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Preprint 2018-24

Raimund Bürger, Paul E. Méndez, Carlos Parés:

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

Abstract:

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection-diffusion equations were recently proposed in [S. Jerez, C. Parés, Entropy stable schemes for degenerate convection-diffusion equations, SIAM J. Numer. Anal. vol. 55 (2017) pp. 240-264]. These schemes extend the theoretical framework by E. Tadmor [ The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comp. vol. 49 (1987) pp. 91-103] to convection-diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations.It is demonstrated that this formulation can naturally be extended to initial-boundary value problems with zero-flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy.The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first-order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. These assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero-flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.

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