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A linear, unconditionally stable, second order decoupled method for the Ericksen-Leslie model with SAV approach

Ruonan Cao, Nianyu Yi

TL;DR

The paper addresses efficient and stable numerical simulation of nematic liquid crystal flows governed by the simplified Ericksen-Leslie model. It develops a linear, fully decoupled, second-order scheme (PCSAV) using pressure-correction, a Lagrange multiplier approach, and scalar auxiliary variables to handle nonlinear terms, and proves unconditional energy stability. A variant with explicit convection (PCSAV-ECT) maintains stability while enabling constant-coefficient linear solves. Numerical experiments confirm second-order convergence in time and space, show robust energy decay for varying penalties, and demonstrate computational advantages of the PCSAV-ECT formulation.

Abstract

In this paper, we present a second order, linear, fully decoupled, and unconditionally energy stable scheme for solving the Erickson-Leslie model. This approach integrates the pressure correction method with a scalar auxiliary variable technique. We rigorously demonstrate the unconditional energy stability of the proposed scheme. Furthermore, we present several numerical experiments to validate its convergence order, stability, and computational efficiency.

A linear, unconditionally stable, second order decoupled method for the Ericksen-Leslie model with SAV approach

TL;DR

The paper addresses efficient and stable numerical simulation of nematic liquid crystal flows governed by the simplified Ericksen-Leslie model. It develops a linear, fully decoupled, second-order scheme (PCSAV) using pressure-correction, a Lagrange multiplier approach, and scalar auxiliary variables to handle nonlinear terms, and proves unconditional energy stability. A variant with explicit convection (PCSAV-ECT) maintains stability while enabling constant-coefficient linear solves. Numerical experiments confirm second-order convergence in time and space, show robust energy decay for varying penalties, and demonstrate computational advantages of the PCSAV-ECT formulation.

Abstract

In this paper, we present a second order, linear, fully decoupled, and unconditionally energy stable scheme for solving the Erickson-Leslie model. This approach integrates the pressure correction method with a scalar auxiliary variable technique. We rigorously demonstrate the unconditional energy stability of the proposed scheme. Furthermore, we present several numerical experiments to validate its convergence order, stability, and computational efficiency.

Paper Structure

This paper contains 7 sections, 2 theorems, 72 equations, 11 figures, 7 tables.

Table of Contents

  1. Introduction
  2. PCSAV method for the modified Ericksen-Leslie model
  3. PCSAV semi-discrete scheme
  4. Discrete energy dissipation property
  5. Implementation of the PCSAV scheme
  6. PCSAV method with explicit scheme for convection term
  7. Numerical Experiments

Key Result

Theorem 2.1

The scheme eq:2stPCSAV2-eq:2stPCSAV6 is unconditionally energy stable in the sense that where and $\{g^n,H^n\}$ are defined by

Figures (11)

  • Figure 1: Example \ref{['62']}, images of the director field $\mathrm{\textbf{d}}$ (first row) and $\lvert\mathrm{\textbf{d}}\rvert$ (second row) at $t=0.01, 0.1, 0.2, 0.4$ computed by PCSAV scheme.
  • Figure 2: Example \ref{['62']}, images of the velocity $\mathrm{\textbf{u}}$ (first row) and $\lvert\mathrm{\textbf{u}}\rvert$ (second row) at $t=0.01, 0.1, 0.2, 0.4$ computed by PCSAV scheme.
  • Figure 3: Example \ref{['62']}, images of the director field $\mathrm{\textbf{d}}$ (first row) and $\lvert\mathrm{\textbf{d}}\rvert$ (second row) at $t=0.01, 0.1, 0.2, 0.4$ computed by PCSAV-ECT scheme.
  • Figure 4: Example \ref{['62']}, images of the velocity $\mathrm{\textbf{u}}$ (first row) and $\lvert\mathrm{\textbf{u}}\rvert$ (second row) at $t=0.01, 0.1, 0.2, 0.4$ computed by PCSAV-ECT scheme.
  • Figure 5: Example \ref{['62']}, kinetic energy, elastic energy, penalty energy and modified energy computed by the PCSAV scheme.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Example 4.1
  • Example 4.2
  • Example 4.3