An accurate, fast, mathematically robust, universal, non-iterative algorithm for computing multi-component diffusion velocities

TL;DR

This paper introduces a novel, mathematically robust algorithm that computes multi-component diffusion velocities in combustion studies with linear complexity, significantly improving speed while maintaining high accuracy.

Contribution

The authors develop the first provably accurate, non-iterative algorithm for diffusion velocities that scales as O(N), leveraging low-rank matrix properties and advanced matrix inversion techniques.

Findings

Algorithm achieves O(N) computational complexity.

Numerical benchmarks confirm high accuracy and efficiency.

Matrix of reciprocal diffusivities is proven to be low rank.

Abstract

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.