Higher-Order Spatio-Temporal Discretization and Efficient Solution Algorithms for Variable-Order Fractional-Order Damping Equations

针对时间分数阶变阻尼方程的数值求解问题，本研究构建了高效离散格式并系统分析了其理论特性。时间方向采用L2_1σ格式进行离散，空间方向应用二阶中心差分方法，建立了完整的数值离散格式，并证明了格式的收敛性与稳定性。为提升计算效率，重点研究了预处理共轭梯度法与多重网格法两类快速算法，详细对比了二者的计算性能。数值实验不仅验证了理论分析，更通过三种求解方法的耗时对比揭示：预处理共轭梯度法因其优化的算法复杂度和高效的预处理技术，在计算效率方面展现出显著优势。For the numerical solution of the time-fractional order variable damping equation, this study constructs an efficient discretization format and systematically analyzes its theoretical properties. The time direction is discretized by the L2_1σ formula, and the spatial direction is discretized by the second-order center difference method, which establishes a complete numerical discretization format and proves the convergence and stability of the format. In order to improve the computational efficiency, two types of fast algorithms, namely the preprocessing conjugate gradient method and the multiple grid method, are emphasized, and the computational performances of the two are compared in detail. The numerical experiments not only verify the theoretical analysis, but also through the time-consuming comparison of the three solution methods show that the preprocessing conjugate gradient method has a significant advantage in computational efficiency due to its optimized algorithmic complexity and efficient preprocessing technology.