A second order convergent difference scheme for the initial‐boundary value problem of Korteweg–de Vires equation

Abstract

AbstractIn this article, we present a two‐level implicit difference scheme for Korteweg–de Vires equation with the initial and boundary conditions by the method of order reduction. The truncation error of the difference scheme is analyzed in detail. In the practical computation, the introduced intermediate variable is decoupled in order to reduce the computational cost. It is proved that the difference scheme is solvable by the Browder theorem. The conservation, boundedness, and the unconditional convergence of the numerical solution are also analyzed at length. The convergence order is two both in space and in time in L2‐norm. The numerical solution is proved to be unique under the optimal step ratio condition. Numerical examples demonstrate that the theoretical analysis is correct.