Numerical analysis of Burgers’ equation with uncertain boundary conditions using the stochastic Galerkin method

Burgers’ equation with stochastic initial and boundary conditions is investigated by a polynomial chaos expansion approach where the solution is represented as a series of stochastic, orthogonal polynomials. The analysis of wellposedness for the stochastic Burgers’ equation follows the pattern of that of the deterministic Burgers’ equation. We use dissipation and spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. Similar to the deterministic case, the time step for hyperbolic stochastic problems solved with explicit methods is proportional to the inverse of the largest eigenvalue of the system matrix. The time step naturally decreases compared to the deterministic case since the spectral radius of the continuous problem grows with the number of polynomial chaos coefficients. Analysis of the characteristics of a truncated system gives a qualitative description of the development of the system over time for different initial and boundary conditions. Knowledge of the initial and boundary expected value and variance is not enough to get a unique solution. Also, the sign of the polynomial chaos coefficients must be known. The deterministic component (expected value) of the solution is affected by the modeling of uncertainty. A shock discontinuity in a purely deterministic problem can be made smooth by assuming uncertain boundary conditions. Division of Scientific Computing, Department of Information Technology, Uppsala University, Sweden. Center for Turbulence Research, Department of Mechanical Engineering, Stanford University, USA. Division of Scientific Computing, Department of Information Technology, Uppsala University, Sweden.