A higher‐order finite element approach to the Kuramoto‐Sivashinsky equation

Abstract

Abstract The Kuramoto‐Sivashinsky equation has emerged as a fundamental evolution equation to describe highly nonlinear physical processes in unstable systems. In general, it constitutes a nonlinear initial‐valued problem involving fourth‐order spatial derivatives. Finite element solutions for the Kuramoto‐Sivashinsky equation are not common because the primal variational formulation of fourth‐order operators requires finite element basis functions which are piecewise smooth and globally at least C 1 ‐continuous. In this paper a novel B‐spline based finite element approach to the solution of the one‐dimensional Kuramoto‐Sivashinsky equation is presented. Extensive numerical studies of different scenarios in the Kuramoto‐Sivashinsky equation will illustrate the quality of our approach.