Commun. Comput. Phys., 10 (2011), pp. 1132-1160.

Numerical Entropy and Adaptivity for Finite Volume Schemes

Gabriella Puppo 1*, Matteo Semplice 2

1 Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italia.
2 Dipartimento di Fisica e Matematica, Universita dell'Insubria, Via Valleggio 11, 22100 Como, Italia.

Received 25 September 2009; Accepted (in revised version) 21 January 2011
Available online 2 August 2011


We propose an a-posteriori error/smoothness indicator for standard semi-discrete finite volume schemes for systems of conservation laws, based on the numerical production of entropy. This idea extends previous work by the first author limited to central finite volume schemes on staggered grids. We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement. We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator. The adaptive scheme uses a single nonuniform grid with a variable timestep. We show how to implement a second order scheme on such a space-time non uniform grid, preserving accuracy and conservation properties. We also give an example of a p-adaptive strategy.

AMS subject classifications: 65M08, 65M50, 76M12

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Key words: Finite volume schemes, hyperbolic systems, local grid refinement, entropy.

*Corresponding author.
Email: (G. Puppo), (M. Semplice)

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