M2 MF — Numerics for transport

The lecture (24h) will be given during fall 2025 for the students of the Master 2 in Fundamental Mathematics (french website). The first sessions take place on Wednesday, November 12. 2025. The complete schedule is available on ADE.

Abstract

This course aims to study the various numerical strategies of approximation of solutions to evolution partial differential equations. Finite difference methods are analyzed through a symbolic approach. This approach allows to revisit the traditional convergence study from a frequency point of view. We will see how to explain and improve the “physics” of the numerical solutions via properties of the discrete dispersion relation. We will quickly tackle spectral and pseudo-spectral methods. In the second part of the course, we will study finite volume strategies. They will be analyzed mainly for the approximation of weak entropy solutions of hyperbolic scalar conservation laws in one space dimension.

Selected references

• Finite difference methods
  • 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
  • Trefethen L. N. (1996) Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, unpublished text, available at LINK
• Finite volumes methods
  • 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

Evaluation

• CC1 on Wednesday, December 3. 09:45-11:15
• CC2 on Wednesday, December 17. 13:15-16:15

The final note is the mean value of CC1 and CC2.

Exercises and numerical experiments

To come.