A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation

Abstract

In this article, we use a multilevel quartic spline quasi-interpolation scheme to solve the one-dimensional nonlinear Korteweg–de Vries (KdV) equation which exhibits a large number of physical phenomena. 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 proposed quasi-interpolation operator. Compared to other numerical methods, the main advantages of our scheme are the higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement. Numerical experiments in this article also show that our scheme is feasible and valid.

Keywords:

• Korteweg–de Vries equation
• soliton
• dispersion
• multilevel B-spline quasi-interpolation

Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant Nos U0935004, 11071031, 10801024), the Fundamental Research Funds for the Central Universities (DUT10ZD112, DUT11LK34), and the National Engineering Research Center of Digital Life, Guangzhou, China.