Analytical and Numerical Methods for the Equations of Dynamic Density Functional Theory

密度泛函理论（DFT）及其动态扩展（DDFT）已成为研究非均质流体微观结构的一种广泛应用的方法。与分子动力学等其他方法相比，DDFT的主要优势在于其较低的计算成本，同时保留了系统的微观细节。由Serafim Kalliadasis教授领导的伦敦帝国理工学院复杂多尺度系统（CMS）小组在DDFT的数值和理论方面都取得了重大进展。例如，最近的两个贡献是将DDFT扩展到可定向胶体，包括惯性和流体动力相互作用，或者是波动DDFT的正式理论框架。尽管DDFT得到了相当大的关注，但仍有几个问题尚未解决。例如，目前最先进的数值方法还不能准确有效地模拟多维DDFT问题和/或涉及各种几何形状。特别是CMS组，主要是通过伪谱方法求解DDFT方程，这对非平凡几何有明显的限制。本博士的主要目标是探索基于有限体积格式的新型数值方法的发展，这些方法可以在广泛的具有挑战性的环境中成功地解决DDFT控制方程。

Density Functional Theory (DFT) and its dynamic extensions (DDFT) have become a widely-employed method in the study of the microscopic structure of non-homogeneous fluids. The main advantage of DDFT over other approaches such as Molecular Dynamics relies on its lower computational cost, while it retains the microscopic details of the system. Significant progress has been accomplished both in numerical and theoretical aspects of DDFT by the Complex Multiscale Systems (CMS) group at Imperial College London, led by Professor Serafim Kalliadasis. For instance, two of the most recent contributions have been the extension of DDFT to orientable colloids including inertia and hydrodynamic interactions or the formal theoretical framework for fluctuating DDFT.Despite the considerable attention that DDFT has received, there are still several issues that remain unresolved. For instance, the current state-of-the-art in numerical methodologies is still some way from allowing the accurate and efficient simulations of multidimensional DDFT problems and/or involving varied geometries. The CMS group in particular, has been mainly solving the DDFT equations via pseudospectral methods, which present clear limitations for non-trivial geometries. The primal objective of this PhD is to explore the development of novel numerical methodologies based on finite volume schemes that can successfully solve the DDFT governing equations in a wide range of challenging settings.