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

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