# Numerical methods for nonlinear hyperbolic balance laws

#### Abstract

In this series of lectures we will get to know modern numerical techniques for nonlinear hyperbolic balance laws. It is well-known that these partial differential equations are fundamental for many physical applications. For example, conservations of mass, momentum and energy yield the Euler (or the Navier-Stokes) equations of gas dyamics. Other typical applications are the shallow water equations used in oceanography or MHD equations in astrophysics.

We will present basic mathematical concepts of weak entropic solutions and point out open questions concerning the existence and uniqueness of weak solutions.
Afterwards the finite volume methods, that are typically the method of choice for this type of PDEs, will be presented and discussed. We will get to know genuinely multidimensional techniques based on the theory of bicharacteristics, discuss the application of the finite volume methods for nonconservative systems and for multiscale problems appearing in the so-called low Mach number flows. The latter is a typical application arising in meteorology. Finally, we will also discuss asymptotic preserving property of our schemes for singular limits of fluid flows.

#### Date and Place