Wu X, van Zwieten G J, van der Zee K G
Multiscale Engineering Fluid Dynamics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
Int J Numer Method Biomed Eng. 2014 Feb;30(2):180-203. doi: 10.1002/cnm.2597. Epub 2013 Sep 10.
We present unconditionally energy-stable second-order time-accurate schemes for diffuse-interface (phase-field) models; in particular, we consider the Cahn-Hilliard equation and a diffuse-interface tumor-growth system consisting of a reactive Cahn-Hilliard equation and a reaction-diffusion equation. The schemes are of the Crank-Nicolson type with a new convex-concave splitting of the free energy and an artificial-diffusivity stabilization. The case of nonconstant mobility is treated using extrapolation. For the tumor-growth system, a semi-implicit treatment of the reactive terms and additional stabilization are discussed. For suitable free energies, all schemes are linear. We present numerical examples that verify the second-order accuracy, unconditional energy-stability, and superiority compared with their first-order accurate variants.
我们提出了用于扩散界面（相场）模型的无条件能量稳定的二阶时间精确格式；具体而言，我们考虑了Cahn-Hilliard方程以及一个由反应性Cahn-Hilliard方程和一个反应扩散方程组成的扩散界面肿瘤生长系统。这些格式属于Crank-Nicolson型，具有自由能的新的凹凸分裂和人工扩散稳定化。非恒定迁移率的情况通过外推法处理。对于肿瘤生长系统，讨论了反应项的半隐式处理和额外的稳定化。对于合适的自由能，所有格式都是线性的。我们给出了数值例子，验证了二阶精度、无条件能量稳定性以及与一阶精确变体相比的优越性。
