Generalized polynomial chaos for the convection diffusion equation with uncertainty

In this paper, several numerical algorithms are presented for solving the convection diffusion equation with random diffusivity and periodic boundary conditions. Based on the generalized polynomial chaos expansion and Galerkin projection, the stochastic convection diffusion equation is turned into a set of coupled deterministic equations. Then the implicit-explicit scheme and the fully implicit scheme are employed to temporal discretization respectively, while the Fourier spectral method is used for spatial discretization. We place emphasis on the study of the two kinds of numerical schemes with different distribution of random inputs. Numerical results show that the Uniform random inputs is special, it is that the statistical error of solution will increase rapidly after reaches the minimum as the polynomial chaos expansion growth. And the implicit-explicit scheme doesn't work well for the two-dimensional model problems. Moreover, numerical simulations by Monte Carlo method are also shown to demonstrate the efficiency and robustness of the proposed algorithms.