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NUMERICAL ENTROPY PRODUCTION AS SMOOTHNESS INDICATOR FOR SHALLOW WATER EQUATIONS

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Basic methods in fluid mechanics

Published online by Cambridge University Press:  28 November 2019

SUDI MUNGKASI Open the ORCID record for SUDI MUNGKASI [Opens in a new window]  and
STEPHEN GWYN ROBERTS Open the ORCID record for STEPHEN GWYN ROBERTS [Opens in a new window]
SUDI MUNGKASI*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Sanata Dharma University, Yogyakarta, Indonesia email sudi@usd.ac.id
STEPHEN GWYN ROBERTS
Affiliation:
Mathematical Sciences Institute, College of Physical and Mathematical Sciences, Australian National University, Canberra, Australia email stephen.roberts@anu.edu.au
*
✉sudi@usd.ac.id
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Abstract

The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.

Keywords

smoothness indicatorshock detectornumerical entropy productionfinite volume methodshallow water equations

MSC classification

Primary: 35L65: Conservation laws
Secondary: 65M08: Finite volume methods 76M12: Finite volume methods
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 4 , October 2019 , pp. 398 - 415
Copyright
© 2019 Australian Mathematical Society

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References

Altazin, T., Ersoy, M., Golay, F., Sous, D. and Yushchenko, L., “Numerical investigation of BB-AMR scheme using entropy production as refinement criterion”, Int. J. Comput. Fluid Dyn. 30 (2016) 256–271; doi:10.1080/10618562.2016.1194977.Google Scholar
Audusse, E., Bouchut, F., Bristeau, M. O., Klein, R. and Perthame, B., “A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows”, SIAM J. Sci. Comput. 25 (2004) 2050–2065; doi:10.1137/S1064827503431090.Google Scholar
Bollermann, A., Noelle, S. and Lukacova-Medvidova, M., “Finite volume evolution Galerkin methods for the shallow water equations with dry beds”, Commun. Comput. Phys. 10 (2011) 371–404; doi:10.4208/cicp.220210.020710a.Google Scholar
Bouchut, F., Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources (Birkhauser, Basel, 2004); doi:10.1007/b93802.Google Scholar
Bouchut, F., “Efficient numerical finite volume schemes for shallow water models”, in: Nonlinear dynamics of rotating shallow water: methods and advances, Volume 2 of Edited Ser. Adv. Nonlinear Sci. Complexity (ed. Zeitlin, V.), (Elsevier, Amsterdam, 2007) 189–256; doi:10.1016/S1574-6909(06)02004-1.Google Scholar
Cravero, I., Puppo, G., Semplice, M. and Visconti, G., “CWENO: uniformly accurate reconstructions for balance laws”, Math. Comp. 87 (2018) 1689–1719; doi:10.1090/mcom/3273.Google Scholar
Dumbser, M., Boscheri, W., Semplice, M. and Russo, G., “Central weighted ENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes”, SIAM J. Sci. Comput. 39 (2017) A2564–A2591; doi:10.1137/17M1111036.Google Scholar
Ersoy, M., Golay, F. and Yushchenko, L., “Adaptive multiscale scheme based on numerical density of entropy production for conservation laws”, Cent. Eur. J. Math. 11 (2013) 1392–1415; doi:10.2478/s11533-013-0252-6.Google Scholar
Golay, F., “Numerical entropy production and error indicator for compressible flows”, C. R. Méc. 337 (2009) 233–237; doi:10.1016/j.crme.2009.04.004.Google Scholar
Harten, A., “The artificial compression method for computation of shocks and contact discontinuities I: single conservation laws”, Comm. Pure Appl. Math. 30 (1977) 611–638; doi:10.1002/cpa.3160300506.Google Scholar
Kalita, P., Dass, A. K. and Sarma, A., “Effects of numerical diffusion on the computation of viscous supersonic flow over a flat plate”, Int. J. Appl. Comput. Math. 2 (2016) 663–678; doi:10.1007/s40819-015-0094-y.Google Scholar
Kurganov, A., Noelle, S. and Petrova, G., “Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations”, SIAM J. Sci. Comput. 23 (2001) 707–740; doi:10.1137/S1064827500373413.Google Scholar
Larrouturou, B., “How to preserve the mass fractions positivity when computing compressible multi-component flows”, J. Comput. Phys. 95 (1991) 59–84; doi:10.1016/0021-9991(91)90253-H.Google Scholar
LeVeque, R. J., Finite volume methods for hyperbolic problems (Cambridge University Press, Cambridge, 2002); doi:10.1017/CBO9780511791253.Google Scholar
