We start by arguing through numerical examples as to why entropy measure
valued solutions are the appropriate solution concept for systems of conservation
laws in
several space dimensions. Two classes of numerical schemes are presented that can be
shown to converge
to entropy measure valued solutions. The first class are finite volume schemes based
on entropy conservative
fluxes and numerical diffusion operators, using a ENO reconstruction. The second
class are space-time shock capturing discontinuous Galerkin (STDG) schemes. The
schemes are compared on a set of numerical experiments. The lecture concludes with a
discussion of efficient ways to compute measure valued solutions using multi-level
monte carlo methods, that were originally developed for uncertainty quantification
in conservation laws.