In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.
Giesselmann, J., Makridakis, C., & Pryer, T. (2015). A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws. SIAM Journal on Numerical Analysis, 53(3), 1280-1303. https://doi.org/10.1137/140970999