Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data

TL;DR

This paper establishes the optimal convergence rate of the multitype sticky particle approximation for one-dimensional diagonal hyperbolic systems with monotonic initial data, providing theoretical error bounds and numerical validation.

Contribution

It introduces a detailed analysis of the L1 error for the multitype sticky particle method, proving the optimal convergence rate and examining the effects of iterative schemes.

Findings

Error at time t is bounded by (1 + t)/n, which is proven to be optimal.

The convergence rate of the approximation is established as optimal.

Numerical simulations confirm the theoretical convergence results.

Abstract

Brenier and Grenier [SIAM J. Numer. Anal., 1998] proved that sticky particle dynamics with a large number of particles allow to approximate the entropy solution to scalar one-dimensional conservation laws with monotonic initial data. In [arXiv:1501.01498], we introduced a multitype version of this dynamics and proved that the associated empirical cumulative distribution functions converge to the viscosity solution, in the sense of Bianchini and Bres-san [Ann. of Math. (2), 2005], of one-dimensional diagonal hyperbolic systems with monotonic initial data of arbitrary finite variation. In the present paper, we analyse the L 1 error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time t is bounded from above by a term of order (1 + t)/n, where n denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations.