时间分数阶阻尼波动方程的差分方法

Difference Methods for Time-Fractional Damped Wave Equations

摘要: 本文研究了一类时间分数阶阻尼波动方程问题的数值解法。针对该模型，引入了L1公式离散Caputo分数阶导数，并将原二阶时间方程改写为一阶系统。在此基础上，对普通时间导数采用半层平均隐式离散，对空间二阶导数分别采用二阶中心差分和四阶紧致差分，建立标准二阶格式和四阶紧致格式。进一步给出两类格式的矩阵形式，证明每一时间层线性系统均唯一可解。利用L1权重的正定性和离散能量法，证明所得格式在离散范数意义下无条件稳定。最后，通过数值算例验证不同分数阶阶数下的时间收敛阶和空间收敛阶，计算结果与理论分析一致。

Abstract: This paper investigates an effective numerical method for a class of time-fractional damped wave equations. For the considered model, the L1 formula is first introduced to discretize the Caputo fractional derivative, and the original second-order-in-time equation is reformulated as a first-order system. On this basis, a half-time-level averaged implicit discretization is applied to the standard time derivative, while the second-order spatial derivative is approximated by the second-order central difference scheme and the fourth-order compact difference scheme, respectively. Consequently, a standard second-order scheme and a fourth-order compact scheme are constructed. The matrix forms of the two schemes are further derived, and it is proved that the linear system at each time level admits a unique solution. By using the positive definiteness of the L1 weights and the discrete energy method, the unconditional stability of the proposed schemes is established in the sense of discrete norms. Finally, numerical examples are presented to verify the temporal and spatial convergence orders for different fractional orders. The computational results are consistent with the theoretical analysis.