A bivariate spline collocation solution of the Korteweg-de Vries equation

Abstract

The Korteweg-deVries (KdV) equation is solved numerically using bivariate spline collocation methods. Our methods permit one or two collocation points in time with an arbitrary number of collocation points in space. The basis functions for the underlying spline spaces are β-splines (in space) and Lagrange polynomials (in time). Numerical experiments show that collocation at two Gauss points in both space and time yields accurate solutions very efficiently. There is numerical evidence for fourth-order convergence in time (at the mesh points), but this is not proved. Using a suite of well known test problems, the methods are compared with the classical method of Zabusky and Kruskal, a convenient and frequently used reference standard for numerical KdV solvers.