  Quantifying uncertainties in hyperbolic equations is a source of several
challenges. First, the solution forms shocks leading to oscillatory behaviour
in the numerical approximation of the solution. Second, the number of unknowns
required for an effective discretization of the solution grows exponentially
with the dimension of the uncertainties, yielding high computational costs and
large memory requirements. An efficient representation of the solution via
adequate basis functions permits to tackle these difficulties. The generalized
polynomial chaos (gPC) polynomials allow such an efficient representation when
the distribution of the uncertainties is known. These distributions are usually
only available for input uncertainties such as initial conditions, therefore
the efficiency of this ansatz can get lost during runtime. In this paper, we
make use of the dynamical low-rank approximation (DLRA) to obtain a memory-wise
efficient solution approximation on a lower dimensional manifold. We
investigate the use of the matrix projector-splitting integrator and the
unconventional integrator for dynamical low-rank approximation, deriving
separate time evolution equations for the spatial and uncertain basis
functions, respectively. This guarantees an efficient approximation of the
solution even if the underlying probability distributions change over time.
Furthermore, filters to mitigate the appearance of spurious oscillations are
implemented, and a strategy to enforce boundary conditions is introduced. The
proposed methodology is analyzed for Burgers' equation equipped with uncertain
initial values represented by a two-dimensional random vector. The numerical
results show a reduction of the memory requirements, and that the important
characteristics of the original system are well captured.
