对流-扩散方程数值解的四次B样条方法

Quartic b-spline methods for numerical solutions convective-diffusion equations

基于四次B样条函数,提出一种求解一类对流-扩散方程的四次B样条方法。首先利用光滑余因子协调法,得到有界闭区间上具有均匀节点的一元四次B样条基函数表达式。接着计算在有界闭区间两端点处具有重节点的几种不同情况下的B样条基函数表达式,这些样条基函数具有非负性、单位分解性等良好的性质。然后将一元四次B样条函数应用于求解一类一维对流-扩散方程,其中对于对流-扩散方程的离散过程,对于时间变量的离散采用向前有限差分,而对于空间变量的离散,引入参数δ,建立四次样条逼近格式。之后利用四次B样条函数去求解该对流-扩散方程。最后通过具体算例,将四次样条逼近方法与有限差分方法进行比较,且给出直观的数值误差对比,由此说明样条逼近方法更加简便实用。

Based on the quartic B spline function,a quartic B spline method for solving a class of convection-diffusion equations was presented.By means of the conformality of smoothing cofactor method,univariate quartic B spline bases were firstly obtained over uniform knots.Secondly,representations of B splines were calculated at the endpoints with multiple knots on the bounded closed interval.In addition,all the quartic B spline bases possessed good properties,such as non-negative property and partition of unity.Thirdly,univariate quartic B spline functions were applied in solving a class of one-dimensional convection-diffusion equations,where discrete in time was realized by forward finite differences and discrete in space was by quartic spline approximation with the parameterδintroduced.Then the convection-diffusion equation was solved by the quartic B spline functions.Finally,according to the numerical example,the comparison was made between the quartic spline approximation method and the finite difference method,and an intuitive comparison of numerical errors was given,which indicates that the former is more convenient and practical than the latter.