An energy stable Runge–Kutta method for convex gradient problems

Abstract

We propose a class of Runge–Kutta methods which provide a simple unified framework to solve the gradient flow of a convex functional in an unconditionally energy stable manner. Stiffly accurate Runge–Kutta methods are high order accurate in terms of time and also assure the energy stability for any time step size when they satisfy the positive definite condition. We provide a detailed proof of the unconditional energy stability as well as unique solvability of the proposed scheme. We demonstrate the accuracy and stability of the proposed methods using numerical experiments for a specific example.