Numerical Entropy and Adaptivity for Finite Volume Schemes (Communications in Computational Physics, 2011)
We propose an a-posteriori error/smoothness indicator for standard semi-discretefinite volume schemes for systems of conservation laws, based on the numericalproduction of entropy. This idea extends previous work by the first author limitedto central finite volume schemes on staggered grids. We prove that the indicator convergesto zero with the same rate of the error of the underlying numerical scheme onsmooth flows under grid refinement. We construct and test an adaptive scheme forsystems of equations in which the mesh is driven by the entropy indicator. The adaptivescheme uses a single nonuniform grid with a variable timestep. We show howto implement a second order scheme on such a space-time non uniform grid, preservingaccuracy and conservation properties. We also give an example of a p-adaptivestrategy.