Project R-16556

CAT-FLOW: A novel arbitrary-order implicit/explicit method for two-phase flow (Research)

Mathematical modelling is the science of taking a real-life problem and converting it into a mathematical equation. For realistic cases, this mathematical equation is difficult, depends on various parameters, and has no closed-form analytical solution, to just mention a few challenges. Hence, the mathematical equation needs to be solved through an algorithm that computes an approximate solution numerically. In the current proposal, we focus on this last aspect. The underlying equations stem from the modelling of two-phase flow, describing a physical system where two fluids (gases or liquids) move in a joint environment. These equations are differential equations, with up to fourth-order derivatives. We first rewrite these equations as differential equations with at most first-order derivatives, as these equations are easier to analyze and to approximate. We will mathematically analyze the properties of these equations. Then, we devise a completely novel discretization paradigm suitable for first-order equations. This discretization is implemented and thoroughly analyzed mathematically and numerically. The outcome of this proposal will be first a good numerical algorithm not only suited for two-phase flow applications, and second a lot of insight on the equations and their discretization.

01 October 2026 - 30 September 2030