Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures

TL;DR

This paper introduces explicit projective and telescopic projective integration methods for solving multispecies Boltzmann and BGK equations efficiently, especially in stiff, hyperbolic regimes with small Knudsen numbers.

Contribution

It develops a hierarchical extrapolation scheme that reduces computational complexity and enables efficient simulation of complex kinetic mixtures.

Findings

Effective handling of extreme mass ratios.

Robustness in fluid instability simulations.

Computational complexity independent of stiffness.

Abstract

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and Bhatnagar-Gross-Krook (BGK) equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.