A Godunov type scheme for a class of LWR traffic flow models with non-local flux

TL;DR

This paper introduces a Godunov type numerical scheme for non-local traffic flow models that improves accuracy over traditional methods, with proven stability and well-posedness.

Contribution

It develops a novel Godunov scheme for scalar conservation laws with non-local flux, including stability analysis and numerical validation.

Findings

The scheme outperforms Lax-Friedrichs in accuracy.

Provides $L^inity$ and BV estimates for solutions.

Numerical examples confirm improved performance.

Abstract

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide $L^\infty$ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.