Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs
Provides rigorous theoretical guarantees for uncertainty quantification in kinetic equations with multiple scales, benefiting computational scientists modeling rarefied gas dynamics with random parameters.
This paper develops a general framework for sensitivity analysis and spectral convergence of stochastic Galerkin approximations for linear and nonlinear kinetic equations with random inputs, proving exponential convergence to equilibrium and spectral accuracy in weighted Sobolev norms.
In this paper, we provide a general framework to study general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin approximations, in both incompressible Navier-Stokes and Euler (acoustic) regimes. First, we show that the general framework put forth in [C. Mouhot and L. Neumann, Nonlinearity, 19, 969-998, 2006, M. Briant, J. Diff. Eqn., 259, 6072-6141, 2005] based on hypocoercivity for the deterministic kinetic equations can be easily adopted for sensitivity analysis for random kinetic equations, which gives rise to an exponential convergence of the random solution toward the (deterministic) global equilibrium, under suitable conditions on the collision kernel. Then we use such theory to study the stochastic Galerkin (SG) methods for the equations, establish hypocoercivity of the SG system and regularity of its solution, and spectral accuracy and exponential decay of the numerical error of the method in a weighted Sobolev norm.