A Multiscale Finite Element Framework for Coupled Continuity and Momentum Equations in the Saint-Venant System

Abstract

This thesis develops a multiscale finite element method for solving the one-dimensional non-linear Saint-Venant equations. The approach combines a coarse-scale treatment of the continuity equation with a fine-scale resolution of the momentum equation. A semi-Lagrangian reconstruction is used to generate time-dependent multiscale basis functions that accurately represent the evolving fine-scale dynamics and capture the mixing effect induced by the velocity field. An IMEX (Implicit-Explicit) time-stepping scheme is used to efficiently integrate the coupled system. Numerical experiments demonstrate that the proposed method delivers robust and accurate results making it a promising tool for simulating transport-dominated non-linear systems.