Energy stable arbitrary order ETD-MS method for gradient flows with Lipschitz nonlinearity

TL;DR

This paper introduces a high-order, energy-stable numerical scheme for gradient flows with Lipschitz nonlinearities, combining ETD, multi-step, stabilization, and interpolation techniques, validated on a thin film growth model.

Contribution

The paper develops a generic $k^{th}$ order in time linear scheme with energy stability for gradient flows with Lipschitz nonlinearity, using an innovative regularization approach.

Findings

The scheme achieves high-order temporal accuracy.

Numerical results confirm energy stability and convergence.

Validated on a thin film epitaxial growth model.

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 $A\tau^k\frac{\partial}{\partial t}\mathcal{L}^{p(k)}u$ 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.