An accurate, fast, mathematically robust, universal, non-iterative algorithm for computing multi-component diffusion velocities
For researchers in numerical combustion and gas dynamics, this algorithm drastically reduces the computational cost of multi-component diffusion, enabling more efficient simulations.
This paper presents an algorithm for computing multi-component diffusion velocities that achieves O(N) complexity, a significant improvement over the standard O(N^3) exact solution, by exploiting the low-rank structure of the binary diffusivity matrix. Numerical benchmarks confirm the algorithm's accuracy and scaling.
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.