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.
Highlights
Mixtures of rarefied gases are found in a wide variety of systems, ranging from the re-entry of an interplanetary probe in the upper atmosphere [21] to microscale flows in pumps which use no moving parts
Kinetic models, such as the seminal Boltzmann equation, are favoured to describe these systems because they are able to reflect the non-equilibrium character of the gases, retaining information about the microscopic many-particle dynamics while avoiding the sheer complexity of the microscopic approach
Whereas the mathematical properties of the classical Boltzmann equation for a single species gas are well known, many questions remain open for the multispecies case
Summary
Mixtures of rarefied gases are found in a wide variety of systems, ranging from the re-entry of an interplanetary probe in the upper atmosphere [21] to microscale flows in pumps which use no moving parts (viz. the Knudsen compressor, [37]). Such complex gases cannot be described by classical fluid models, such as the compressible Euler or the Navier-Stokes systems, because of their non-equilibrium behaviour, induced by their rarefaction. Kinetic models, such as the seminal Boltzmann equation, are favoured to describe these systems because they are able to reflect the non-equilibrium character of the gases, retaining information about the microscopic many-particle dynamics while avoiding the sheer complexity of the microscopic approach. Whereas the mathematical properties of the classical Boltzmann equation for a single species gas are well known (crucially, its derivation from Newtonian dynamics was addressed in [28]), many questions remain open for the multispecies case. The method was not asymptotic preserving (AP), namely, stable in
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