Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity (CSIAM Transactions on Applied Mathematics, 2021)
We present a methodology to construct efficient high-order in time accuratenumerical schemes for a class of gradient flows with appropriate Lipschitz continuousnonlinearity. There are several ingredients to the strategy: the exponential time differencing (ETD), the multi-step (MS) methods, the idea of stabilization, and the techniqueof interpolation. They are synthesized to develop a generic $k^{th}$ order in time efficientlinear numerical scheme with the help of an artificial regularization term of the form $Aτ^k\frac{∂}{∂t}\mathcal{L}^{p(k)}u$ where $\mathcal{L}$ is the positive definite linear part of the flow, $τ$ is the uniformtime 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 slopeselection model is examined with a fourth-order ETD-MS discretization in time andFourier pseudo-spectral in space discretization. Our numerical results on convergenceand energy stability are in accordance with our theoretical results.