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

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