# An Efficient and Accurate Adaptive Time-Stepping Method for the Landau-Lifshitz Equation
###### Abstract
[PERSON] and [PERSON] [7] contributed to the discourse by presenting a semi-implicit finite difference method (FDM) designed for the mathematical model that encapsulates the dynamics of magnetization, incorporating inertial effects into their frame work. [PERSON] et al. proposed a Gauss-Seidel projection method with unconditional stability for the numerical solution of the LL equation, employing the Gauss-Seidel method [8]. [PERSON] [9] proposed an adaptive procedure with a step size selection function to solve time-fractional advection-diffusion-reaction models. Step sizes are adapted according to the behavior of the solution. [PERSON] et al. devised a methodology combining a Gauss-Seidel method of an implicit fractional-step solver for the gyromagnetic term with the projection scheme to address the heat flow of harmonic maps [10]. Adaptive methods can be applied to other areas such as the adaptive signal for time-frequency domain to conduct quilted frames [11]. Meanwhile, [PERSON] et al. explored the numerical solution using the finite element method (FEM) in their works [12; 13; 14]. A Fourier spectral method was employed by [PERSON] et al. to approximate the solution of the LL equation, showcasing the versatility of different numerical approaches [15]. For a comprehensive exploration of numerical methods applicable to the LL equation, a plethora of references are available. [PERSON] and [PERSON]'s work [16], [PERSON]'s stability analysis [17], [PERSON]'s insights [18], [PERSON] et al. 's magnetic domain modeling [19], and [PERSON] et al. 's exploration of error analysis [20] provide detailed insights into the diverse methodologies employed to tackle the challenges posed by the LL equation. [PERSON] et al. [21] proposed a second-order semi-implicit method based on the backward difference method for the LL equation. They conducted a thorough analysis of their proposed scheme and presented the results of convergence tests. These references collectively offer a rich tapestry of numerical strategies, laying the groundwork for a deeper understanding and development of effective solution methodologies for the LL equation. In real-world applications, finding solutions to boundary value problems involving partial differential equations often necessitates the application of numerical techniques. Among the predominant methodologies utilized are the FEM [22], finite volume method (FVM), FDM [23; 24; 25; 26], and spectral method (SM) [27]. [PERSON] [26] proposed an optimized Runge-Kutta scheme with higher-order derivatives to solve parabolic and hyperbolic partial differential equations. Numerical simulations compared the Mead and Renaut methods with the proposed method by solving the linear advection equation, showing superior stability and applicability. [PERSON] et al. [27] proposed an SM using the Christov functions to solve the problem formulated by the LL and magnetostatic equations. Numerical simulations showed good agreement between the exact and numerical solution. These numerical approaches typically employ a fixed step size, which may not be optimal across a diverse array of problems. Recent developments have introduced alternative techniques, particularly those based on adaptive time-stepping methods, offering distinct advantages over traditional approaches such as FEM, FVM, FDM, and SM. Adaptive time-stepping methods automatically adjust the size of temporal step according to the local characteristics of the problem, allowing for dynamic adaptation to the varying complexities inherent in the solution [28]. [PERSON] and [PERSON] [29] proposed an adaptive time stepping method based on the energy decreasing scheme, which is an unconditionally energy stable method for the LL equation. As shown through the numerical results, the adaptive time-stepping method is computationally more efficient than a typical semi-implicit method. Recently, [PERSON] et al.