Ｌａｎｄａｕ‐Ｌｉｆｓｈｉｔｚ‐Ｇｉｌｂｅｒｔ方程式のＣｒａｎｋ‐Ｎｉｃｏｌｓｏｎ法による陰解法

Abstract

The efficiency of implicit methods in comparison with the classical Runge-Kutta method is investigated when the former are applied to the Landau-Lifshitz-Gilbert equation. The Crank-Nicolson and the two-stage fourth-order implicit Runge-Kutta methods were chosen and implemented by means of Newton's method. Implicit solutions using Newton's method call for the solution of large-scale simultaneous algebraic equations. To decrease the computing time, only the nearest-neighbor interaction was taken into consideration in the coefficient matrix of the simultaneous algebraic equations, which were solved by using the preconditioned Bi-CGSTAB method. Through calculations of three-dimensional magnetization reversal in a fine magnetic particle, it was confirmed that the examined methods yield a much larger maximum usable time step than the classical Runge-Kutta method. It was found that the Crank-Nicolson method takes about a quarter of the computing time required by the classical Runge-Kutta method when a dissipation constant α of 0.01 and a cell size of 8 nm are used. The maximum usable time step yielded by the two-stage fourth-order Runge-Kutta method was found to be almost four times as large as that yielded by the Crank-Nicolson method, though the former method required a longer computing time than the latter in the present calculation.