Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient
1 Institut für Mathematik, Julius-Maximilians-Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Strasse 30, 97074, Würzburg, Germany.
2 Centre For Applicable Mathematics (CAM), Tata Institute of Fundamental Research, P.O. Box 6503, GKVK post office, 560065 Bangalore, India.
Received: 6 June 2015
Revised: 5 September 2015
Accepted: 20 October 2015
We are concerned with fully-discrete schemes for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension. More precisely, we show the convergence of approximate solutions, generated by the scheme corresponding to vanishing diffusive-dispersive scalar conservation laws with a discontinuous coefficient, to the corresponding scalar conservation law with discontinuous coefficient. Finally, the convergence is illustrated by several examples. In particular, it is delineated that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kružkov−Oleĭnik entropy solutions, but contain nonclassical undercompressive shock waves.
Mathematics Subject Classification: 35L65 / 35L67 / 35L25 / 65M12 / 65M06 / 65M08
Key words: Conservation laws / discontinuous flux / diffusive-dispersive approximation / finite difference scheme / convergence / entropy condition / nonclassical shock
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