Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unconditionally energy stable numerical schemes for the three-dimensional magneto-micropolar equations

Hailong Qiu

TL;DR

The paper tackles unconditionally energy stable numerical schemes for the nonstationary 3D magneto-micropolar equations by formulating two primary time discretizations: Euler semi-implicit and Crank–Nicolson with extrapolated nonlinear terms. It develops fully discrete first-order schemes with stabilized finite elements, proves unconditional energy stability, and derives error estimates for velocity, magnetic, micro-rotation, and pressure fields. It further introduces second-order Crank–Nicolson based schemes and stabilized decoupled variants, with corresponding energy stability results and error analysis, while noting the need for future work on stability proofs for some decoupled nonlinear schemes. Numerical tests in FreeFem++ validate convergence rates, energy stability, and long-time behavior in 3D, including convergence tests and a Lid-driven cavity flow, underscoring the practical robustness of the proposed methods for complex magneto-micropolar dynamics.

Abstract

In this paper we consider unconditionally energy stable numerical schemes for the nonstationary 3D magneto-micropolar equations that describes the microstructure of rigid microelements in electrically conducting fluid flow under some magnetic field. The first scheme is comprised of the Euler semi-implicit discretization in time and conforming finite element/stabilizedfinite element in space. The second one is based on Crank-Nicolson discretization in time and extrapolated treatment of the nonlinear terms such that skew-symmetry properties are retained. We prove that the proposed schemes are unconditionally energy stable. Some error estimates for the velocity field, the magnetic field, the micro-rotation field and the fluid pressure are obtained. Furthermore, we establish some first-order decoupled numerical schemes. Numerical tests are provided to check the theoretical rates and unconditionally energy stable.

Unconditionally energy stable numerical schemes for the three-dimensional magneto-micropolar equations

TL;DR

The paper tackles unconditionally energy stable numerical schemes for the nonstationary 3D magneto-micropolar equations by formulating two primary time discretizations: Euler semi-implicit and Crank–Nicolson with extrapolated nonlinear terms. It develops fully discrete first-order schemes with stabilized finite elements, proves unconditional energy stability, and derives error estimates for velocity, magnetic, micro-rotation, and pressure fields. It further introduces second-order Crank–Nicolson based schemes and stabilized decoupled variants, with corresponding energy stability results and error analysis, while noting the need for future work on stability proofs for some decoupled nonlinear schemes. Numerical tests in FreeFem++ validate convergence rates, energy stability, and long-time behavior in 3D, including convergence tests and a Lid-driven cavity flow, underscoring the practical robustness of the proposed methods for complex magneto-micropolar dynamics.

Abstract

In this paper we consider unconditionally energy stable numerical schemes for the nonstationary 3D magneto-micropolar equations that describes the microstructure of rigid microelements in electrically conducting fluid flow under some magnetic field. The first scheme is comprised of the Euler semi-implicit discretization in time and conforming finite element/stabilizedfinite element in space. The second one is based on Crank-Nicolson discretization in time and extrapolated treatment of the nonlinear terms such that skew-symmetry properties are retained. We prove that the proposed schemes are unconditionally energy stable. Some error estimates for the velocity field, the magnetic field, the micro-rotation field and the fluid pressure are obtained. Furthermore, we establish some first-order decoupled numerical schemes. Numerical tests are provided to check the theoretical rates and unconditionally energy stable.
Paper Structure (11 sections, 11 theorems, 105 equations, 6 figures)

This paper contains 11 sections, 11 theorems, 105 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Continuous problem
  3. First-order numerical schemes
  4. Error analysis
  5. Second-order/decoupled numerical schemes
  6. Second-order numerical schemes
  7. Decoupled numerical schemes
  8. Numerical results
  9. 3D convergence test
  10. 3D stability test
  11. 3D Lid-driven cavity flow

Key Result

theorem 1

The weak formulation eq9.1 has at least one solution $(\textbf{u}, p, \textbf{B}, \textbf{w})$ such that Moreover, problem eq9.1 has an unique weak solution in two spatial dimension.

Figures (6)

  • Figure 1: (a) Discrete energy $E_n$ with Scheme 3.1; (b) Discrete energy $E_n$ with Scheme 3.2.
  • Figure 2: (a) Discrete energy $E_n$ with Scheme 5.1; (b) Discrete energy $E_n$ with Scheme 5.2.
  • Figure 3: (a) Discrete energy $E_n$ with Scheme 5.3; (b) Discrete energy $E_n$ with Scheme 5.4.
  • Figure 4: The velocity field (a) and (b) of flow by the Scheme 5.1 and Scheme 5.3.
  • Figure 5: The magnetic field (a) and (b) of flow by the Scheme 5.1 and Scheme 5.3.
  • ...and 1 more figures

Theorems & Definitions (11)

  • theorem 1
  • lemma 1
  • lemma 2
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • theorem 6
  • theorem 7
  • theorem 8
  • ...and 1 more