A stochastic asymptotic-preserving scheme for a kinetic-fluid model for disperse two-phase flows with uncertainty

Abstract

In this paper we consider a kinetic-fluid model for disperse two-phase flows with uncertainty. We propose a stochastic asymptotic-preserving (s-AP) scheme in the generalized polynomial chaos stochastic Galerkin (gPC-sG) framework, which allows the efficient computation of the problem in both kinetic and hydrodynamic regimes. The sAP property is proved by deriving the equilibrium of the gPC version of the Fokker-Planck operator. The coefficient matrices that arise in a Helmholtz equation and a Poisson equation, essential ingredients of the algorithms, are proved to be positive definite under reasonable and mild assumptions. The computation of the gPC version of a translation operator that arises in the inversion of the Fokker-Planck operator is accelerated by a spectrally accurate splitting method. Numerical examples illustrate the sAP property and the efficiency of the gPC-sG method in various asymptotic regimes.

Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling

“…By simply changing dµ in the proof above to dµ = e −φ M (v) dvdxπ(z)dz, one can reach the same result as Theorem 3.2-the uniform regularity of f in the random space. However, proving the uniform convergence of the stochastic Galerkin method for (20) is more complicated and remains a further investigation.…”

“…Compared with the classical Monte-Carlo method, the gPC-SG approach enjoys a spectral accuracy in the random space-if the solution is sufficient regular-while the Monte-Carlo method converges with only half-th order accuracy. For recent activities for uncertainty quantificaiton in kinetic theory, we refer to a recent review article [11], which surveyed recent results in the study of kinetic equations with random inputs, [15,3,10,33,14,27,22,16,31,20], including their mathematical properties such as regularity and long-time behavior in the random space, construction of efficient stochastic Galerkin methods and handling of multiple scales by s-AP schemes.…”

“…Compared with the classical Monte-Carlo method, the gPC-SG approach enjoys a spectral accuracy in the random space-if the solution is sufficient regular-while the Monte-Carlo method converges with only half-th order accuracy. For recent activities for uncertainty quantificaiton in kinetic theory, we refer to a recent review article [11], which surveyed recent results in the study of kinetic equations with random inputs, [15,3,10,33,14,27,22,16,31,20], including their mathematical properties such as regularity and long-time behavior in the random space, construction of efficient stochastic Galerkin methods and handling of multiple scales by s-AP schemes.Recently, the authors in [28] have provided a general framework using hypocoercivity to study general class of linear and nonlinear kinetic equations with uncertainties from the initial data or collision kernels in both incompressible Navier-Stokes and Euler (acoustic) regimes. For initial data near the global Maxwellian, an exponential convergence of the random solution toward the deterministic global equilibrium, a spectral accuracy and exponential decay of the numerical error of the SG system have been established.…”

Hypocoercivity Based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs

“…While uncertainty quantification has been a hot topic in scientific and engineering computing in the last two decades, research on uncertainty quantification for kinetic equations has been relatively recent. We refer to recent review articles [23,11] and some recent works [30,22,10,27,26,8,33,34,28,31,29,39,1] in this direction. The first sensitivity analysis similar to this paper for the linear transport equation, with uniform (in the Knudsen number) spectral convergence of the gPC-SG approximation, was given by Jin, Liu and Ma in [26].…”

A computational method for solving stochastic Itô–Volterra integral equation with multi-stochastic terms

“…Nowadays, modelling different problems in different issues of science leads to stochastic equations [1]. These equations arise in many fields of science such as mathematics and statistics [2][3][4][5][6][7], finance [8][9][10], physics [11][12][13], mechanics [14,15], biology [16][17][18], and medicine [19,20]. Whereas most of them do not have an exact solution, the role of numerical methods and finding a reliable and accurate numerical approximation have become highlighted [21].…”