A family of numerical schemes for kinematic flows with discontinuous flux

Abstract

Multiphase flows of suspensions and emulsions are frequently approximated by spatially one-dimensional kinematic models, in which the velocity of each species of the disperse phase is an explicitly given function of the vector of concentrations of all species. The continuity equations for all species then form a system of conservation laws which describes spatial segregation and the creation of areas of different composition. This class of models also includes multi-class traffic flow, where vehicles belong to different classes according to their preferential velocities. Recently, these models were extended to fluxes that depend discontinuously on the spatial coordinate, which appear in clarifier-thickener models, in duct flows with abruptly varying cross-sectional area, and in traffic flow with variable road surface conditions. This paper presents a new family of numerical schemes for such kinematic flows with a discontinuous flux. It is shown how a very simple scheme for the scalar case, which is adapted to the "concentration times velocity" structure of the flux, can be extended to kinematic models with phase velocities that change sign, flows with two or more species (the system case), and discontinuous fluxes. In addition, a MUSCL-type upgrade in combination with a Runge-Kutta-type time discretization can be devised to attain second-order accuracy. It is proved that two particular schemes within the family, which apply to systems of conservation laws, preserve an invariant region of admissible concentration vectors, provided that all velocities have the same sign. Moreover, for the relevant case of a multiplicative flux discontinuity and a constant maximum density, it is proved that one scalar version converges to a BV t entropy solution of the model. In the latter case, the compactness proof involves -R. Bürger et al. a novel uniform but local estimate of the spatial total variation of the approximate solutions. Numerical examples illustrate the performance of all variants within the new family of schemes, including applications to problems of sedimentation, traffic flow, and the settling of oil-in-water emulsions.

Hyperbolicity Analysis of Polydisperse Sedimentation Models via a Secular Equation for the Flux Jacobian

“…Other multispecies kinematic flow models of the type (1.1), (1.2), which are amenable to a similar hyperbolicity analysis, include multi-class vehicular traffic [7,15,17,24,48,51,52] and the creaming of emulsions [15,39].…”

Well-posedness of general boundary-value problems for scalar conservation laws

“…The technique of this section relies upon translation arguments for proving localized BV estimates. It goes back to Bürger et al [13,14] where the idea was introduced in the context of finite volume numerical approximations.…”

“…Second, we give several stability results for entropy solutions on the hyperbolic problem (H ϕ,β (u 0 , f )), with a focus on stability with respect to different approximations of the BC graphs β. In Section 7, first we improve the existence results in the one-dimensional case, dropping most of the assumptions on ϕ and β with the help of the BV loc estimates due to Bürger, Karlsen, García and Towers [13,14]. Second, following Eymard, Gallouët and Herbin [19] we present a notion of entropy-process solution that is useful in order to prove convergence of approximations with only weak compactness properties; it can be exploited under the additional, quite restrictive assumption that an entropy solution exists already.…”

“…Then we can use the idea of [13,Lemma 4.2] and [14,Lemma 5.4]: for a > 0, using the mean-value theorem for each ε > 0 we can find a contour (0, T ) × {c ε } with 0 < c ε < a such that TotVar u ε along these contours is uniformly bounded by C a . The variation of u 0 is also bounded, therefore using the classical estimate of Bardos, LeRoux and Nédélec [9] for the Dirichlet problem for viscous conservation law (with boundary datum given by the values of u ε on our contour), we get the bound…”

Optimal results on TV bounds for scalar conservation laws with discontinuous flux

“…Due to the lack of the total variation bound the convergence theory for the discontinuous flux has been studied by using different technique, namely, singular mapping technique in [43,52,6,39], [7]. It has been noticed in [12] that the solution is of total variation bounded away from the interface x = 0. But the information at the interface x = 0 was completely unknown.…”