Discrete implementation of korteweg de vries equation based on a modified “Cabaret” scheme

Abstract

The work is focused on creation and investigation of the scheme applied to solving the problems related to non-linear wave processes described by Korteweg de Vries model equation. The behavior of the numerical solution in this case depends mostly on the difference scheme choice. The article proposes to implement a problem numerical solution on the basis of the improved “cabaret” scheme. Its difference operator is a linear combination of difference schemes operators called “cross” and “cabaret”, while the modified scheme is obtained from schemes with optimal weight coefficients. Subject to certain values of the weighting coefficients, the combination leads to mutual compensation of approximation error, and the resulting scheme acquires better properties than the original schemes. The scheme called “left corner” was considered as a test finite-difference scheme for the results assessment in a model problem numerical solution. The results of stability and accuracy investigation showed that the proposed difference scheme has the same limitations as the “left corner” scheme with an approximation error equal O(τ2+ h3).