Numerical Approximation of a Macroscopic Model of Pedestrian Flows

Abstract

This paper is concerned with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians. Solutions of the associated Riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate the Lax inequalities, which makes their numerical approximation very sensitive. This study proposes to extend the transport‐equilibrium strategy proposed in [C. Chalons, C. R. Acad. Sci. Paris Ser. I, 342 (2006), pp. 623–626] and [C. Chalons, Transport‐equilibrium schemes for computing nonclassical shocks. I. Scalar conservation laws, submitted] for computing the nonclassical solutions of scalar conservation laws with either a concave‐convex or a convex‐concave flux function and supplemented with an invertible kinetic function. These strong properties are not fulfilled in the present setting, since the flow function admits several inflection points and the kinetic function is not invertible. We nevertheless succeed in obtaining an efficient numerical scheme. Numerical evidence is provided.