报告问题 (Title)： Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model（液态薄膜粗化模子的保结构，，，能量稳固的数值名堂）

报告人 (Speaker)： 王成 教授（University of Massachusetts Dartmouth）

报告时间 (Time)：2025年1月9日(周四）10:00-11:00

报告所在 (Place)：校本部 F309

约请人(Inviter)：段成华

主理部分：理学院数学系

报告摘要：

Positivity preserving, energy stable numerical schemes are proposed and analyzed for the droplet liquid film model, with a singular Leonard-Jones energy potential involved. Both the first and second order accurate temporal algorithms are considered. In the first order scheme, the convex potential and the surface diffusion terms are implicitly, while the concave potential term is updated explicitly. Furthermore, we provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity is always preserved for the phase variable at a point-wise level. Moreover, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. In the second order numerical scheme, the BDF temporal stencil is applied, and an alternate convex-concave decomposition is derived, so that the concave part corresponds to a quadratic energy. In turn, the combined Leonard-Jones potential term is treated implicitly, and the concave part the is approximated by a second order Adams-Bashforth explicit extrapolation, and an artificial Douglas-Dupont regularization term is added to ensure the energy stability. The unique solvability and the positivity-preserving property for the second order scheme are similarly established.

In addition, optimal rate convergence analysis is derived for both numerical schemes. A few numerical simulation results are also presented.