@TechReport{ it:2008-011,
author = {Per Pettersson and Gianluca Iaccarino and Jan
Nordstr{\"o}m},
title = {Numerical Analysis of {B}urgers' Equation with Uncertain
Boundary Conditions Using the Stochastic {G}alerkin
Method},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2008},
number = {2008-011},
month = mar,
abstract = {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.}
}