A Posteriori Error Estimation via Nonlinear Error Transport with Application to Shallow Water

Numerical error estimation for time dependent hyperbolic problems is challenging for theoretical and practical reasons. In these systems, error can propagate long distances and produce effects far from the point of generation. In addition, nonlinear interactions of error, as well as discretization nonlinearities can play important roles and must be addressed. In this work, we investigate the use of error transport equations for a posteriori error estimation. We discuss the inclusion of nonlinearities in the error equations, which are particularly important for situations where local errors become large, such as near shocks.