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

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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  • 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.

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


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