Numerical Solution of Landau-Lifshitz-Gilbert Equation with Two Space Variables for Vertical Bloch Lines

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Abstract
Detailed structures of two-dimensional vertical Bloch lines (VBL's) are obtained from first-principle calculations in which the Euler equation is solved by means of Newton's method. Starting from the thus-obtained equilibrium state, the Gilbert equation is solved using various numerical methods to examine their accuracy and extent of stability. The examined methods include the forward difference, the original and the so-called modified Dufort-Frankel, ADI, backward difference, and Crank-Nicolson methods. The results of the examination show the advantage of using the backward difference method and the risk of using the modified Dufort-Frankel method. Although both methods are found to be very stable, the modified Dufort-Frankel method has the possibility of yielding results that are completely different from what is believed to be an accurate solution. This paper describes the principles of the numerical methods examined and the results of the examination together with the derived detailed structures of two-dimensional VBL's.