Efficient Stochastic Asymptotic-Preserving Implicit-Explicit Methods for Transport Equations with Diffusive Scalings and Random Inputs

Abstract

For linear transport and radiative heat transfer equations with random inputs, we develop new generalized polynomial chaos based Asymptotic-Preserving stochastic Galerkin schemes that allow efficient computation for the problems that contain both uncertainties and multiple scales. Compared with previous methods for these problems, our new method use the implicit-explicit (IMEX) time discretization to gain higher order accuracy, and by using a modified diffusion operator based penalty method, a more relaxed stability condition-a hyperbolic, rather than parabolic, CFL stability condition, is achieved in the case of small mean free path in the diffusive regime. The stochastic Asymptotic-Preserving property of these methods will be shown asymptotically, and demonstrated numerically, along with computational cost comparison with previous methods.

Multi-fidelity methods for uncertainty propagation in kinetic equations

“…Linear transport equations. -Another class of kinetic models that will be considered in the sequel are the linear transport equation under diffusive scaling and with random parameters [45,49,54]. Let f (t, x, v, z) be the probability density distribution of particles at time t > 0, position x ∈ D ⊆ R, and with v ∈ [−1, 1] the cosine of the angle between the particle velocity and its position.…”

“…In spite of the vast amount of existing research on the approximation of Boltzmann and related equations, the study of kinetic equations with stochastic terms has been considered only in recent years [30,41,45,53,78,84,86,87,92,108]. See in particular the recent collection [46] and the survey [69].…”

“…Most of the literature on quantification of uncertainties in kinetic equations is based on the use of Stochastic-Galerkin methods built via generalized Polynomial Chaos (gPC) expansions [19,40,45,53,92,101,108]. Only recently these problems have been analyzed in the framework of statistical sampling methods based on Monte Carlo (MC) techniques [23,24,26,33,41].…”

Multi-scale control variate methods for uncertainty quantification in kinetic equations

“…Kinetic equations have been applied to model a variety of phenomena whose multiscale nature cannot be described by a standard macroscopic approach [4,9,43]. In spite of the vast amount of existing research, both theoretically and numerically (see [12,42] for recent surveys), the study of kinetic equations has mostly remained deterministic and only recently a systematic study of the effects of uncertainty has been undertaken [16,45,23,13,20,37]. In reality, there are many sources of uncertainties that can arise in these equations, like incomplete knowledge of the interaction mechanism between particles/agents, imprecise measurements of the initial and boundary data and other sources of uncertainty like forcing and geometry, etc.…”

“…The simplest class of numerical methods for quantifying uncertainty in partial differential equations are the stochastic collocation methods. In contrast to stochastic Galerkin (SG) methods based on generalized Polynomial Chaos (gPC) (see [1,16,45,23,13,20,37] and the volume [19] for applications to kinetic equations and the references therein), stochastic collocation methods are non-intrusive, so they preserve all features of the deterministic numerical scheme, and easy to parallelize [44]. Here we consider the closely related class of statistical sampling methods based on Monte Carlo (MC) techniques.…”