A new stochastic approach for advection-diffusion problems with uncertain parameters

Abstract

We propose a new numerical method for solving advection-diffusion equations with uncertainty. The stochastic effects are introduced either in the velocity field or in the initial data resulting into a class of stochastic partial differential equations. Chaos polynomials are used to represent the stochastic processes in the considered model. This procedure, known also by spectral decomposition, results into a system of deterministic advection-diffusion equations. For each chaos coefficient, the method of characteristics is used to integrate in time its associated advection diffusion equation. Central finite differencing is implemented for the space discretization. The proposed method is verified for an advection-diffusion equation with known analytical solution.
We also apply the method for simulation of the one-dimensional viscous Burgers equation. In both examples, the method demonstrates its ability to better maintain the shape of the solution in the presence of uncertainty and shocks.