Main Article Content
Abstract
Numerical entropy production can be used as a smoothness indicator of solutions to conservation laws. By definition the entropy production is non-positive. However some authors, using a finite volume method framework, showed that positive overshoots of the numerical entropy production were possible for conservation laws (no source terms involved). Note that the one-and-a-half-dimensional shallow water equations without source terms are conservation laws. A report has been published regarding the behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations without source terms. The main result of that report was that positive overshoots of the numerical entropy production were avoided by use of a modified entropy flux which satisfies a discrete numerical entropy inequality. In the present article we consider an extension problem of the previous report. We take the one-and-a-half-dimensional shallow water equations involving topography. The topography is a source term in the considered system of equations. Our results confirm that a modified entropy flux which satisfies a discrete numerical entropy inequality is indeed required to have no positive overshoots of the entropy production.
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References
- Bouchut, F., ”Efficient numerical finite volume schemes for shallow water models”, In V. Zeitlin (editor), Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances, Edited Series on Advances in Nonlinear Science and Complexity, Volume 2, pages 189–256, Elsevier, Amsterdam, 2007.
- Ersoy, M., Golay, F. and Yushchenko, L., ”Adaptive multiscale scheme based on numerical density of entropy production for conservation laws”, Central European Journal of Mathematics, 11 (2013), 1392–1415.
- Golay, F., ”Numerical entropy production and error indicator for compressible flows”, Comptes Rendus Mecanique, 337 (2009), 233–237.
- LeVeque, R.J., Numerical methods for conservation laws, Birkhauser Verlag, Basel, Second edition, 1992.
- LeVeque, R.J., Finite-volume methods for hyperbolic problems, Cambridge University Press, Cambridge, 2002.
- Mungkasi, S., ”A study of well-balanced finite volume methods and refinement indicators for the shallow water equations”, Bulletin of the Australian Mathematical Society, 88 (2013), 351–352.
- Mungkasi, S. and Roberts, S.G., ”Behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations”, ANZIAM Journal, 54 (2013), C18–C33.
- Mungkasi, S. and Roberts, S.G., ”Weak local residuals in an adaptive finite volume method for one-dimensional shallow water equations”, Journal of the Indonesian Mathematical Society, 20 (2014), 11–18.
- Mungkasi, S., Li, Z. and Roberts, S.G., ”Weak local residuals as smoothness indicators for the shallow water equations”, Applied Mathematics Letters, 30 (2014), 51–55.
- Puppo, G. and Semplice, M., ”Numerical entropy and adaptivity for finite volume schemes”, Communications in Computational Physics, 10 (2011), 1132–1160.
- Puppo, G. and Semplice, M., ”Entropy and the numerical integration of conservation laws”, submitted to Proceedings of the 2nd Annual Meeting of the Lebanese Society for Mathematical Sciences 2011.
- Puppo, G., ”Numerical entropy production on shocks and smooth transitions”, Journal of Scientific Computing, 17 (2002), 263–271.
- Puppo, G., ”Numerical entropy production for central schemes”, SIAM Journal on Scientific Computing, 25 (2003), 1382–1415.
- Zhang, Y. and Zhou, C.H., ”An immersed boundary method for simulation of inviscid compressible flows”, International Journal for Numerical Methods in Fluids, 74 (2014), 775–793.
References
Bouchut, F., ”Efficient numerical finite volume schemes for shallow water models”, In V. Zeitlin (editor), Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances, Edited Series on Advances in Nonlinear Science and Complexity, Volume 2, pages 189–256, Elsevier, Amsterdam, 2007.
Ersoy, M., Golay, F. and Yushchenko, L., ”Adaptive multiscale scheme based on numerical density of entropy production for conservation laws”, Central European Journal of Mathematics, 11 (2013), 1392–1415.
Golay, F., ”Numerical entropy production and error indicator for compressible flows”, Comptes Rendus Mecanique, 337 (2009), 233–237.
LeVeque, R.J., Numerical methods for conservation laws, Birkhauser Verlag, Basel, Second edition, 1992.
LeVeque, R.J., Finite-volume methods for hyperbolic problems, Cambridge University Press, Cambridge, 2002.
Mungkasi, S., ”A study of well-balanced finite volume methods and refinement indicators for the shallow water equations”, Bulletin of the Australian Mathematical Society, 88 (2013), 351–352.
Mungkasi, S. and Roberts, S.G., ”Behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations”, ANZIAM Journal, 54 (2013), C18–C33.
Mungkasi, S. and Roberts, S.G., ”Weak local residuals in an adaptive finite volume method for one-dimensional shallow water equations”, Journal of the Indonesian Mathematical Society, 20 (2014), 11–18.
Mungkasi, S., Li, Z. and Roberts, S.G., ”Weak local residuals as smoothness indicators for the shallow water equations”, Applied Mathematics Letters, 30 (2014), 51–55.
Puppo, G. and Semplice, M., ”Numerical entropy and adaptivity for finite volume schemes”, Communications in Computational Physics, 10 (2011), 1132–1160.
Puppo, G. and Semplice, M., ”Entropy and the numerical integration of conservation laws”, submitted to Proceedings of the 2nd Annual Meeting of the Lebanese Society for Mathematical Sciences 2011.
Puppo, G., ”Numerical entropy production on shocks and smooth transitions”, Journal of Scientific Computing, 17 (2002), 263–271.
Puppo, G., ”Numerical entropy production for central schemes”, SIAM Journal on Scientific Computing, 25 (2003), 1382–1415.
Zhang, Y. and Zhou, C.H., ”An immersed boundary method for simulation of inviscid compressible flows”, International Journal for Numerical Methods in Fluids, 74 (2014), 775–793.