Arbitrarily high-order linear energy stable schemes for gradient flow models

We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for general gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the gradient flow model into an equivalent one with a quadratic free energy and a modified mobility. For any positive integer k > 0 and t(n) = n Delta t, where Delta t is the time step size, we linearize the EQ-reformulated model in (t(n), t(n+1)] through an extrapolation from numerical solutions already obtained in [0,t(n)] so that the approximation error is in the order of O(Delta t(k)). Then we employ an s-stage algebraically stable Runge-Kutta method to discretize the linearized model in (t(n), t(n+1)]. For the spatial discretization, we use the Fourier pseudo-spectral method to match the order of accuracy in time. The resulting fully discrete scheme is linear, unconditionally energy stable, uniquely solvable, and can reach arbitrarily high order. Furthermore, we present a family of linear schemes based on prediction-correction methods to complement the new linear schemes. Some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the schemes. (C) 2020 Elsevier Inc. All rights reserved.