DYNAMICAL LOW-RANK APPROXIMATION FOR BURGERS' EQUATION WITH UNCERTAINTY

摘要

Quantifying uncertainties in hyperbolic equations is a source of several challenges. First, the solution forms shocks leading to oscillatory behavior 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. The number of unknowns can be reduced through generalized polynomial chaos polynomials, which allow for 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 be lost during runtime. In this paper, we make use of the dynamical low-rank approximation (DLRA) to obtain a memorywise efficient solution approximation on a lower-dimensional manifold for Burgers' equation. We investigate and compare the use of the matrix projector-splitting integrator and the unconventional integrator for DLRA, deriving efficient time evolution equations for the spatial and uncertain basis functions, respectively. This guarantees an accurate approximation of the solution even if the underlying probability distributions change over time. The proposed methodology is analyzed for Burgers' equation equipped with uncertain initial values represented by a two-dimensional random vector. The numerical experiments show a significant reduction of the memory requirements, while important characteristics of the original system are well captured.