Volterra积分-微分方程的高精度数值算法研究

Research on high-precision numerical algorithm of Volterra integro-differential equation

提出了求解Volterra积分-微分方程的一种高精度数值解法:重心插值配点法,即重心有理插值配点法和重心Lagrange插值配点法.该方法分为两步,首先,对Volterra积分-微分方程采用重心插值配点法进行离散,构造出相应的离散格式;其次,依次选取第二类Chebyshev节点和等距节点进行数值计算.对Volterra积分-微分的积分项中的未知函数作出改变,将其替换为相应的导数.数值结果表明,当选取第二类Chebyshev节点时,重心有理插值配点法和重心Lagrange插值配点法对修改后的方程仍然具有较高的计算精度;然而,选取等距节点时,重心有理插值配点法依然保持着很高的计算精度,但是重心Lagrange插值配点法的计算精度有明显下降.

In this paper,a numerical method for solving Volterra integro-differential equation is proposed:barycentric interpolation collocation method(Barycentric Lagrange interpolation collocation method and Barycentric rational interpolation collocation method).There are two steps in this method.Firstly,Volterra integro-differential equation is discretized by the mentioned method,and the corresponding discretization scheme is constructed.Secondly,the second type of Chebyshev nodes and equidistant nodes are selected for numerical calculation.In this paper,the unknown function in the integral term of Volterra integral differential is replaced by the corresponding derivative.The numerical results show that both the barycentric rational interpolation collocation method and barycentric Lagrange interpolation collocation method still have good computational effect on the modified equation.However,when equidistant nodes are selected,barycentric rational interpolation collocation method still has a high calculation accuracy,but the calculation accuracy of barycentric Lagrange interpolation collocation method has dropped significantly.