Dynamical low-rank approximation for Burgers' equation with uncertainty
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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.
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