Error estimate for the first order energy stable scheme of Q-tensor nematic model

TL;DR

This work addresses the gap in rigorous error analysis for hydrodynamic $Q$-tensor models of nematic liquid crystals by developing a linear, first-order SAV-based scheme that is unconditionally energy-stable and uniquely solvable. By introducing a scalar auxiliary variable $r$ and reformulating the energy, the authors derive a semi-discrete system with a stable energy-dissipation law and provide a detailed $L^2$-error analysis showing $O(\delta t)$ convergence under suitable regularity assumptions. The analysis hinges on establishing a discrete $L^{\infty}$ bound for ${\bf Q}^n$ and leveraging discrete Gronwall arguments to control error growth. Numerical experiments corroborate the theoretical findings, demonstrating first-order temporal accuracy and energy decay consistent with the theory, thereby validating the practical reliability of the scheme for 3D hydrodynamic Q-tensor simulations.

Abstract

We present rigorous error estimates towards a first-order unconditionally energy stable scheme designed for 3D hydrodynamic Q-tensor model of nematic liquid crystals. This scheme combines the scalar auxiliary variable (SAV), stabilization and projection method together. The unique solvability and energy dissipation of the scheme are proved. We further derive the boundness of numerical solution in L^{\infty} norm with mathematical deduction. Then, we can give the rigorous error estimate of order O(δt) in the sense of L2 norm, where δt is the time step.Finally, we give some numerical simulations to demonstrate the theoretical analysis.

Figures (2)

• Figure 1: Dynamical evolution of director.
• Figure 2: Time evolution of modified energy.