Time-Dependent Polynomial Chaos

Conventional generalized polynomial chaos is known to fail for long time integration, loosing its optimal convergence behaviour and developing unacceptable error levels. The reason for this loss of convergence is the assumption that the probability density function is constant in time. By allowing a probability density function to evolve in time the optimal properties of polynomial chaos are retrieved without resorting to high polynomial degrees. This time-dependent approach is applied to a system of coupled non-linear differential equations. These results are compared to the conventional generalized polynomial chaos solutions and Monte Carlo simulations.