Linear Transport Under Uncertainty

Abstract

The aim of this chapter, based on [18], is to present accurate and stable numerical schemes for the solution of a class of linear diffusive transport problems. The advection-diffusion equation subject to uncertain viscosity with known statistical description is represented by a spectral expansion in the stochastic dimension. The gPC framework and the stochastic Galerkin method are used to obtain an extended system which is analyzed to find discretization constraints on monotonicity, stiffness and stability. A comparison of stochastic Galerkin versus methods based on repeated evaluations of deterministic solutions, such as stochastic collocation, is provided but this is not our primary focus. However, we do include a few examples on relative performance and numerical properties with respect to monotonicity requirements and convergence to steady-state, to encourage the use of stochastic Galerkin methods.Special care is exercised to ensure that the stochastic Galerkin projection results in a system with a positive semidefinite diffusion matrix. The sign of the eigenvalues of a pure advection problem is not a stumbling block as long as the boundary conditions are properly adjusted to match the number of ingoing characteristics, as shown in [7]. Unlike the case of stochastic advection, the sign of the eigenvalues of the diffusion matrix of the advection-diffusion problem is crucial. A negative eigenvalue leads to the growth of the solution norm and hence numerical instability. The source of the growth is in the volume term, and no treatment of the boundary conditions can eliminate it.Advection-diffusion problems with uncertainty have been investigated by several authors. Ghanem and Dham