Numerical Solutions to the Rosenau–Kawahara Equation for Shallow Water Waves via Pseudo–Compact Methods

Panasun Manorot, Phakdi Charoensawan, Supreedee Dangskul

Abstract

This paper presents two linear finite difference schemes for the so--called Rosenau--Kawahara equation, modified from a linear scheme by Hu et al. in 2014, under a pseudo--compact method. Existence and uniqueness of solutions generated by both schemes are proved. It is shown that the first scheme possesses some conservation properties for mass and energy, whereas the other proposed scheme provides only mass conservation. Some discussions on stability are given, which reveal that numerical solutions are stable with respect to $\|\cdot\|_{\infty}$. It is also shown that pseudo--compactness allows some terms in the schemes to reach fourth--order convergence, even though the numerical solutions is of second--order convergence overall.  Furthermore, numerical simulations are illustrated confirming that our schemes induce some improvements over the existing scheme by Hu et al. on precision and cost.