# An Efficient and Accurate Adaptive Time-Stepping Method for the Landau-Lifshitz Equation
## 4 Conclusions  
The article introduced a new adaptive time-stepping FDM for solving the LL equation, which is crucial for investigating magnetic materials and processes. The method strategically selects time steps to keep displacement within a tolerance and allows larger steps near equilibrium states for faster computations. Benchmark tests demonstrated the algorithm's efficiency and accuracy and showed its potential for advancing magnetic phenomena analysis. We expect that the proposed time-stepping adaptive method can be applied to various equations which can be numerically approximated. Reaction-diffusion equations such as the Allen-Cahn equation can make good use of the proposed method because the displacement of the numerical solution is determined by the time step size. Furthermore, the evolution of the numerical solution varies during the numerical simulation, which indicates that the adaptive time step can be an efficient solution. In this paper, we considered a one-dimensional LL equation. Extending the proposed method to multi-dimensional problems presents challenges for explicit methods in terms of stability, which is a limitation of this study. Oscillations have already been reported in one-dimensional problems. We expect that adopting an implicit method combined with the adaptive time-stepping method could provide a solution for the multi-dimensional LL problem. In future work, we will present an efficient adaptive time-stepping method for the LL Equation (1) in two- and three-dimensional spaces. As a preliminary model, we consider the two-dimensional case as follows. In two-dimensional space, Equation (1) on \(\Omega=(L_{x},R_{x})\times(L_{y},R_{y})\) is defined as  
\[\frac{\partial\mathbf{m}(x,y,t)}{\partial t}=-\mathbf{m}(x,y,t)\times\Delta \mathbf{m}(x,y,t)+\mathbf{f}(x,y,t),\ \ (x,y)\in\Omega,\ \ 0<t\leq T. \tag{15}\]  
Figure 12: Simulation of constant forcing term. Numerical solution at (**a**) \(t=0.001\), (**b**) \(t=0.007\), and (**c**) \(t=0.01\). Numerical solutions of (**d**) \(u\), (**e**) \(v\), and (**f**) \(w\) at \(t=0.01\). Let \(N_{x}\) and \(N_{y}\) be positive integers. The discrete computational domain is defined as \(\Omega_{h}=\{(x_{i}=L_{x}+(i-0.5)h,y_{j}=L_{y}+(j-0.5)h)\mid i=1,2,\ldots,N_{x}\), \(j=1,2,\ldots,N_{y}\}\), where the spatial step size \(h=(R_{x}-L_{x})/N_{x}=(R_{y}-L_{y})/N_{y}\). We discretize Equation (15) as  
\[\frac{\mathbf{m}_{ij}^{n+1}-\mathbf{m}_{ij}^{n}}{\Delta t^{n+1}}=-\mathbf{m}_{ ij}^{n}\times\frac{\mathbf{m}_{i+1,j}^{n}+\mathbf{m}_{i-1,j}^{n}+\mathbf{m}_{i,j+1 }^{n}+\mathbf{m}_{i,j-1}^{n}-4\mathbf{m}_{ij}^{n}}{h^{2}}+\mathbf{f}_{ij}^{n}.\]  
We consider the case without the forcing term, where \(\mathbf{f}\equiv 0\). An exact solution of Equation (15) is given by [4]  
\[u^{\epsilon}(x,y,t) = \sin\Bigl{(}\frac{\pi}{24}\Bigr{)}\cos\Bigl{(}2\pi(x+y)+8\pi t\cos \Bigl{(}\frac{\pi}{24}\Bigr{)}\Bigr{)},\] \[v^{\epsilon}(x,y,t) = \sin\Bigl{(}\frac{\pi}{24}\Bigr{)}\sin\Bigl{(}2\pi(x+y)+8\pi t\cos \Bigl{(}\frac{\pi}{24}\Bigr{)}\Bigr{)},\] \[w^{\epsilon}(x,y,t) = \cos\Bigl{(}\frac{\pi}{24}\Bigr{)}.\]  
The initial condition on the \(\Omega=(0,1)\times(0,1)\) is given by  
\[(u(x,y,0),v(x,y,0),w(x,y,0))=\Bigl{(}\sin\Bigl{(}\frac{\pi}{24}\Bigr{)}\cos(2 \pi(x+y)),\sin\Bigl{(}\frac{\pi}{24}\Bigr{)}\sin(2\pi(x+y)),\cos\Bigl{(}\frac {\pi}{24}\Bigr{)}\Bigr{)}.\]  
We used the parameters \(N_{x}=N_{y}=32\), \(tol=10^{-6}\), \(\Delta t_{\text{max}}=0.2h^{2}\), and the final time \(T=0.1\). For visualization of the numerical solution at time \(t=0.1\), we used the scaled numerical solution \(0.25\mathbf{m}_{ij}\) for \(i=1,3,5,\ldots,31\), \(j=1,3,5,\ldots,31\). Figure 13 shows the scaled numerical solutions at \(t=0\) and \(t=0.1\) using the adaptive time-stepping method.