M2 MF — Discretization of evolutionary problems

The lecture will be given during fall 2024 for the students of the Master 2 in Fundamental Mathematics (french website). It consists in two complementary parts.

• The first part is given by Adrien Laurent dedicated to geometric numerical integration, from November 5. to November 22.
• The second part given by myself, from November 27. to December 13., dedicated to finite difference and finite volume methods.

Evaluation schedule

Devoir maison $(DM)$ à rendre le vendredi 13 décembre.
Evaluation $(E)$ le jeudi 19 décembre 8h-10h (voir planning ADE).

La notation pour cette seconde partie du cours est $P2 = \max(E,\tfrac{DM+E}{2})$.
La note globale pour l’UE est la moyenne $\tfrac{P1+P2}{2}$, où $P1$ désigne la note délivrée pour la première partie “Intégration numérique géométrique”.

Abstract

This course covers various aspects of the discretization of evolutionary problems, particularly the numerical preservation of the geometry and physics of dynamics. When the flow of an evolution problem preserves certain quantities (such as energy, volume, geometric constraint, etc.), it is natural to want to use numerical schemes that also preserve these quantities.

In the first part of the course, we will explore geometric numerical integration, an original blend of advanced algebraic and geometric tools (Hopf algebras, differential complexes, Lie groups, etc.) used for the construction of efficient numerical methods (see teaching webpage of Adrien Laurent).

In the second part of the course, finite difference methods for evolution PDEs are studied, notably through the symbolic approach. This approach, in addition to their convergence study, allows one to access the “physics” of their solution through the discrete dispersion relation. Complementing the theoretical course “Hyperbolic PDEs”, given by Miguel Rodrigues in parallel, finite volume methods will be studied for approximating the weak entropic solutions of nonlinear hyperbolic PDEs (with the study limited to the scalar case in one spatial dimension). The importance of having invariant or dissipated quantities in the approximation will be particularly emphasized.

Selected references for the second part

• Strikwerda JC (2004) Finite difference schemes and partial differential equations, 2nd ed., Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). DOI
• Gustafsson B, Kreiss H-O, Oliger J (2013) Time-dependent problems and difference methods, John Wiley & Sons, Inc., Hoboken, NJ. DOI
• LeVeque RJ (2002) Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge. DOI
• Bouchut F (2004) Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkhäuser. DOI