Convergence analysis of orthogonal spline collocation method for Landau-Lifshitz-Gilbert equation

We develop an orthogonal spline collocation method for the Landau-Lifshitz-Gilbert equation describing magnetization dynamics in ferromagnetic materials. The method employs Gaussian collocation points, leading to a simple implementation and sparse linear systems that can be solved efficiently. A complete and rigorous error analysis is presented, demonstrating optimal-order convergence in H 1 -norm. We provide some numerical experiments to confirm our theoretical analysis and show the temporal and spatial convergence order in various norms. Comparisons with existing methods demonstrate the accuracy and efficiency of the proposed scheme and additional simulations investigate the possible blowup phenomena. The magnetic microstructures captured by this method agree well with established techniques, supporting the reliability of the method for micromagnetic simulations.