  We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve
time-dependent stochastic partial differential equations (SPDEs) with white
noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos
(PC) methods is notoriously difficult as the dimension of the stochastic
variables increases linearly with time. Exploiting the markovian property of
white noise, DgPC [1] implements a restart procedure that allows us to expand
solutions at future times in terms of orthogonal polynomials of the measure
describing the solution at a given time and the future white noise. The
dimension of the representation is kept minimal by application of a
Karhunen--Loeve (KL) expansion. Using frequent restarts and low degree
polynomials on sparse multi-index sets, the method allows us to perform long
time simulations, including the calculation of invariant measures for systems
which possess one. We apply the method to the numerical simulation of
stochastic Burgers and Navier--Stokes equations with white noise forcing. Our
method also allows us to incorporate time-independent random coefficients such
as a random viscosity. We propose several numerical simulations and show that
the algorithm compares favorably with standard Monte Carlo methods.
