Finite volumes for conservation law systems

The aim of this course is to present numerical methods adapted to hyperbolic systems of conservation laws, found in particular in compressible fluid mechanics. The methods covered will be of the finite volume type (extended in the case of schemes of order at least 2), and we will endeavor to understand their construction, their interest, as well as the difficulties raised by the nature of the problems (non-uniqueness of weak solutions, non-regular solutions, etc.).

- Approximation of the 1D convection equation
- Finite volume schemes for scalar 1D conservation laws
- General concepts for hyperbolic systems (shocks, relaxations, contact discontinuities, Riemann invariants, etc.)
- Classical finite volume schemes for hyperbolic systems of 1D conservation laws: Godunov, approximate Riemann solvers (HLL-type schemes), Roe schemes.
- 2nd-order schemes: problems, MUSCL, ENO and MOOD methods,
- Extensions for unstructured 2D meshes.
- Applications: compressible Euler, Saint Venant, etc.