Linear Numerical Schemes for a $\textbf{Q}$-Tensor System for Nematic Liquid Crystals

Abstract

In this work, we present three linear numerical schemes to model nematic liquid crystals using the Landau-de Gennes $\textbf{Q}$-tensor theory. The first scheme is based on using a truncation procedure of the energy, which allows for an unconditionally energy stable first order accurate decoupled scheme. The second scheme uses a modified second order accurate optimal dissipation algorithm, which gives a second order accurate coupled scheme. Finally, the third scheme uses a new idea to decouple the unknowns from the second scheme which allows us to obtain accurate dynamics while improving computational efficiency. We present several numerical experiments to offer a comparative study of the accuracy, efficiency and the ability of the numerical schemes to represent realistic dynamics.

Figures (13)

• Figure 1: Convergence rate reference solution $\textbf{Q}^{\hbox{exact}}$ obtained using scheme OD1D. Colors indicate alignment with the dominant eigenvector. Dominant eigenvector is shown with black lines. The color bar for the remaining figures is given on the right.
• Figure 2: Defect dynamics using scheme OD1D in 2D with Neumann boundary conditions at times $t=0.01,0.1,0.25,0.35,0.5,1.0$. Color represents the difference of the two largest eigenvalues of $\textbf{Q}$ and indicates the alignment of the nematic with the dominant eigenvector shown as black lines.
• Figure 3: Top row: the energy of the system computed with scheme UES1D for different time steps. Left shows the energy over the whole time interval, and right a zoomed in view of the energy on the time interval $[0.3, 0.5]$. Bottom row: the numerical dissipation for different time steps. Left shows the numerical dissipation over the whole time interval, and right shows a zoomed in view over the time interval $[0.3, 0.5]$.
• Figure 4: Top row: the energy of the system computed with scheme OD2C for different time steps. Left shows the energy over the whole time interval, and right a zoomed in view of the energy on the time interval $[0.3, 0.5]$. Bottom row: the numerical dissipation for different time steps. Left shows the numerical dissipation over the whole time interval, and right shows a zoomed in view over the time interval $[0.3, 0.5]$.
• Figure 5: Top row: the energy of the system computed with scheme OD1D for different time steps. Left shows the energy over the whole time interval, and right a zoomed in view of the energy on the time interval $[0.3, 0.5]$. Bottom row: the numerical dissipation for different time steps. Left shows the numerical dissipation over the whole time interval, and right shows a zoomed in view over the time interval $[0.3, 0.5]$.

Theorems & Definitions (38)

• Remark 1
• Remark 2
• Theorem 1: Well-Posedness
• Lemma 1: Traceless Property
• Lemma 2: Maximum Principle
• Lemma 3: Energy Law
• Lemma 4: Discrete Energy Law
• proof
• Definition 1
• Remark 3