Solution of Landau-Lifshitz-Gilbert Equation by Newton's Method

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  • Ｎｅｗｔｏｎ法によるＬａｎｄａｕ－Ｌｉｆｓｈｉｔｚ－Ｇｉｌｂｅｒｔ方程式の解法
  • Newtonホウ ニ ヨル Landau Lifshitz Gilbert ホウテイシキ ノ カイホウ

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The use of Newton’s method to solve the Landau-Lifshitz-Gilbert (LLG) equation discretized according to the Crank-Nicolson method is examined from the viewpoint of accuracy and computation time. Magnetization reversal and Bloch wall motion in a thin Permalloy film were calculated as an example. Although the convergence speed of Newton’s method is high, the solution of simultaneous algebraic equations requires a computation time of O(N3) in principle, when we denote the numbers of computing cells as N. If we take only the nearest-neighbor magnetostatic interaction into consideration in the implicit treatment, the coefficient matrix for the algebraic equation and the computation time are reduced to tridiagonal and O(N), respectively, in the case of one-dimensional calculation. Newton’s iteration can be repeated several times without increasing the order of computing time since the time required to calculate the demagnetizing field is O(N log N) if we use FFT. On detailed comparison with the Runge-Kutta method, a typical method for ordinary differential equations, the proposed method is found useful from the viewpoint of time-step dependence of numerical error and reliability.

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