  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.
