The Landau--Lifshitz--Baryakhtar equation describes the evolution of magnetic spin field in magnetic materials at elevated temperature below the Curie temperature, when long-range interactions and longitudinal dynamics are taken into account. We propose two linear fully-discrete $C^1$-conforming methods to solve the problem, namely a semi-implicit Euler method and a semi-implicit BDF method, and show that these schemes are unconditionally stable. Error analysis is performed which shows optimal convergence rates in each case. Numerical results corroborate our theoretical results.

翻译：Landau--Lifshitz--Baryakhtar方程描述了磁性材料在居里温度以下、高温时，计入长程相互作用和纵向动力学情况下磁自旋场的演化过程。我们提出两种线性全离散$C^1$-协调方法求解该问题，即半隐式Euler方法和半隐式BDF方法，并证明了这些格式是无条件稳定的。误差分析表明每种情况下均达到最优收敛速率。数值结果验证了我们的理论结论。