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

TL;DR

This paper develops a rigorous error estimate for a first-order energy stable numerical scheme for 3D hydrodynamic Q-tensor models of nematic liquid crystals, combining SAV, stabilization, and projection methods.

Contribution

It introduces a novel unconditionally energy stable scheme with proven solvability, energy dissipation, and error bounds for simulating nematic liquid crystals.

Findings

Proved unique solvability and energy dissipation of the scheme.

Derived an O(Δt) error estimate in L2 norm.

Numerical simulations confirm theoretical results.

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({\delta}t) in the sense of L2 norm, where {\delta}t is the time step.Finally, we give some numerical simulations to demonstrate the theoretical analysis.