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arXiv · 2305.11532

Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations

Marco Sutti·Bart Vandereycken
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Abstract

We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen-Cahn and the Fisher-KPP equation on the manifold of fixed-rank matrices. Furthermore, our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

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Marco Sutti, Bart Vandereycken. 2023-05-19. Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations. https://doi.org/10.1007/s10915-024-02629-8

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