A note on the central-upwind scheme for nonlocal conservation laws

The central-upwind flux is a widely used numerical flux function for local conservation laws. It has been investigated by Kurganov and Polizzi (2009) for a specific nonlocal conservation law and can be derived from a fully-discrete second-order scheme. Here, we derive this fully-discrete scheme in detail with a particular focus on the occurring nonlocal terms. In addition, we derive the central-upwind flux for a class of nonlocal conservation laws and use an estimate on the nonlocal speed which fixes the nonlocality at the cell interfaces. We prove that the resulting first-order numerical scheme converges to the correct solution. Under additional assumptions on the analytical flux we present a similar result for a second-order central-upwind scheme. Numerical examples compare the central-upwind schemes to Godunov-type schemes and the fully-discrete scheme.

💡 Research Summary

The paper addresses the numerical treatment of one‑dimensional scalar conservation laws with a spatial nonlocal term, namely
 ∂ₜρ(t,x) + ∂ₓ F(ρ(t,x), (ω_η * ρ)(t,x)) = 0,
where the flux is of the product form F(ρ,R)=g(ρ)v(R) and the convolution (ω_η * ρ) uses a forward kernel of compact support