Energy-decreasing exponential time differencing Runge–Kutta methods for phase-field models

Abstract

Gradient flow models attract much attention these years. The energy naturally decreases along the direction of gradient flows. In order to preserve this property, various numerical schemes have been developed and among them, a very significant approach is the exponential time differencing Runge–Kutta method (ETDRK). In this paper we prove that the second order ETDRK (ETDRK2) scheme unconditionally preserves the energy dissipation law for a family of phase field models, such as the Allen–Cahn equation, the Cahn–Hilliard equation and the molecular beam epitaxy (MBE) model. As far as we know, this is the first work to show that a second-order linear scheme can guarantee the dissipation of the original energy unconditionally, instead of dissipation of modified energy for most existing works. Furthermore, we present some numerical simulations to demonstrate the accuracy and stability of the ETDRK2 scheme by applying the spectral method.