The intimate connection between well-posedness theory and a posteriori error estimates for numerical methods on the example of hyperbolic systems

In this talk we will review some results on rigorous a posteriori error estimates for numerical approximations of systems of hyperbolic conservation laws, i.e. bounds for discretization errors that can be computed from numerical solutions without making assumption of the properties of the exact solution. We will explain the fundamental link between a posterirori error estimates and stability properties of the PDE that is to be approximated.

We will describe a posteriori error estimates that have been derived a few years ago based on relative entropy stability estimates. We will outline recent progress in a posteriori error estimates for one-dimensional hyperbolic conservation laws based on two approaches: Firstly, results using Bressan's stability theory and, secondly, results using a-contraction estimates based on work of Vasseur and Krupa.