Numerical analysis of Cahn-Hilliard type equations on evolving surfaces

演化表面上 Cahn-Hilliard 型方程的数值分析

• 批准号: 2593373
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2593373
• 关键词: Numerical; analysis; Cahn; Hilliard; type

There has been much research into the Cahn-Hilliard equation since it was initially proposed by John W. Cahn and John E. Hilliard in 1958. Typically, this is studied on a stationary Euclidean domain. This research considers a twofold improvement, by looking at hypersurfaces (i.e. a non-Euclidean domain) which may be evolving in time. This adds non-trivial complications to the existing theory. Beyond mathematical curiosity, there is genuine reason to consider this set up, as there are physically relevant models coupling the Cahn-Hilliard equation to surface evolution equations (for example by a forced mean curvature flow). In this research I aim to: adapt results of the numerical analysis of the Cahn-Hilliard equation with a logarithmic potential energy to the case of (stationary) surface finite elements, and study similar equations on an evolving domain. This includes a fully discrete numerical scheme for the Cahn-Hilliard equation on an evolving surface with a sufficiently smooth potential - as current research only has considered semi-discrete schemes for this problem. Similarly, one can couple the Cahn-Hilliard equations to the Navier-Stokes equations to obtain a system which can be used to model richer phenomena. The evolving surface Navier-Stokes equations have been a recent topic of interest, but there is currently much less research regarding Navier-Stokes-Cahn-Hilliard systems on evolving surfaces. I aim to study a Navier-Stokes-Cahn-Hilliard system, similar to the well known "Model H" of Hohenberg and Halperin, on an evolving surface both analytically and numerically. These problems been studied for a stationary, flat domain but there is considerably less literature for evolving domains or hypersurfaces. There is currently much interest in partial differential equations on evolving hypersurfaces, this adds extra complexity and accuracy to the underlying physical model. As such, the numerical analysis of such models is necessary for practical schemes which will be implemented. Similarly, this serves nicely as an example of a nonlinear partial differential equation on an evolving surface for which we can obtain quantitative information. An example being recent work on the phenomenon of "separation from the pure phases". Funding is through the Engineering and Physical Sciences Research Council (EPSRC), which covers research in mathematical sciences. Moreover, the Cahn-Hilliard equation finds applications, outside of mathematics, to many real world problems such as in metallurgy, modelling chemotaxis, RNA/protein dynamics, and in image processing analysis. Likewise the Navier-Stokes-Cahn-Hilliard system has been used to model chemotaxis (with advection) and tumour growth, and the kinetics of lipid biomembranes.

自 John W. Cahn 和 John E. Hilliard 于 1958 年首次提出 Cahn-Hilliard 方程以来，人们对其进行了大量研究。通常，这是在稳态欧几里德域上进行研究的。这项研究通过观察可能随时间演化的超曲面（即非欧几里得域）来考虑双重改进。这给现有理论增加了不小的复杂性。除了数学上的好奇心之外，还有真正的理由考虑这种设置，因为存在将 Cahn-Hilliard 方程与表面演化方程耦合的物理相关模型（例如通过强制平均曲率流）。 在这项研究中，我的目标是：将具有对数势能的 Cahn-Hilliard 方程的数值分析结果应用于（静止）表面有限元的情况，并研究演化域上的类似方程。这包括具有足够平滑势的演化表面上的 Cahn-Hilliard 方程的完全离散数值方案 - 因为当前研究仅考虑了该问题的半离散方案。类似地，我们可以将 Cahn-Hilliard 方程与 Navier-Stokes 方程耦合起来，以获得一个可用于模拟更丰富现象的系统。演化表面纳维-斯托克斯方程是最近人们感兴趣的话题，但目前关于演化表面纳维-斯托克斯-卡恩-希利亚德系统的研究要少得多。我的目标是在不断演化的表面上进行分析和数值研究，研究 Navier-Stokes-Cahn-Hilliard 系统，类似于 Hohenberg 和 Halperin 著名的“H 模型”。这些问题是针对静止、平坦的域进行研究的，但关于演化域或超曲面的文献要少得多。目前人们对演化超曲面的偏微分方程很感兴趣，这给底层物理模型增加了额外的复杂性和准确性。因此，此类模型的数值分析对于将要实施的实际方案是必要的。类似地，这很好地作为演化表面上的非线性偏微分方程的示例，我们可以获得定量信息。一个例子是最近关于“纯相分离”现象的研究。 资金来自工程和物理科学研究委员会 (EPSRC)，该委员会涵盖数学科学研究。此外，Cahn-Hilliard 方程在数学之外还可以应用于许多现实世界的问题，例如冶金、趋化性建模、RNA/蛋白质动力学和图像处理分析。同样，Navier-Stokes-Cahn-Hilliard 系统已用于模拟趋化性（平流）和肿瘤生长以及脂质生物膜的动力学。