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Numerical Entropy and Adaptivity for Finite Volume Schemes

Published online by Cambridge University Press:  20 August 2015

Gabriella Puppo and
Matteo Semplice
Gabriella Puppo*
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italia
Matteo Semplice*
Affiliation:
Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italia
*
Corresponding author.Email:gabriella.puppo@polito.it
Email:matteo.semplice@uninsubria.it
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Abstract

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.

Keywords

65M0865M5076M12Finite volume schemeshyperbolic systemslocal grid refinemententropy

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Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011, pp. 1132 - 1160
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Berger, M. J. and Leveque, R. J., Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J. Numer. Anal., 35(6) (1998), 2298–2316.CrossRefGoogle Scholar
[2]Chiavassa, G. and Donat, R., Point value multiscale algorithms for 2D compressible flows, SIAM J. Sci. Comput., 23(3) (2001), 805–823.CrossRefGoogle Scholar
[3]Cockburn, B., An introduction to the discontinuous Galerkin method for convection-dominated problems, in Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997), Volume 1697, Lect. Notes. Math., 151–268, Springer, Berlin, 1998.Google Scholar
[4]Dawson, C. and Kirby, R., High resolution schemes for conservation laws with locally varying time steps, SIAM J. Sci. Comput., 22(6) (2001), 2256–2281.Google Scholar
[5]Dedner, A., Makridakis, C. and Ohlberger, M., Error control for a class of Runge-Kutta Discontinuos Galerkin methods for non linear conservation laws, SIAM J. Numer. Anal., 45 (2007), 514–538.Google Scholar
[6]Fjordholm, U., Mishra, S. and Tadmor, E., Arbitrarily high-order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws, SAM Report 2001-39, ETH Zurich, 2011.Google Scholar
[7]Giles, M. B. and Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta. Numer., 11 (2002), 145–236.Google Scholar
[8]Godlewski, E. and Raviart, P. A., Numerical approximation of hyperbolic systems of conservation laws, Volume 118, Applied Mathematical Sciences, Springer-Verlag, New York, 1996.Google Scholar
[9]Gottlieb, S., Shu, C.-W. and Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43(1) (2001), 89–112.Google Scholar
[10]Guermond, J.-L., Pasquetti, R. and Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230(11) (2011), 4248–4267.CrossRefGoogle Scholar
[11]Harten, A., The artificial compression method for computation of shocks and contact discontinuities I: single conservation laws, Commun. Pure Appl. Math., 30(5) (1977), 611–638.CrossRefGoogle Scholar
[12]Harten, A., Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Commun. Pure Appl. Math., 48(12) (1995), 1305–1342.CrossRefGoogle Scholar
[13]Hartmann, R. and Houston, P., Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, J. Comput. Phys., 183(2) (2002), 508–532.Google Scholar
[14]Karni, S. and Kurganov, A., Local error analysis for approximate solutions of hyperbolic conservation laws, Adv. Comput. Math., 22(1) (2005), 79–99.Google Scholar
[15]Karni, S., Kurganov, A. and Petrova, G., A smoothness indicator for adaptive algorithms for hyperbolic systems, J. Comput. Phys., 178(2) (2002), 323–341.Google Scholar
[16]Kröner, D. and Ohlberger, M., A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions, Math. Comput., 69(229) (2000), 25–39.Google Scholar
[17]Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160(1) (2000), 241–282.Google Scholar
[18]Le Veque, R. J., Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 1992.CrossRefGoogle Scholar
[19]Le Veque, R. J., CLAWPACK Version 4.3 Users Guide, 2006. http://www.amath.washington.edu/∼claw/.Google Scholar
[20]Levy, D., Puppo, G. and Russo, G., Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22(2) (2000), 656–672.Google Scholar
[21]Marobin, D. and Puppo, G., An Error Indicator for Semidiscrete Schemes, Springer, editor, Computational Fluid Dynamics 2004, pages 103–108, 2006.Google Scholar
[22]Müller, S. and Stiriba, Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping, J. Sci. Comput., 30(3) (2007), 493–531.Google Scholar
[23]Ohlberger, M., A review of a posteriori error control and adaptivity for approximations of non-linear conservation laws, Int. J. Numer. Methods Fluids, 59(3) (2009), 333–354.Google Scholar
[24]Osher, S. and Sanders, R., Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput., 41(164) (1983), 321–336.CrossRefGoogle Scholar
[25]Puppo, G., Numerical entropy production on shocks and smooth transitions, in Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), 17 (2002), 263–271.Google Scholar
[26]Puppo, G., Numerical entropy production for central schemes, SIAM J. Sci. Comput., 25(4) (2003), 1382–1415.Google Scholar
[27]Qiu, J. and Shu, C.-W., On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes, J. Comput. Phys., 183(1) (2002), 187–209.Google Scholar
[28]Shu, C.-W., High order ENO and WENO schemes for computational fluid dynamics, In High-order methods for computational physics, Volume 9, Lect. Notes Comput. Sci. Eng., pages 439–582, Springer, Berlin, 1999.Google Scholar
[29]Shu, C.-W. and Osher, S., Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77(2) (1988), 439–471.Google Scholar
[30]Tadmor, E., The numerical viscosity of entropy stable schemes for systems of conservation laws I, Math. Comput., 49(179) (1987), 91–103.CrossRefGoogle Scholar
[31]Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta. Numer., 12 (2003), 451–512.Google Scholar
[32]Tang, H. Z. and Warnecke, G., High resolution schemes for conservation laws and convection-diffusion equations with varying time and space grids, J. Comput. Math., 24(2) (2006), 121–140.Google Scholar
[33]Verfürth, R., A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner Verlag and J. Wiley, Stuttgart, 1996.Google Scholar