A time-adaptive finite volume method for the Cahn–Hilliard and Kuramoto–Sivashinsky equations

Abstract

This paper presents a complete finite volume method for the Cahn–Hilliard and Kuramoto–Sivashinsky type of equations. The spatial discretization is high-order accurate and suitable for general unstructured grids. The time integration is addressed by means of implicit an implicit–explicit fourth order Runge–Kutta schemes, with error control and adaptive time-stepping. The outcome is a practical, accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations. The use of adaptive time-stepping is of paramount importance in problems governed by the Cahn–Hilliard model; an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps. In addition to driving the simulations efficiently, the time-adaptive procedure provides a quantitative (not just qualitative) characterization of the rich temporal scales present in phase separation processes governed by the Cahn–Hilliard phase-field model.