CONVERGENCE OF A CONSERVATIVE CRANK-NICOLSON FINITE DIFFERENCE SCHEME
FOR THE KDV EQUATION
Abstract
In this paper, we study the stability and convergence of a conservative
Crank-Nicolson finite difference scheme applied to the Korteweg-De Vries
(KdV) equation endowed with initial data. We design a three-point
average scheme associated to the convective term and the dispersion term
is discretized by certain discrete operators along with the
Crank-Nicolson scheme for the temporal discretization to establish that
the proposed scheme is L 2 -conservative. The convergence analysis
reveals that utilizing inherent Kato’s local smoothing effect,
the proposed scheme converges to a classical solution for sufficiently
regular initial data u 0 ∈ H 3 ( R ) and to a weak solution in L 2 ( 0 ,
T ; L loc 2 ( R ) ) for non-smooth initial data u 0 ∈ L 2 ( R ) .
Optimal convergence rates in both space and time for the devised scheme
are derived. The theoretical results are justified through several
numerical illustrations.