Stochastic variational multiscale analysis of the advection–diffusion equation: Advective–diffusive regime and multi-dimensional problems

Abstract

We present the variational multiscale (VMS) formulation for the stochastic advection–diffusion equation in which the source of uncertainty may arise from random advective and diffusive fields (or material parameters), a random source term, or a combination among them. In this formulation, a stabilization parameter arises that, for the advection–diffusion equation, is a complicated function of the uncertain advection and diffusion parameters. To efficiently incorporate the stochastic stabilization parameter in the numerical method, we develop a projection-based approach to obtain an approximation of the stabilization parameter in the form of a generalized polynomial chaos (gPC) expansion. We demonstrate the current approach for problems which span the advective and diffusive regimes in the stochastic domain. Specifically, four cases are considered with multi-dimensional physical and stochastic domains. These cases include an advection-dominated case with an uncertain advection parameter, a case spanning both the advective and diffusive regimes with uncertain advection and diffusion parameters, a case with an uncertain source term, and a case with scalar transport under an uncertain advection field in a channel with multiple branches. For the latter case uniform and non-uniform meshes are employed. The expectation and variance from the stochastic Galerkin and VMS formulations are compared against their counterparts from analytical or Monte Carlo sampling methods where appropriate. In summary, an accurate and stable numerical solution is obtained using the VMS formulation with projection-based approximation of the stochastic stabilization parameter for multi-dimensional problems including non-trivial physical discretization.