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

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.