On the formulation of energy conservation in the eeKdV equation

Abstract

The Korteweg-de Vries (KdV) equation is a well-known model equation for unidirectional shallow-water (long) surface waves. The equation includes dispersion and weak non-linearity. The derivation of the equation originates in assuming that the velocity potential takes the form of an asymptotic expansion, and applying this in the classical surface wave problem. While a typical assumption on the relative size of non-dimensional key parameters introduced in the derivation will give the KdV equation as a final result, one can change the assumption on the relative size of parameters, and end up with an equation including terms in higher orders in desired parameters. The present article presents the derivation of an extended form referred to as the eeKdV equation.Information regarding various properties of the flow can be found by studying the derivation of the eeKdV equation itself, and some of the relations found can be used for studying the energy balance of a system modeled by the equation. In line with previous work for the KdV equation, we present here corresponding formulations of energy balance laws for an inertial reference frame, in context of the eeKdV equation. We also present a partial verification of the KdV and eeKdV energy flux expressions by looking at a far-field, uniform flow situation, as well as performing a numerical study to confirm the assumed behaviour of the error in the eeKdV energy equation in the context of a undular bore flow setup. Further, conserved integrals for the eeKdV equation are presented and numerically checked.