# Advanced numerical methods for hyperbolic equations and applications

**WEEK 1:**

Theoretical aspects of hyperbolic conservation laws. Review of basic numerical concepts for hyperbolic equations. Finite volume methods for one-dimensional systems. Godunov's method. The Riemann problem for linear systems. The Riemann problem for the shallow water equations. Approximate Riemann solvers. Godunov-type finite volume methods for non-linear systems. Centred numerical fluxes. Construction of higher order non-oscillatory methods via non-linear schemes: TVD, ENO and WENO reconstruction procedures. Discontinuous Galerkin Finite Element methods for one-dimensional problems. Robust and accurate discretization of source terms: stiff and non-stiff cases. The well-balanced property and numerical methods for non-conservative hyperbolic systems. Extension to multiple space dimensions.

WEEK 2:

The second week is dedicated to the extension of the methods introduced in the first week to complex geometries using unstructured triangular meshes in two space dimensions and using mesh-free approaches.

Mesh-based algorithms: Unstructured meshes for two-dimensional geometries. High-order reconstruction on unstructured meshes in multiple space dimensions. High Order Finite volume and discontinuous Galerkin finite element methods on unstructured meshes. Applications to the shallow water equations and the Euler equations of compressible gas dynamics.

Mesh-free algorithms: Introduction to particle methods. Guidelines for implementation of smooth particle hydrodynamics (SPH) based on the Riemann solvers introduced in the first week.

High Performance Computing: Parallelization of the above-mentioned methods using the MPI (Message Passing Interface) standard.

At the end of the second week, the course is rounded-off by advanced seminar-style lectures with outlooks on the following topics: extension to 3D tetrahedral meshes, compressible multi-phase flows, electromagnetic, acoustic and seismic wave propagation.