Computation of the self-diffusion coefficient with low-rank tensor methods: application to the simulation of a cross-diffusion system

TL;DR

This paper introduces a novel deterministic low-rank tensor method to compute the self-diffusion coefficient for cross-diffusion systems, enabling efficient simulation of multi-species particle models.

Contribution

It proposes a new low-rank tensor approach to compute the self-diffusion coefficient, replacing traditional long-time limit estimations.

Findings

Efficient computation of the self-diffusion coefficient using low-rank tensor methods.

Successful simulation of cross-diffusion systems with the new approach.

Demonstrated advantages over classical algorithms in accuracy and computational cost.

Abstract

Cross-diffusion systems arise as hydrodynamic limits of lattice multi-species interacting particle models. The objective of this work is to provide a numerical scheme for the simulation of the cross-diffusion system identified in [J. Quastel, Comm. Pure Appl. Math., 45 (1992), pp. 623--679]. To simulate this system, it is necessary to provide an approximation of the so-called self-diffusion coefficient matrix of the tagged particle process. Classical algorithms for the computation of this matrix are based on the estimation of the long-time limit of the average mean square displacement of the particle. In this work, as an alternative, we propose a novel approach for computing the self-diffusion coefficient using deterministic low-rank approximation techniques, as the minimum of a high-dimensional optimization problem. The computed self-diffusion coefficient is then used for the simulation of the cross-diffusion system using an implicit finite volume scheme.