Numerical schemes for a class of nonlocal conservation laws: a general approach

TL;DR

The paper addresses nonlocal conservation laws where a nonlocal transport term complicates standard local solvers. It introduces a general finite-volume framework that first discretizes the nonlocal term via quadrature and then advances with a flux of the form $F^n_{j+1/2}(a,b)=G(a,b)\,V_j^n$, with verifiable conditions on $G$ ensuring convergence to the unique weak entropy solution. The authors prove convergence under a CFL condition, establish a discrete maximum principle and BV/time-continuity estimates, and demonstrate the approach on an Arrhenius-type look-ahead traffic model and a sedimentation model, illustrating versatility and accuracy relative to existing methods. The work provides a flexible, general toolkit for nonlocal conservation laws beyond the tested cases and highlights potential extensions to higher-order schemes and asymptotic regimes as the nonlocal horizon vanishes.

Abstract

In this work we present a rather general approach to approximate the solutions of nonlocal conservation laws. In a first step, we approximate the nonlocal term with an appropriate quadrature rule applied to the spatial discretization. Then, we apply a numerical flux function on the reduced problem. We present explicit conditions which such a numerical flux function needs to fulfill. These conditions guarantee the convergence to the weak entropy solution of the considered model class. Numerical examples validate our theoretical results and demonstrate that the approach can be applied to other nonlocal problems.

Figures (3)

• Figure 1: Space discretization and approximation of the nonlocal term for ${\eta}=3\Delta x$ with the densities used to calculated $V_j^n$ highlighted in gray.
• Figure 2: Approximate solutions obtained by the different schemes from chiarello2018global and \ref{['eq:LxFmod']}--\ref{['eq:EOflux']} with $\Delta x=0.01$ and a reference solution (ref. sol.), left, and a zoom into the spatial domain, right.
• Figure 3: Approximate solutions obtained by the different schemes from betancourt2011nonlocal and \ref{['eq:LxFmod']}--\ref{['eq:EOflux']} with $\Delta x=0.2\cdot 2^{-5}$ and a reference solution (ref. sol.) at $T=3$ for the first initial condition, left and at $T=1$ for the second initial condition, right.

Theorems & Definitions (23)

• Definition 2.2
• Theorem 2.3: compare to Theorem 1.2 of chiarello2018global
• Definition 3.1: Approximation of the nonlocal term
• Definition 3.2: Numerical flux function
• Remark 3.3
• Remark 3.4: Generalization of the flux
• Remark 3.5: Comparison to other general approaches in the literature
• Remark 3.6: Higher order extensions
• Theorem 4.1
• Proposition 4.2