Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity

Abstract

We present a methodology to construct efficient high-order in time accurate numerical schemes for a class of gradient flows with appropriate Lipschitz continuous nonlinearity. There are several ingredients to the strategy: the exponential time differencing (ETD), the multi-step (MS) methods, the idea of stabilization, and the technique of interpolation. They are synthesized to develop a generic k^th  order in time efficient linear numerical scheme with the help of an artificial regularization term of the form $A{\tau }^{k}\frac{\partial }{\partial t}{\mathcal{L}}^{p\left(k\right)}u$ where $\mathcal{L}$ is the positive definite linear part of the flow, $\tau $ is the uniform time step-size. The exponent p(k) is determined explicitly by the strength of the Lipschitz nonlinear term in relation to $\mathcal{L}$ together with the desired temporal order of accuracy k. To validate our theoretical analysis, the thin film epitaxial growth without slope selection model is examined with a fourth-order ETD-MS discretization in time and Fourier pseudo-spectral in space discretization. Our numerical results on convergence and energy stability are in accordance with our theoretical results.

Keywords

Gradient flow / epitaxial thin film growth / exponential time differencing / long time energy stability / arbitrary order scheme / multi-step method