Numerical methods for hyperbolic systems: Application to MHD

The lecture will be devoted to the numerical approximation of hyperbolic conservation laws using finite difference, finite volumes and discontinuous Galerkin discretization in space. Explicit and implicit time discretisation will be considered. After introducing the methods in the scalar case, they will be applied to systems with increasing difficulties (Maxwell, Euler, MHD) in the 1D case. We will focus on the Magnetohydrodynamics system (MHD) which is widely used in astrophysics and plasma physics, point out its specific difficulties and discuss some standard test problems. An exercise class is associated to the lecture where as well analytical exercises as coding exercises in Matlab will be proposed.
Lectures Professor : Prof. Dr. Eric Sonnendrucker
Informations : Lectures and exercises of Master, Technic University of Munich.
Key words : hyperbolic systems, MHD, finite volumes methods, Galerkin discontinuous methods, Maxwell systems, Euler systems.
Publications :
Numerical Approximation of Hyperbolic Systems of Conservation Laws, E. Godlewski and P.A. Raviart, Springer, 1996 .
Finite Volume Methods for Hyperbolic Problems, R.J. Leveque, Cambridge Texts in Applied Mathematics, 2002 .
Mathematical Aspects of Discontinuous Galerkin Methods, D. A. Di Pietro and A. Ern, vol. 69 SMAI Mathématiques et Applications, Springer, 2012.
Exercises sheets :
Exercises sheet 1: Advection equation and finite volumes schemes.
Exercises sheet 2: Galerkin Discontinuous for the advection equation.
Exercises sheet 3: Numerical schemes for wave equations
Exercises sheet 4: Finite volumes scheme for scalar nonlinear equations
Exercises sheets correction :
Exercises sheet 1: Advection equation and finite volumes schemes. Part 1, Part 2.
Exercises sheet 2: Discontinuous Galerkin for advection equation. Part 1, Part 2.
Exercises sheet 3: Numerical schemes for wave equations, Part 1, Part 2
Exercises sheet 4: Finite volumes scheme for scalar nonlinear equations, Part 1, Part 2.
Numerical exercises:
Numerical exercises sheet 1: Discontinuous Galerkin solvers for hyperbolic linear systems.
Numerical exercises sheet 2: Discontinuous Galerkin solvers for hyperbolic nonlinear systems.


Matlab program: 1D Discontinuous Galerkin code for hyperbolic systems.
Description: 1D Discontinuous Galerkin code with arbitrary order Lagrange Polynomial + SSP Runge Kutta method (order one, two and three) for the advection, Maxwell, Euler equations and the P1 model. Non uniform grids.
Work in progress: Variable advection velocity, MHD and Burgers equations, AP schemes.