Numerical Computation of Korteweg-de Vries (KdV) Equation Using Finite Difference Approximation

Abstract

In this paper, we study a general form of third order nonlinear partial differential equation known as Korteweg-de Vries (KdV) equation. A traveling wave solution method is discussed for analytic solution of the general form of KdV equation. In order to understand the effect of convection and dispersion terms of the equation we present a numerical evaluation of the analytical solution for various values of convection and dispersion coefficients. Finite difference scheme for the numerical solution of the KdV equation is investigated and stability condition for a first-order scheme using convex combination method is determined. Von Neumann stability analysis is performed to determine the stability condition for a second order scheme. We present error estimation of the numerical schemes and verify qualitative behavior of the KdV equation.