A nonlocal model for heterogeneous material flow on conveyor belts

Jul 16, 2026

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A finite volume approximation scheme is developed to solve a two-dimensional nonlocal macroscopic material flow model with explicit treatment of boundaries appearing in the nonlocal terms. Building on a scalar result, the framework is extended to a heterogeneous multi-class system on bounded domains. Convergence of Roe-type approximate solutions with dimensional splitting is proved, focusing on handling the flux discontinuity via a regularized flux and establishing BV bounds. Numerical experiments validate agreement with microscopic simulations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages30

Reading Time1 h 16 min

Introduction

Nonlocal conservation laws and Cauchy/initial-boundary value formulation

Conveyor-belt velocity fields and collision operator

Boundary modeling via augmented convolution kernel

Extension to the whole-space reformulation

arXiv :2510 . 17500v2 [math .NA] 12 Jul 2026 A nonlocal model for heterogeneous material flow on conveyor belts Paola Goatin 1 Simone G¨ottlich2 Fabian Ziegler2 July 14, 2026 Abstract In this paper, a finite volume approximation scheme is used to solve a nonlocal macroscopic material flow model in two space dimensions, accounting for the presence of boundaries in the nonlocal terms. Based on a previous result for the scalar case, we extend the setting to a system of heterogeneous material on bounded domains. We prove the convergence of the approximate solutions constructed using the Roe scheme with dimensional splitting, where the major challenge lies in the treatment of the discontinuity occurring in the flux function. We consider a regularized version of the flux function and establish BV bounds for a Roe-type scheme. Numerical tests show a good agreement with microscopic simulations. Keywords: nonlocal systems of conservation laws; heterogeneous material flow; Roe scheme; boundary conditions. 2010 Mathematics Subject Classification: 35L65, 65M12 1 Introduction In this paper, we consider the Cauchy problem for a system of nonlocal conservation laws in two space dimensions describing the movement of N ∈ N classes of objects of different sizes and corresponding maximal densities Rc > 0, c = 1 , ... , N, on a domain Ω ⊆ R2 . This extends a previous result for the scalar case [14] . The initial-boundary value problem for the particle densities ρ = (ρ1 , ... ,ρN ) as a function of time t and position (x, y) ∈ Ω is given by 􀀸 􀀾 􀀼 􀀾 􀀺 ∂tρc + ∇ · 􀀒ρc 􀀐 vstat(x, y) + vdync[ρ](x, y)􀀑􀀓 = 0 , (t, x, y) ∈[0, T] × Ω , c = 1 , ... , N, ρc (0, x, y) = ρco(x, y), (x, y) ∈ Ω , (1.1) 1 Universit´e Cˆote d’Azur, Inria, CNRS, LJAD, 2004 route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France. E-mail: [paola.goatin@inria.fr](paola.goatin@inria.fr) 3 University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany. Email: [goettlich@uni-mannheim.de](goettlich@uni-mannheim.de) for any T > 0, where for weighting factors εc > 0 Nr = X αcρc , c=1 vdync[ρ] = H(η˜ ∗ r − rmax) Ic[r] with Ic[r] = −εc ∇ (ηc∗ r) q 1 + 􀀍 ∇ (ηc∗ r) 􀀍 2 . (1.2) Above, H denotes the Heaviside function, which becomes active whenever the maximal density rmax > 0 is exceeded, while η˜,ηc are positive, smooth mollifiers depending on the object sizes. The maximal density rmax is expressed in terms of the smallest object density. This concept, referred to as pce, originates from traffic flow modeling, where it is employed to account for the varying sizes of vehicles on the road [15] . Smaller objects occupy less space, and thus their maximal density is higher than that of larger objects. The weighted total density r = P αcρc is defined as the sum of the individual particle densities ρc , rescaled by a weight αc. The weights αc are determined by comparing the maximum density rcmax of particle class c to that of the smallest particle class. Without loss of generality, we assume that c = 1 denotes the class of the smallest particles, so that r1max > rcmax for all c > 1. We then set rmax = r1max and αc = r1max/rcmax. The scalar version of 1.1 was introduced in [10], to model the flow of particles on a conveyor belt. The velocity vector field induced by the conveyor belt is denoted by vstat and is independent of both time and the density ρ, and common to all particle classes. For an illustration of the geometry of the conveyor belt, we refer to Section 4, Figure 1 . Below the maximal density rmax, parts are transported with the velocity of the conveyor belt. If the density exceeds rmax, the dynamic velocity vector field vdync becomes active and models the collision of particles via the operator Ic[r] . The negative gradient of the convolution ηc ∗ r pushes the mass towards lower density regions. The particular choice of the collision operator Ic(r) was introduced in [5] to describe crowd dynamics. In particular

FAQs

What problem does the paper address?

The paper studies a two-dimensional nonlocal conservation law system modeling heterogeneous material flow on conveyor belts, including boundary effects through nonlocal terms.

How are boundary conditions incorporated into the nonlocal model?

Boundaries are handled by modifying the convolution kernel using an augmented total density defined on the whole space, ensuring the domain remains invariant via an outward normal constraint.

How is the numerical method constructed and analyzed for convergence?

A finite volume scheme is used with Roe-type updates and dimensional splitting; the main difficulty is the flux discontinuity, which is managed using a regularized flux function and BV bounds.

Jul 16, 2026

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