Modelling and numerical analysis of energy-dissipating systems with nonlocal free energy

The broad objective of this thesis is to design finite-volume schemes for a family of energy-dissipating systems. All the systems studied in this thesis share a common property: they are driven by an energy that decreases as the system evolves. Such decrease is produced by a dissipation mechanism, which ensures that the system eventually reaches a steady state where the energy is minimised. The numerical schemes presented here are designed to discretely preserve the dissipation of the energy, leading to more accurate and cost-effective simulations. Most of the material in this thesis is based on the publications [16, 54, 65, 66, 243]. The research content is structured in three parts. First, Part II presents well-balanced first-, second- and high-order finite-volume schemes for a general class of hydrodynamic systems with linear and nonlinear damping. These well-balanced schemes preserve stationary states at machine precision, while discretely preserving the dissipation of the discrete free energy for first- and second-order accuracy. Second, Part III focuses on finite-volume schemes for the Cahn-Hilliard equation that unconditionally and discretely satisfy the boundedness of the phase eld and the free-energy dissipation. In addition, our Cahn-Hilliard scheme is employed as an image inpainting filter before passing damaged images into a classification neural network, leading to a significant improvement of damaged-image prediction. Third, Part IV introduces nite-volume schemes to solve stochastic gradient-flow equations. Such equations are of crucial importance within the framework of fluctuating hydrodynamics and dynamic density functional theory. The main advantages of these schemes are the preservation of non-negative densities in the presence of noise and the accurate reproduction of the statistical properties of the physical systems. All these fi nite-volume schemes are complemented with prototypical examples from relevant applications, which highlight the bene fit of our algorithms to elucidate some of the unknown analytical results.Open Acces