题目： A positivity preserving, energy stable finite difference scheme for the Flory -Huggins -Cahn- Hilliard - Navier-Stokes system

报告人：王成教授（美国马塞诸塞州立大学达特茅斯分校）

时间：2022年11月25日（周五）10:00-12:00

线上报告：Zoom网址 Zoom网址：https://umassd.zoom.us/j/6339263022

摘 要 ： A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the singular chemical potential, the logarithmic term and the surface diffusion term are implicitly updated, while an explicit computation is applied to the concave expansive term. Moreover, the convective term in the phase field evolutionary equation is approximated in a semi-implicit manner. Similarly, the fluid momentum equation is computed by a semi-implicit algorithm: implicit treatment for the kinematic diffusion term,explicit update for the pressure gradient, combined with semi-implicit approximations to the fluid convection and the phase field coupled term, respectively. Such a semi-implicit method gives an intermediate velocity field. Subsequently, a Helmholtz projection into the divergence-free vector field yields the velocity vector and the pressure variable at the next time step. This approach decouples the Stokes solver, which in turn drastically improves the numerical efficiency. The positivity-preserving property and the unique solvability of the proposed numerical scheme is theoretically justified, i.e., the phase variable is always between -1 and 1, following the singular nature of the logarithmic term as the phase variable approaches the singular limit values. In addition, an iteration construction technique is applied in the positivity-preserving and unique solvability analysis, motivated by the non-symmetric nature of the fluid convection term. The energy stability of the proposed numerical scheme could be derived by a careful estimate. A few numerical results are presented to validate the
robustness of the proposed numerical scheme.

个人简介：王成，1993年毕业于中国科技大学获数学学士学位，2000年在美国坦普尔大学获得博士学位,。2000-2003年在美国印尼安纳大学做博士后，2003-2008年在美国田纳西大学任助理教授，2008-2012年在美国麻省大学达特茅斯分校任助理教授，2012年晋升为副教授，2019年晋升为教授。主要研究领域是应用数学，包括数值分析、偏微分方程、流体力学、计算电磁学等。在Journal of Computational Physics，SIAM Journal on Numerical Analysis，IMA Journal of Numerical Analysis等期刊上发表论文五十多篇。