Behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations

Abstract

This article reports the behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations. The one-and-a-half-dimensional shallow water equations are the one-dimensional shallow water equations with a passive tracer or transverse velocity. The studied behaviour is with respect to the choice of numerical fluxes to evolve the mass, momentum, tracer-mass (transverse momentum), and entropy. When solving the one-and-a-half-dimensional shallow water equations using a finite volume method, we recommend the use of a double sided stencil flux for the mass and momentum, and in addition, a single sided stencil (upwind) flux for the tracer-mass. Having this recommended combination of fluxes, we use a double sided stencil entropy flux to compute the numerical entropy production, but this flux generates positive overshoots of the numerical entropy production. Positive overshoots of the numerical entropy production are avoided by use of a modified entropy flux, which satisfies a discrete numerical entropy inequality. References F. Bouchut. Efficient numerical finite volume schemes for shallow water models. In V. Zeitlin (editor), Nonlinear dynamics of rotating shallow water: methods and advances, Volume 2 of Edited series on advances in nonlinear science and complexity, pages 189--256. Elsevier, Amsterdam, 2007. http://dx.doi.org/10.1016/S1574-6909(06)02004-1 R. J. LeVeque. Finite-volume methods for hyperbolic problems. Cambridge University Press, Cambridge, 2002. http://dx.doi.org/10.1017/CBO9780511791253 S. Mungkasi and S. G. Roberts. Numerical entropy production for shallow water flows. ANZIAM Journal, 52(CTAC2010):C1--C17, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3786/1410 G. Puppo and M. Semplice. Numerical entropy and adaptivity for finite volume schemes. Communications in Computational Physics, 10(5):1132--1160, 2011. http://dx.doi.org/10.4208/cicp.250909.210111a J. J. Stoker. Water Waves: The Mathematical Theory with Application. Interscience Publishers, New York, 1957. http://onlinelibrary.wiley.com/book/10.1002/9781118033159