Orthogonal Decomposition of Stochastic Transport Equation

Uncertainty quantification seeks to provide a quantitative means to understand complex systems that are impacted by parametric uncertainty. The generalized polynomial chaos (gPC), see[3] and [2], as a tool of stochastic Galerkin method, is used for numerical solving of those complex systems which are described by stochastic ordinary differential equations (SODE) and stochastic partial differential equations (SPDE). We take transport equation as an example, and show that the Galerkin procedure results in a system of deterministic partial differential equations whose solving require an additional effort. Moreover, taking as a model various examples of SPDEs, numerical results with appropriate errors and deviations from exact solutions are presented. To show these results in a clearly visible manner, we used a number of figures, obtained using programming language Matlab.