Invariant-region-preserving WENO schemes for one-dimensional multispecies kinematic flow models

TL;DR

This paper introduces high-order invariant-region-preserving WENO schemes for multispecies kinematic flow models, ensuring physically relevant solutions in applications like traffic flow and sedimentation.

Contribution

It develops a new family of high-order finite volume schemes with linear scaling limiters that preserve the invariant region for coupled conservation laws.

Findings

Schemes preserve nonnegativity and sum constraints of solutions.

Theoretical proof of IRP property under CFL condition.

Numerical simulations confirm effectiveness in traffic and sedimentation models.

Abstract

Multispecies kinematic flow models are defined by systems of N strongly coupled, nonlinear first-order conservation laws, where the solution is a vector of N partial volume fractions or densities. These models arise in various applications including multiclass vehicular traffic and sedimentation of polydisperse suspensions. The solution vector should take values in a set of physically relevant values (i.e., the components are nonnegative and sum up at most to a given maximum value). It is demonstrated that this set, the so-called invariant region, is preserved by numerical solutions produced by a new family of high-order finite volume numerical schemes adapted to this class of models. To achieve this property, and motivated by [X. Zhang, C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys. 229 (2010) 3091--3120], a pair of linear scaling limiters is applied to a high-order weighted essentially non-oscillatory (WENO) polynomial reconstruction to obtain invariant-region-preserving (IRP) high-order polynomial reconstructions. These reconstructions are combined with a local Lax-Friedrichs (LLF) or Harten-Lax-van Leer (HLL) numerical flux to obtain a high-order numerical scheme for the system of conservation laws. It is proved that this scheme satisfies an IRP property under a suitable Courant-Friedrichs-Lewy (CFL) condition. The theoretical analysis is corroborated with numerical simulations for models of multiclass traffic flow and polydisperse sedimentation.