Finite difference scheme for the Landau–Lifshitz equation

• Published in: Japan journal of industrial and applied mathematics Vol. 29; no. 1; pp. 83 - 110
• Main Authors: Fuwa, Atsushi, Ishiwata, Tetsuya, Tsutsumi, Masayoshi
• Format: Journal Article
• Language: English
• Published: Japan Springer Japan 01-02-2012; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Original Paper; Finite difference scheme; Error estimates; Landau–Lifshitz equation; 65M06; 35K55; 35Q60
• ISSN: 0916-7005; 1868-937X
• DOI: 10.1007/s13160-011-0054-9

In this paper we propose a finite difference scheme for the Landau–Lifshitz equation and the Heisenberg equation. These equations describe the evolution of spin fields in continuum ferromagnetism and have the following properties:
length preserving,
energy conservation or dissipation property.
We show that our scheme inherits these characteristic properties. The proposed scheme is implicit nonlinear, so we check an unique solvability of our schemes. Furthermore, we establish the error estimate for this problem. Finally, we demonstrate numerical examples in order to show the effectiveness of our scheme.