Numerical solution of Korteweg-de vries equation

Abstract

The Kotteweg-de Vrr-es(KdV)is a mathematical model of waves on shallow water surfaces. The mathematical theory behind the KdV equation is rich and interesting, and, in the broad sense, is a topic of active mathematical research. The equation is named after Diederik Korteweg and Gustav de Vries, It has long been known that conservative discretization schemes for the KdV and other nonlinear equations tend to become numericrtlly unstable. Although finite difference approximations have been used, there are always instabilities of the solutions obtained, In this work we solved the Korteweg-ds Vries (KdV) equation using an explicit finite difference method, subject. to various boundery conditions which are travelling wave solutions to the KdV equation. The methodology involved carefully designing conservative finite difference discretization that can remain stable and deliver sharp solution profiles fora long time. We then determined the accuracy of the finite diffurence scheme by comparing the graphical outputs of the numerical results.