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Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations
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Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations
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Journal Article
Implicit Low-Rank Riemannian Schemes for the Time Integration of Stiff Partial Differential Equations
2024
Overview
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 equations on the manifold of fixed-rank matrices. 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.
Publisher
Springer US,Springer Nature B.V
