MQ拟插值求解KdV方程

Applying Multiquadric Quasi-Interpolation to Solve KdV Equation

中文关键词:  KdV方程  MQ拟插值  数值解.

英文关键词:KdV equation  multiquadric(MQ) quasi-interpolation  numerical solution.

中文摘要:

      陈荣华和吴宗敏提出了一种推广的$\mathscr{L_D}$拟插值算子, 并且将其用于求解Burgers方程. 在其思想的启发下, 本文提出了一种求解KdV方程的数值方法. 在本文的方法中, 空间变量的3阶导数用空间变量1阶导数的2阶中心差商近似, 时间变量的1阶导数用其1阶向前差商近似, 而对于空间变量的1阶导数, 则用广义的$\mathscr{L_D}$拟插值算子的1阶导数来近似. 由于不需要求解线性方程组, 所以算法简单并且易于实现. 数值试验表明本文的方法是可行, 有效的.

英文摘要:

      Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations. Based on the good performance, Chen and Wu presented a kind of multiquadric (MQ) quasi-interpolation, which is generalized from the $\mathscr{L_D}$ operator, and used it to solve hyperbolic conservation laws and Burgers' equation. In this paper, a numerical scheme is presented based on Chen and Wu's method for solving the Korteweg-de Vries (KdV) equation. The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the derivative of the generalized $\mathscr{L_D}$ quasi-interpolation operator. The algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.