Lozano, C., “Entropy production by explicit Runge–Kutta schemes”, J. Sci. Comput. 76 (2018) 521–564; doi:10.1007/s10915-017-0627-0.Google Scholar
Mungkasi, S., “A study of well-balanced finite volume methods and refinement indicators for the shallow water equations”, Bull. Aust. Math. Soc. 88 (2013) 351–352; Ph.D. Thesis, The Australian National University, Canberra, 2012; http://hdl.handle.net/1885/10301.Google Scholar
Mungkasi, S., “An accurate smoothness indicator for shallow water flows along channels with varying width”, Appl. Mech. Mater. 771 (2015) 157–160; doi:10.4028/www.scientific.net/AMM.771.157.Google Scholar
Mungkasi, S., “Numerical entropy production of the one-and-a-half-dimensional shallow water equations with topography”, J. Indones. Math. Soc. 21 (2015) 35–43; doi:10.22342/jims.21.1.198.35-43.Google Scholar
Mungkasi, S., “Adaptive finite volume method for the shallow water equations on triangular grids”, Adv. Math. Phys. 2016 (2016); 7528625; doi:10.1155/2016/7528625.Google Scholar
Mungkasi, S. and Roberts, S. G., “On the best quantity reconstructions for a well balanced finite volume method used to solve the shallow water wave equations with a wet/dry interface”, ANZIAM J. 51 (2010) C48–C65; doi:10.21914/anziamj.v51i0.2576.Google Scholar
Mungkasi, S. and Roberts, S. G., “Numerical entropy production for shallow water flows”, ANZIAM J. 52 (2011) C1–C17; doi:10.21914/anziamj.v52i0.3786.Google Scholar
Mungkasi, S. and Roberts, S. G., “Analytical solutions involving shock waves for testing debris avalanche numerical models”, Pure Appl. Geophys. 169 (2012) 1847–1858; doi:10.1007/s00024-011-0449-1.Google Scholar
Mungkasi, S. and Roberts, S. G., “Behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations”, ANZIAM J. 54 (2013) C18–C33; doi:10.21914/anziamj.v54i0.6243.Google Scholar
Mungkasi, S. and Roberts, S. G., “Weak local residuals in an adaptive finite volume method for one-dimensional shallow water equations”, J. Indones. Math. Soc. 20 (2014) 11–18; doi:10.22342/jims.20.1.176.11-18.Google Scholar
Noelle, S., Pankratz, N., Puppo, G. and Natvig, J. R., “Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows”, J. Comput. Phys. 213 (2006) 474–499; doi:10.1016/j.jcp.2005.08.019.Google Scholar
Nogueira, X., Ramirez, L., Khelladi, S., Chassaing, J. C. and Colominas, I., “A high-order density-based finite volume method for the computation of all-speed flows”, Comput. Methods Appl. Math. 298 (2016) 229–251; doi:10.1016/j.cma.2015.10.004.Google Scholar
Puppo, G., “Numerical entropy production on shocks and smooth transitions”, J. Sci. Comput. 17 (2002) 263–271; doi:10.1023/A:1015117118157.Google Scholar
Puppo, G., “Numerical entropy production for central schemes”, SIAM J. Sci. Comput. 25 (2003) 1382–1415; doi:10.1137/S1064827502386712.Google Scholar
Puppo, G. and Semplice, M., “Numerical entropy and adaptivity for finite volume schemes”, Commun. Comput. Phys. 10 (2011) 1132–1160; doi:10.4208/cicp.250909.210111a.Google Scholar
Puppo, G. and Semplice, M., “Well-balanced high order 1D schemes on non-uniform grids and entropy residuals”, J. Sci. Comput. 66 (2016) 1052–1076; doi:10.1007/s10915-015-0056-x.Google Scholar
Semplice, M., Coco, A. and Russo, G., “Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction”, J. Sci. Comput. 66 (2016) 692–724; doi:10.1007/s10915-015-0038-z.Google Scholar
Semplice, M. and Loubere, R., “Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions”, J. Comput. Phys. 354 (2018) 86–110; doi:10.1016/j.jcp.2017.10.031.Google Scholar
Srinivasan, B. and Kumar, V., “The versatility of an entropy inequality for the robust computation of convection dominated problems”, Procedia Comput. Sci. 108 (2017) 887–896; doi:10.1016/j.procs.2017.05.099.Google Scholar
Yushchenko, L., Golay, F. and Ersoy, M., “Entropy production and mesh refinement – application to wave breaking”, Mech. Ind. 16 (2015) 301; doi:10.1051/meca/2015003.Google Scholar
Zhang, H., Guo, Y. Y., Jin, Y. and Li, Y., “An entropy production method to investigate the accuracy and stability of numerical simulation of one-dimensional heat transfer”, Heat Transfer Res. 43 (2012) 669–693; doi:10.1615/HeatTransRes.2012005925.Google Scholar