A fully decoupled scheme for the simplified Ericksen–Leslie system with variable density

Jiaoyan Hou; Zhaowei Wang; Danxia Wang; Jiaoyan Hou; Zhaowei Wang; Danxia Wang

doi:10.3934/math.20261236

AIMS Mathematics

2026, Volume 11, Issue 9: 31294-31328. doi: 10.3934/math.20261236

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A fully decoupled scheme for the simplified Ericksen–Leslie system with variable density

1.
Shanxi College of Technology, Shuozhou, 036000, Shanxi, China
2.
School of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, China

Received: 10 May 2026 Revised: 12 September 2026 Accepted: 15 September 2026 Published: 23 September 2026
MSC : 35Q35, 65M60

In this work, we develop a fully decoupled sphere-constraint-preserving scheme for the simplified Ericksen–Leslie system with variable density. To begin, the original system is reformulated so that the constraint condition $ |{{\mathbf b}}| = 1 $ can be maintained in the discrete numerical scheme. Next, by employing the Gauge–Uzawa method together with the scalar auxiliary variable (SAV) approach, we construct a first-order time discretization scheme that is fully decoupled and unconditionally energy stable. Combining this with the finite element method (FEM) yields a first-order fully discrete scheme along with its implementation details. A rigorous error analysis is carried out for the temporal semi-discretization. Subsequently, the convergence of the proposed numerical method is established through a rigorous error analysis. Finally, several numerical experiments are carried out to demonstrate the effectiveness of the scheme.
- Ericksen–Leslie,
- variable density,
- Gauge–Uzawa method,
- SAV method
Citation: Jiaoyan Hou, Zhaowei Wang, Danxia Wang. A fully decoupled scheme for the simplified Ericksen–Leslie system with variable density[J]. AIMS Mathematics, 2026, 11(9): 31294-31328. doi: 10.3934/math.20261236

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Abstract

In this work, we develop a fully decoupled sphere-constraint-preserving scheme for the simplified Ericksen–Leslie system with variable density. To begin, the original system is reformulated so that the constraint condition $ |{{\mathbf b}}| = 1 $ can be maintained in the discrete numerical scheme. Next, by employing the Gauge–Uzawa method together with the scalar auxiliary variable (SAV) approach, we construct a first-order time discretization scheme that is fully decoupled and unconditionally energy stable. Combining this with the finite element method (FEM) yields a first-order fully discrete scheme along with its implementation details. A rigorous error analysis is carried out for the temporal semi-discretization. Subsequently, the convergence of the proposed numerical method is established through a rigorous error analysis. Finally, several numerical experiments are carried out to demonstrate the effectiveness of the scheme.

References

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