Boundary procedures for the time-dependent stochastic Burgers' equation

The stochastic Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system which gives a qualitative description of the development of the system over time for different initial and boundary conditions is presented. The numerical discretization is based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. Since the system is quite new, boundary conditions and related data are not known but has a weak connection to the original Burgers' equation. Time-dependent boundary conditions require knowledge about the time development of high order moments of the boundary values. We will discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions.