Coarse graining, dynamic renormalization and the kinetic theory of shock clustering

TL;DR

This paper applies an equation-free approach to study shock clustering in scalar conservation laws with disordered initial conditions, successfully modeling the kinetics via coagulation equations and developing a particle scheme for self-similar solutions.

Contribution

It introduces a novel application of the equation-free methodology to scalar conservation laws, enabling the computation of shock clustering dynamics and self-similar solutions.

Findings

Validated the numerical scheme against exact solutions in Burgers turbulence.

Demonstrated the coagulation model describes shock clustering kinetics.

Developed a particle scheme for self-similar solutions with fat tails.

Abstract

We demonstrate the utility of the equation free methodology developed by one of the authors (I.G.K) for the study of scalar conservation laws with disordered initial conditions. The numerical scheme is benchmarked on exact solutions in Burgers turbulence corresponding to Levy process initial data. For these initial data, the kinetics of shock clustering is described by Smoluchowski's coagulation equation with additive kernel. The equation free methodology is used to develop a particle scheme that computes self-similar solutions to the coagulation equation, including those with fat tails.