Applying Explicit Schemes to the Korteweg-de Vries Equation

Abstract

The water wave soliton is a result of a dynamic balance between dispersion and nonlinear effects. It brings together many branches of mathematics, some of which touch on deep ideas. The Korteweg-de Vries equation is typical of all model equations of nonlinear waves in the soliton phenomena. Four explicit difference schemes are used in order to approximate the Korteweg-de Vries equation, namely; (a) a First order upwind scheme, (b) the Zabusky-Kruskal scheme, (c) the Lax-Wendroff scheme, and (d) the Walkley scheme. Our main interest was to analyse which explicit scheme among the four performs well when implemented to the KdV equation to produce the best soliton solution. Hence, reviewing and considering existing schemes. Three sets of initial data are used to explore the numerical approximations. Two, including the data proposed by Zabusky and Kruskal, involve a single soliton wave, whilst the other involves the separation into two solitons, which will interact in time. These initial conditions and periodic boundary conditions are described in detail taking into account physical, mathematical and computational considerations. Accuracy, consistency and Fourier stability in regard to the four explicit schemes for the Korteweg-de Vries equation are discussed. Numerical results are reported for a single soliton solution and the separation into two solitons with different velocities are investigated. Graphical results are presented to show how well these four schemes agree well with each other. After comparing the four explicit schemes, the best scheme was the Zabusky and Kruskal scheme since it is a two-step scheme, which uses the explicit leapfrog finite difference scheme and was good for low amplitudes and less running time was needed than the other three explicit schemes.