

Meshless Collocation Methods with Barycentric Interpolation for Solving the Helmholtz Equation

本文针对Helmholtz方程,借助Chebyshev插值节点,运用重心Lagrange插值基函数和重心有理插值基函数推导了求解该类方程的两种无网格配点法.首先,将插值基函数应用于空间变量及其偏导数,建立了基于配点法的二阶微分方程组.其次,在给定的插值节点上,利用微分矩阵对其进行了简化.最后通过三种测试节点来计算数值算例,从而验证了本文方法不仅可以计算大波数问题,还可以计算变波数问题,并且算法具有精确稳定、计算量小和高效等优点.

In this paper,Chebyshev interpolation nodes,barycentric Lagrange interpolation basis function and barycentric rational interpolation basis function are used to deduce two type of meshless collocation methods for solving the Helmholtz equation.First of all,the two kinds of interpolation basis functions are applied to treat the spatial variables and its partial derivatives,and the collocation method for solving the second order differential equations are established.Secondly,the differential matrices are used to simplify the given differential equations on a given interpolation node.Finally,based on three kinds of test nodes,numerical experiments show that the present method can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high accuracy,good numerical stability,cost-effective and high efficiency.