Convergence rate for a semidiscrete approximation of scalar conservation laws

We propose a semidiscrete scheme for approximation of entropy solutions of one-dimensional scalar conservation laws with nonnegative initial data. The scheme is based on the concept of particle paths for conservation laws and can be interpreted as a finite-particle discretization. A convergence rate of order $1\mathrm{/}2$ with respect to initial particle spacing is proved. As a special case, this covers the convergence of the Follow–the–Leader model to the Lighthill–Whitham–Richards model for traffic flow.