Energy stable arbitrary order ETD-MS method for gradient flows with Lipschitz nonlinearity
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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 Aτ^k∂/∂ tℒ^p(k)u where ℒ is the positive definite linear part of the flow, τ is the uniform time step-size. The exponent p(k) is determined explicitly by the strength of the Lipschitz nonlinear term in relation to 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.
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