Research On Brycentric Interpolation Collocation Method In Dynamical Diffusion Problems

Many engineering problems and physical problems can be transformed into initial boundary value problems of differential equations,but most cases cannot get exact analytical solutions.Therefore,numerical methods must be used to solve these problems.The common numerical methods for solving such problems include boundary element method,finite difference method,finite element method,etc.Barycentric interpolation collocation method is a completely new method of numerical solution of partial differential equations in recent years.This article introduces this method to convection diffusion equation and the heat conduction equations.Then combines with numerical examples and verifies the characteristic of high accuracy,and further explores the factors which affect calculation precision.The results show that barycentric interpolation collocation method has the advantages of high accuracy and good adaptability of nodes,and can be applied in the heat conduction equation and the convective diffusion equation.The calculation precision of the Chebyshev node is higher than that of the isometric node,but the precision of the equidistant node can be satisfied with the engineering requirement.The following research work is mainly carried out:(1)Using the gravity interpolation to construct approximate functions of time and space variables and using Kronecker integrative discrete differential operator.Then the differential operator is replaced by the same order differential matrix and the discrete formula is written.(2)Applying the elimination method or the substitution method to add the boundary conditions.The initial conditions and boundary value conditions of the heat conduction equation and the convective diffusion equation are discretized,and the discrete algebraic equations are obtained.The numerical solution is solved by Matlab.(3)Exploring the factors which affect the precision of numerical calculation,such as the number of nodes,the type of nodes and the shape parameter,etc.Seeking the necessary conditions for the highest possible accuracy.(4)On the basis of the center of gravity interpolation of one-dimensional equation,the discrete process of the second barycenter interpolation collocation method is derived.The high accuracy,convenience and good adaptability of the gravity interpolation method are verified by a two-dimensional numerical example.And we get the conclusion that in the two-dimensional,the time direction is unconditional stability.