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A New Fractional Step Structure Preserving Method for The Landau-Lifshitz-Gilbert Equation

Changjian Xie

Abstract

In this paper, we propose a structure preserving method using a Crank-Nicolson's type method with an implicit Gauss-Seidel fractional iteration. Such a method is of first-order accuracy in time and second-order accuracy in space, stable and length preserving. Such a proposed method brings great benefits for the theoretical analysis. The numerical accuracy, norm preserving and stability are verified for 1D and 3D tests.

A New Fractional Step Structure Preserving Method for The Landau-Lifshitz-Gilbert Equation

Abstract

In this paper, we propose a structure preserving method using a Crank-Nicolson's type method with an implicit Gauss-Seidel fractional iteration. Such a method is of first-order accuracy in time and second-order accuracy in space, stable and length preserving. Such a proposed method brings great benefits for the theoretical analysis. The numerical accuracy, norm preserving and stability are verified for 1D and 3D tests.
Paper Structure (11 sections, 1 theorem, 109 equations, 4 figures, 6 tables)

This paper contains 11 sections, 1 theorem, 109 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The governing equation and numerical scheme
  3. Scheme Property
  4. Consistency of our proposed scheme
  5. Stability of our proposed scheme
  6. Convergence of our proposed scheme
  7. Numerical experiments
  8. Accuracy and efficiency tests
  9. Norm preserving tests
  10. Initial conditions and stability tests
  11. Conclusions and discussions

Key Result

Lemma 1

For any $\Delta t > 0$ and mesh size $h > 0$, the operator $(I - \Delta t \Delta_h)^{-1}$ is uniformly bounded in $L^2(\Omega)$: where $C$ is a constant independent of $\Delta t$ and $h$. Moreover,

Figures (4)

  • Figure 1: The solution profile using proposed method in 1D given the initial condition $m_0$ with $T0$ specified without source term, $\alpha=0.01$ and $T=0.1$, $N_x=2000$, $N_t=5$.
  • Figure 2: The solution profile using the proposed methods in 3D given the initial condition $m_0$ with initial condition specified without source term, $\alpha=0$ and $T=0.1$, $N_x=N_y=N_z=20$, $N_t=40$. Top row with initial condition; Bottom row with proposed method. Initial condition given: $\hbox{\boldmath $m$}_0=[\cos(x^2(1-x)^2)\sin(0.01),\sin(x^2(1-x)^2)\sin(0.01),\cos(0.01)]$.
  • Figure 3: The solution profile using the proposed methods in 3D given the initial condition $m_0$ with initial condition specified without source term, $\alpha=0$ and $T=0.1$, $N_x=N_y=N_z=20$, $N_t=40$. Top row with initial condition; Bottom row with proposed method. Initial condition given: $\hbox{\boldmath $m$}_0=[\cos(\cos(\pi x))\sin(0.01),\sin(\cos(\pi x))\sin(0.01),\cos(0.01)]$.
  • Figure 4: The solution profile using the proposed methods in 3D given the initial condition $m_0$ with initial condition specified without source term, $\alpha=0$ and $T=0.1$, $N_x=N_y=N_z=20$, $N_t=40$. Top row with initial condition; Bottom row with proposed method. Initial condition given: $\hbox{\boldmath $m$}_0=[\cos(\cos(\cos(\pi x)))\sin(\pi x+T0),\sin(\cos(\cos(\pi x)))\sin(\pi x+T0),\cos(\pi x+T0)]$.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1: Boundedness of Elliptic Operator
  • proof