Design and Analysis of Numerical Schemes with Characteristic Methods on Generic Grids for Flows in Porous Media

This thesis studies characteristic-based numerical schemes for a mathematical model of miscible fluid flow in porous media, a coupled and advection-dominated system of partial differential equations describing an oil recovery process. In particular, we analyse characteristic-based schemes on generic polygonal meshes, with diffusive component discretised using modern finite volume methods. We also develop a novel algorithm for reconstructing velocity fields suitable for characteristic tracking. The proposed scheme mixes two characteristic-based discretisations, and implements an original post-processing algorithm to ensure local and global mass conservation. Extensive numerical tests are provided, on various polygonal meshes. A rigorous convergence analysis is also performed.