Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

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Kazashi, Yoshihito and Nobile, Fabio and Vidličková, Eva (2021) Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations. Numerische Mathematik, 149. 973–1024. ISSN 0029-599X (https://doi.org/10.1007/s00211-021-01241-4)

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Abstract

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.

  • Item type:Article
    ID code:86370
    Dates:
    Date
    Event
    17 November 2021
    Published
    2 October 2021
    Accepted
    7 August 2020
    Submitted
    Notes:Kazashi, Y., Nobile, F. & Vidličková, E. Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations. Numer. Math. 149, 973–1024 (2021). https://doi.org/10.1007/s00211-021-01241-4 © The Author(s) 2021
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:03 Aug 2023 09:08
    Last modified:20 Nov 2024 09:25
    URI:https://strathprints.strath.ac.uk/id/eprint/86370
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