Abstract
In this paper, the Crank-Nicolson extrapolation scheme is 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 Crank-Nicolson scheme for the linear term and the semi-implicit extrapolation scheme for the nonlinear term. Moreover, in numerical analysis, we split the error function into two parts, one from the spatial discretization and other one from the temporal discretization, by introducing a corresponding time-discrete system. Then, based on some regularity assumptions, we present optimal error estimate and prove that the scheme is almost 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.
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