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

• Published in: arXiv.org 2022-10
• Main Authors: Dabaghi, Jad, Ehrlacher, Virginie, Strössner, Christoph
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Computation; Diffusion coefficient; Finite volume method; Mathematical analysis; Optimization; Self diffusion; Simulation; Tensors
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2111.11349

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.