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This paper is concerned with the construction of high order schemes onirregular grids for balance laws, including a discussion of an a-posteriorierror indicator based on the numerical entropy production. We also imposewell-balancing on non uniform grids for the shallow water equations, which canbe extended similarly to other cases, obtaining schemes up to fourth order ofaccuracy with very weak assumptions on the regularity of the grid. Our resultsshow the expected convergence rates, the correct propagation of shocks acrossgrid discontinuities and demonstrate the improved resolution achieved with alocally refined non-uniform grid.  The error indicator based on the numerical entropy production, previouslyintroduced for the case of systems of conservation laws, is extended to balancelaws. Its decay rate and its ability to identify discontinuities is illustratedon several tests. The schemes proposed in this work naturally can also beapplied to systems of conservation laws.

Well-balanced high order schemes on non-uniform grids and entropy residuals