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

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