# A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping

Darae Jeong, Junseok Kim

Research output: Contribution to journalArticle

5 Citations (Scopus)

### Abstract

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.

Original language English 613-623 11 Journal of Computational and Applied Mathematics 234 2 https://doi.org/10.1016/j.cam.2010.01.002 Published - 2010 May 15

### Fingerprint

Crank-Nicolson Scheme
Landau-Lifshitz Equation
Damping
Ferromagnetism
Finite difference method
Numerical Solution
Crank-Nicolson Method
Crank-Nicolson
Multigrid Method
Time Stepping
Discrete Systems
Numerical Scheme
Difference Method
Parabolic Equation
System of equations
Finite Difference
Nonlinearity

### Keywords

• Crank-Nicolson
• Finite difference method
• Landau-Lifshitz equation
• Nonlinear multigrid method

### ASJC Scopus subject areas

• Applied Mathematics
• Computational Mathematics

### Cite this

In: Journal of Computational and Applied Mathematics, Vol. 234, No. 2, 15.05.2010, p. 613-623.

Research output: Contribution to journalArticle

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N2 - An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.

AB - An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.

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