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arXiv · 2504.11676

Maximum bound principle for Q-tensor gradient flow with low regularity integrators

Wenshuai Hu·Guanghua Ji
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Abstract

The Landau-de Gennes (LdG) theory is a widely used thermodynamic continuum framework for modeling the behavior of ordered states and defects in liquid crystals with a tensor-order parameter $Q$. In this study, we develop and analyze first- and second-order low-regularity integrator (LRI) schemes for the $Q$-tensor gradient flow and prove the maximum bound principle. In particular, through the reformulation of the LRI schemes, we establish rigorous modified energy dissipation laws for the LRI1a and LRI1b schemes, thereby filling a significant theoretical gap in the existing literature on LRI methods. Moreover, this reformulation establishes a structural bridge between the LRI schemes and backward differentiation formula (BDF) methods, which opens up new possibilities for the construction and analysis of LRI-type methods. We then establish first- and second-order temporal convergence under $H^1$ and $H^2$ regularity assumptions, respectively. Several numerical experiments are presented to validate our theoretical results and to simulate the evolution of defect dynamics.

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Wenshuai Hu, Guanghua Ji. 2026-09-17. Maximum bound principle for Q-tensor gradient flow with low regularity integrators. https://arxiv.org/abs/2504.11676

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