Unconditionally stable methods for gradient flow using Convex Splitting Runge-Kutta scheme

We propose a Convex Splitting Runge-Kutta (CSRK) scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner. The key feature of the scheme is a combination of a convex splitting method and a specially designed multi-stage two-additive Runge-Kutta method. Our methods are high order accurate in time and assure the gradient (energy) stability for any time step size. We provide detailed proof of the unconditional energy stability and present issues on the practical implementations. We demonstrate the accuracy and stability of the proposed methods using numerical experiments of the Cahn-Hilliard equation.