Abstract
In this paper we study the general discrete-velocity models of Boltzmann equation with uncertainties from collision kernel and random inputs. We follow the framework of Kawashima and extend it to the case of diffusive scaling in a random setting. First, we provide a uniform regularity analysis in the random space with the help of a Lyapunov-type functional, and prove a uniformly (in the Knudsen number) exponential decay towards the global equilibrium, under certain smallness assumption on the random perturbation of the collision kernel, for suitably small initial data. Then we consider the generalized polynomial chaos based stochastic Galerkin approximation (gPC-SG) of the model, and prove the spectral convergence and the exponential time decay of the gPC-SG error uniformly in the Knudsen number.
Highlights
17 In this paper, we are interested in the discrete-velocity models (DVMs) of the Boltzmann equations with
1 of this paper is to study the general DVMs of the Boltzmann equations under the influence of random 2 uncertainties from the collision kernel and initial data
In this paper, by extending the framework of Kawashima for deterministic models to the case of random DVMs, we carry out the regularity and local sensitivity analysis for DVMs with random inputs in the initial data and collision coe cients, in di↵usive scaling under periodic boundary condition
Summary
17 In this paper, we are interested in the discrete-velocity models (DVMs) of the Boltzmann equations with. 1 of this paper is to study the general DVMs of the Boltzmann equations under the influence of random 2 uncertainties from the collision kernel and initial data. Compared to direct simulation methods 7 such as Monte Carlo [1, 5], gPC-SG is more e cient and accurate if the solution is smooth enough 8 in the random space Another typical di culty in kinetic modeling is due to small scales determined 9 by the mean free path, relaxation time, etc. By extending the framework of Kawashima for deterministic models to the case of random DVMs, we carry out the regularity and local sensitivity analysis for DVMs with random inputs in the initial data and collision coe cients, in di↵usive scaling under periodic boundary condition.
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