A posteriori error estimates for numerical solutions to hyperbolic conservation laws

For general n x n hyperbolic systems of conservation laws in one space dimension, it is well known that the Cauchy problem has a unique entropy-weak solution, depending continuously on the initial data.  Assuming small total variation, a priori estimates on the \L^1 distance between an approximate solution and the exact solution have been obtained in connection with (i) front tracking approximations, (ii) the Glimm scheme, and (iii) vanishing viscosity approximations. However, no a priori estimate is yet known for approximate solutions obtained by fully discrete schemes, such as the Lax-Friedrichs or the Godunov scheme.

In this talk I shall explain the key obstruction toward a priori error bounds for such discrete schemes. Taking a different point of view, I will present some recent results on a posteriori error estimates. These are achieved by a "post-processing algorithm" that checks the total variation of the numerically computed solution, and computes its oscillation on suitable subdomains.

This is a joint work with Maria Teresa Chiri and Wen Shen.