Physically accurate advection

Abstract

Mimetic discretization methods are emerging techniques designed to preserve, as much as possible, properties of the continuous differential equation. In this framework the geometric nature of physics plays a crucial role. Thus, it is necessary to use a new language to model physical problems, Differential Geometry. The calculus of differential forms reveals intrinsic structures that usually, are obscured by metric notions implicitly encoded in vector calculus. In the language of differential forms the unified treatment of scalar and non-scalar advection is simplified by the use of one operator: the Lie Derivative. Much attention has been devoted to the discretization of scalar advection problems in numerical analysis. Therefore, it would be expected that non-scalar advection is not common or relevant. This is not true. The so-called magnetic advection in electromagnetism or the advection of vorticity in fluid mechanics are examples of important non-scalar fields. Furthermore, it is widely known that the classical Galerkin formulation when applied to scalar convection may lead to spurious (non-physical) oscillations. The goal of this thesis is to derive a physically accurate discretization of the generalized advection problem. Thus, very small or even vanishing artificial diffusion should reflect the robustness of these methods. Also in this thesis, geometric structure-preserving semi-Lagrangian and Eulerian discrete schemes are derived.