  Using accurate multi-component diffusion treatment in numerical combustion
studies remains formidable due to the computational cost associated with
solving for diffusion velocities. To obtain the diffusion velocities, for low
density gases, one needs to solve the Stefan-Maxwell equations along with the
zero diffusion flux criteria, which scales as $\mathcal{O}(N^3)$, when solved
exactly. In this article, we propose an accurate, fast, direct and robust
algorithm to compute multi-component diffusion velocities. To our knowledge,
this is the first provably accurate algorithm (the solution can be obtained up
to an arbitrary degree of precision) scaling at a computational complexity of
$\mathcal{O}(N)$ in finite precision. The key idea involves leveraging the fact
that the matrix of the reciprocal of the binary diffusivities, $V$, is low
rank, with its rank being independent of the number of species involved. The
low rank representation of matrix $V$ is computed in a fast manner at a
computational complexity of $\mathcal{O}(N)$ and the Sherman-Morrison-Woodbury
formula is used to solve for the diffusion velocities at a computational
complexity of $\mathcal{O}(N)$. Rigorous proofs and numerical benchmarks
illustrate the low rank property of the matrix $V$ and scaling of the
algorithm.
