7 research outputs found
Monte Carlo stochastic Galerkin methods for the Boltzmann equation with uncertainties: space-homogeneous case
In this paper we propose a novel numerical approach for the Boltzmann
equation with uncertainties. The method combines the efficiency of classical
direct simulation Monte Carlo (DSMC) schemes in the phase space together with
the accuracy of stochastic Galerkin (sG) methods in the random space. This
hybrid formulation makes it possible to construct methods that preserve the
main physical properties of the solution along with spectral accuracy in the
random space. The schemes are developed and analyzed in the case of space
homogeneous problems as these contain the main numerical difficulties. Several
test cases are reported, both in the Maxwell and in the variable hard sphere
(VHS) framework, and confirm the properties and performance of the new methods
An introduction to uncertainty quantification for kinetic equations and related problems
We overview some recent results in the field of uncertainty quantification
for kinetic equations and related problems with random inputs. Uncertainties
may be due to various reasons, such as lack of knowledge on the microscopic
interaction details or incomplete information at the boundaries or on the
initial data. These uncertainties contribute to the curse of dimensionality and
the development of efficient numerical methods is a challenge. After a brief
introduction on the main numerical techniques for uncertainty quantification in
partial differential equations, we focus our survey on some of the recent
progress on multi-fidelity methods and stochastic Galerkin methods for kinetic
equations