Introduction to the finite volume method

This course introduces the finite volume method, a widely used technique for the numerical approximation of partial differential equations known as conservation laws, with particular emphasis on the design of finite volume schemes and their implementation for bi-dimensional problems. Part of the course will also be devoted to the notion of weak solution of partial differential equations within the framework of distribution theory: this theory gives mathematical meaning to the notion of non-regular solution, which is of crucial importance for the finite volume method applied to hyperbolic equations. The methods seen in the course will be programmed during practical sessions in Fortran and C++ (EM7PG201).

First-year mathematics and numerical analysis course

1) PDEs and conservation laws

Classification of PDEs
Conservation laws
Stokes formula
Flux

2) Finite volume method

general principles
General 2D approach: mesh, mean values, flux
Example 1: heat equation, centered flux, properties, link with finite differences, implementation
Example 2: stationary diffusion problem
Example 3: advection-diffusion equation

3) Distribution theory and weak solutions

Motivations
Example of non-regular solutions
Notion of distribution and derivative in the sense of distributions
Jump formula
Weak solution
Riemann problem

4) Finite volume method for linear hyperbolic equations

Godunov's method - Off-center flow
Construction of a 2D diagram from a 1D diagram
Hyperbolic system
Extension to order 2

5) Non-linear conservation laws

Examples : Burgers, road traffic
Existence and uniqueness
Solution of the Riemann problem
Godunov scheme
Approximate Riemann solvers

Numerical analysis - Scientific computing