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 Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. 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. As a new contribution, it is demonstrated in the present work, first, that these schemes 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. Second, it is shown that 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.

A well‐balanced and entropy stable scheme for a reduced blood flow model

“…Let us also recall that a convex scalar function 𝜂 = 𝜂(w) ∈ C 1 is an entropy for the system of conservation laws 𝜕 t w + 𝜕 x f(w) = 0, (6) if there exists an associated entropy flux G = G(w) that satisfies ∇G(w) = v T H(w), (7) where v = ∇𝜂(w) is the vector of entropy variables. If a function G satisfying (7) exists, then (𝜂, G) is called an entropy pair for the conservation law (6).…”

“…Certain multispecies kinematic flow models (e.g., of vehicular traffic [3,6]) lead to first-order systems of conservation laws of arbitrary size that are endowed with an entropy pair and admit the construction of entropy stable schemes [8]. Such schemes can even be applied to certain systems of degenerate parabolic equations [21], although the class of diffusion coefficients compatible with the entropy variable framework is very narrow, as is elaborated in [7].…”