Study On The Unconditional Convergence Schemes For The Korteweg-de Vries Equation

In this paper a full discrete Euler semi-implicit finite element scheme and the Crank-Nicolson extrapolation scheme are proposed for the Korteweg-de Vries equation,where a finite element method is applied for the spatial approximation.The time discretization is based on the Euler scheme and Crank-Nicolson scheme for the linear term and the semi-implicit extrapolation scheme for the nonlinear term,respectively.Moreover,in numerical analysis,we split the error function into two parts,one from the spatial discretizat,ion and other one from the temporal discretization,by introducing a corresponding time-discrete system.Then,based on some regularity assumptions,we present optimal error estimates and prove that the scheme is unconditionally convergent,i.e.,the scheme is convergent when the time step is less than or equal to a constant.Finally,numerical tests confirm the theoretical results of the presented method.