Stochastic Galerkin Methods for Time-Dependent Radiative Transfer Equations with Uncertain Coefficients

The generalized polynomial chaos (gPC) method is one of the most popular method for uncertainty quantification. Being essentially a spectral approach, the gPC method exhibits the spectral convergence rate which heavily depends on the regularity of the solution in the random space. Many regularity studies have been made for stochastic elliptic and parabolic equations while regularities studies of stochastic hyperbolic equations has long been infeasible due to its intrinsic difficulties. In this paper, we investigate the impact of uncertainty on the time-dependent radiative transfer equation (RTE) with nonhomogeneous boundary conditions, which sits somewhere between hyperbolic and parabolic equations. We theoretically prove the a-priori bound of the solution, its continuity with respect to the scattering coefficient, and its regularity in the random space. These studies can serve as a building block in understanding the influence of uncertainties in the passage from hyperbolic to parabolic equations. Moreover, we vigorously justify the validity of the gPC expansion ansatz based on the regularity study. Then the stochastic Galerkin method of the gPC approach is employed to discretize the random variable. We further conduct a delicate analysis to show the exponential decay rate of the gPC coefficients and establish the error estimates of the stochastic Galerkin approximation for both one-dimensional and multi-dimensional random space cases. Numerical tests are presented to verify our analytical results.