Monte Carlo stochastic Galerkin methods for the Boltzmann equation with uncertainties: space-homogeneous case

Abstract

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.

References in corpus (5)

• Multi-scale control variate methods for uncertainty quantification in kinetic equations
• Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertainties
• Boltzmann-type models with uncertain binary interactions
• A bi-fidelity method for the multiscale Boltzmann equation with random parameters
• Uncertainty quantification for the BGK model of the Boltzmann equation using multilevel variance reduced Monte Carlo methods

Cited by in corpus (5)

• Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties
• Multivariate error modeling and uncertainty quantification using importance (re-)weighting for Monte Carlo simulations in particle transport
• Uncertainty quantification for the BGK model of the Boltzmann equation using multilevel variance reduced Monte Carlo methods
• Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties
• Uncertainty quantification and control of kinetic models of tumour growth under clinical uncertainties