Characteristic Finite Difference Scheme For Stochastic Convection-Diffusion Equations

By using the polynomial chaos expansion method to approximate the convection-diffusion equations with uncertainty, a characteristic finite difference scheme is derived for numerically solving the stochastic convection-diffusion equations. The numerical experiments show the scheme is adaptable and effective.The paper is organized as follows.In section one, there are some basic knowledge of convection diffusion equa-tion and some useful methods. For convection-diffusion equations with uncer-tainty, we introduce the equation and the method. Whatever we choose to use the polynomial chaos representation to treat the randomness in the equations, the method transforms the stochastic convection-diffusion equation into a system of deterministic convection-diffusion equations.In section two, consider the stochastic convection diffusion equations(?)tu+v(x,t, ω)(?)xu-v(?)xxu=f(x,t,ω),(x,t,ω)∈[l, L]×(0,T]×Ω, where v is the diffusion coefficient and v(x, t, ω)=v+σζ is a stochastic velocity field, v is the mean velocity and a is a parameter used to control the amplitude of the stochastic perturbation.The equation is complemented with the following boundary and initial con-ditions u(x,0,ω)=u0(x, ω),(x,ω)∈[l, L]×Ω,u(l,t,ω)=g0(t,ω), u(L, t,ω)=g1t, ω),(t,ω)∈[0, T]×Ω. Based on paper [5] we are going to use the generalized polynomial chaos method with the stochastic convection-diffusion equations. Using the polynomial chaos representation to treat the randomness in the equations, the method trans-forms the stochastic convection-diffusion equation into a system of deterministic convection-diffusion equations.In section three of the thesis, we consider classical convection-diffusion e-quationto use the finite difference method combine with the method of characteris-tics.Because normally xi*is not grid nodes, we need to use the three adjacent nodes’quadratic interpolation. So we can get characteristic difference equationSimilar we can get the difference scheme of stochastic convection diffusion equation of matrix equations asWe can prove that the algorithm is second order convergence on space and first order convergence on time. Besides numerical examples can verify the effective-ness of the algorithm.In section four, we can see that algorithm for two-dimensional random con-vection diffusion equation is valid.We use the same method as before.So we can get characteristic difference equation...