On the stability of the low-rank projector-splitting integrators for hyperbolic and parabolic equations

2025/07/21 by Shiheng Zhang, Jingwei Hu, Zhang, Shiheng +1 · 1 citation
Mathematics · Physics and Astronomy · Computer Science · #Numerical methods for differential equations #Model Reduction and Neural Networks #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2507.15192

Abstract

We study the stability of a class of dynamical low-rank methods--the projector-splitting integrator (PSI)--applied to linear hyperbolic and parabolic equations. Using a von Neumann-type analysis, we investigate the stability of such low-rank time integrator coupled with standard spatial discretizations, including upwind and central finite difference schemes, under two commonly used formulations: discretize-then-project (DtP) and project-then-discretize (PtD). For hyperbolic equations, we show that the stability conditions for DtP and PtD are the same under Lie-Trotter splitting, and that the stability region can be significantly enlarged by using Strang splitting. For parabolic equations, despite the presence of a negative S-step, unconditional stability can still be achieved by employing Crank-Nicolson or a hybrid forward-backward Euler scheme in time stepping. While our analysis focuses on simplified model problems, it offers insight into the stability behavior of PSI for more complex systems, such as those arising in kinetic theory.

Citations

• Asymptotic-Preserving Dynamical Low-Rank Method for the Stiff Nonlinear Boltzmann Equation
• A review of low-rank methods for time-dependent kinetic simulations
• A parallel rank-adaptive integrator for dynamical low-rank approximation
• Efficient dynamical low-rank approximation for the Vlasov-Ampère-Fokker-Planck system
• On the stability of robust dynamical low-rank approximations for hyperbolic problems
• A rank-adaptive robust integrator for dynamical low-rank approximation
• Existence of dynamical low-rank approximations to parabolic problems
• A low-rank projector-splitting integrator for the Vlasov--Poisson equation
• A projector-splitting integrator for dynamical low-rank approximation

Cited by

• Dynamical Tensor Train Approximation for Kinetic Equations

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