[Paper Review] A Linear, Decoupled and Energy stable scheme for smectic-A Liquid Crystal Flows

Xiaofeng Yang, A. V. Brylev|arXiv (Cornell University)|Jan 25, 2017

Fluid Dynamics and Turbulent Flows5 references3 citations

TL;DR

This paper presents a linear, decoupled, and unconditionally energy-stable time discretization scheme for simulating smectic-A liquid crystal flows governed by a nonlinear system coupling the incompressible Navier-Stokes equations with two second-order elliptic equations for the director and layer order parameters. By employing explicit-implicit treatments of nonlinear terms, the scheme achieves unconditional energy stability and solves linear, decoupled equations at each time step, enabling efficient and robust long-time simulations with provable energy dissipation.

ABSTRACT

In this paper, we consider numerical approximations for the model of smectic-A liquid crystal flows. The model equation, that is derived from the variational approach of the de Gennes free energy, is a highly nonlinear system that couples the incompressible Navier-Stokes equations, and two nonlinear coupled second-order elliptic equations. Based on some subtle explicit--implicit treatments for nonlinear terms, we develop a unconditionally energy stable, linear and decoupled time marching numerical scheme. We also rigorously prove that the proposed scheme obeys the energy dissipation law at the discrete level. Various numerical simulations are presented to demonstrate the accuracy and the stability thereafter.

Motivation & Objective

• To develop a numerically efficient and thermodynamically consistent scheme for the complex, nonlinear system modeling smectic-A liquid crystal flows.
• To address the challenge of coupling between velocity, pressure, director field, and layer function through convection and stress terms.
• To construct a time discretization that is unconditionally stable and preserves the energy dissipation law at the discrete level.
• To achieve linear and decoupled equations at each time step to avoid costly nonlinear solves.
• To validate the scheme's accuracy, stability, and ability to capture physical phenomena like chevron patterns under magnetic fields and shear flow.

Proposed method

• A stabilized time discretization is designed using explicit treatment for nonlinear terms and implicit treatment for linear terms.
• The scheme decouples the system into separate linear equations for velocity, pressure, director field, and layer function at each time step.
• A subtle splitting strategy is employed to handle the nonlinear convection and stress terms, ensuring energy stability.
• The discrete energy law is rigorously proven, showing unconditional energy dissipation independent of time step size.
• The method is implemented using finite difference or finite element spatial discretization with appropriate boundary conditions.
• Numerical experiments validate the first-order temporal accuracy and energy stability through convergence and long-time simulations.

Experimental results

Research questions

• RQ1 Can a linear and decoupled time discretization be constructed for the highly nonlinear smectic-A liquid crystal model while preserving energy stability?
• RQ2 How can the strong coupling between velocity, pressure, director field, and layer function be handled without solving nonlinear systems at each time step?
• RQ3 What explicit-implicit treatment of nonlinear terms ensures unconditional energy stability in the discrete setting?
• RQ4 Can the scheme accurately capture physical phenomena such as chevron patterns induced by magnetic fields and shear flows?
• RQ5 Does the scheme maintain first-order temporal accuracy and energy dissipation in long-time simulations?

Key findings

• The proposed scheme achieves unconditional energy stability, meaning the discrete energy decreases at every time step regardless of the time step size.
• The scheme is linear and decoupled, solving separate linear systems for velocity, pressure, director, and layer function at each time step, significantly reducing computational cost.
• Numerical convergence tests confirm the scheme is first-order accurate in time for all variables, with error slopes matching theoretical expectations.
• Simulations of the Helfrich-Hurault effect show the formation of chevron patterns under magnetic fields, with energy dissipative curves confirming stability.
• Inclusion of shear flow distorts the symmetry of undulations, demonstrating the scheme's ability to capture complex hydrodynamic effects.
• The energy dissipation curve remains monotonic decreasing over long-time simulations, verifying the scheme's robustness for long-time dynamics.