圣维南方程的3次B样条拟插值数值解

Numerical Solution of Saint-Venant Equation by Cubic B-spline Quasi-interpolation

首先针对不同阶的连续可导函数,对3次B样条拟插值算子进行相应的误差估计。其次将3次B样条拟插值方法用于求解圣维南方程,利用3次B样条拟插值的一阶导数近似圣维南方程的空间导数,同时使用向前差分近似其一阶时间导数,求出其数值解。最后将所得结果与4阶龙格库塔法和蛙跳格式所得数值解进行对比分析,结果表明3次B样条拟插值方法具有一定的优越性。

Firstly,the error estimates of cubic spline quasi-intepolating operators are derived for continuous differential function with different orders.Secondly,cubic B-spline quasi-interpolation is used to get the numerical solution of Saint-Venant equation.Specifically,the derivatives of the quasi-interpolation are used to approximate the spatial derivative of the dependent variable and forward difference method is used to approximate the time derivative of the dependent variable.Finally,the numerical solutions are compared with the solution obtained by the fourth order Runge-Kutta method and the leapfrog scheme.Then numerical examples show that cubic spline quasi-intepolating method has some advantages.