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arXiv · 1611.02967

A Second Order Energy Stable Scheme for the Cahn-Hilliard-Hele-Shaw Equations

Wenbin Chen·Wenqiang Feng·Yuan Liu·Cheng Wang·Steven M. Wise
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Abstract

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an $\ell^2 (0,T; H_h^3)$ stability of the numerical scheme. To overcome the difficulty associated with the convection term $\nabla \cdot (\phi \boldsymbol{u})$, we perform an $\ell^\infty (0,T; H_h^1)$ error estimate instead of the classical $\ell^\infty (0,T; \ell^2)$ one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.

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Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang, Steven M. Wise. 2016-11-09. A Second Order Energy Stable Scheme for the Cahn-Hilliard-Hele-Shaw Equations. https://arxiv.org/abs/1611.02967

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