Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel

Esther S. Daus; Shi Jin; Liu Liu

doi:10.3934/krm.2019034

Article Contents

2019, Volume 12, Issue 4: 909-922. Doi: 10.3934/krm.2019034

This issue Previous Article An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems Next Article Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws

Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel

1.
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria
2.
School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China
3.
Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78705, USA

Received: October 2018

Revised: January 2019

Published: May 2019

The first author acknowledges partial support from the Austrian Science Fund (FWF), grants P27352 and P30000, the second author is supported by NSF grants DMS-1522184, DMS-1819012 and DMS-1107291: RNMS KI-Net, NSFC grants No. 31571071 and No. 11871297, the third author is supported by the funding DOE–Simulation Center for Runaway Electron Avoidance and Mitigation, project No. DE-SC0016283.

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Abstract

In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018], spectral convergence and long-time decay of the numerical solution towards the global equilibrium of the stochastic Galerkin approximation for the Boltzmann equation with random inputs in the initial data and collision kernel for hard potentials and Maxwellian molecules under Grad's angular cutoff were established using the hypocoercive properties of the collisional kinetic model. One assumption for the random perturbation of the collision kernel is that the perturbation is in the order of the Knudsen number, which can be very small in the fluid dynamical regime. In this article, we remove this smallness assumption, and establish the same results but now for random perturbations of the collision kernel that can be of order one. The new analysis relies on the establishment of a spectral gap for the numerical collision operator.

Keywords:

The Boltzmann equation with uncertainties,
multiple scales,
large random perturbation,
hypocoercivity,
gPC stochastic Galerkin method,
sensitivity analysis.

Mathematics Subject Classification: Primary: 35Q20; Secondary: 65M70.

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References

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