变阶变系数时间分数阶阻尼波动方程的有限差分方法

A Finite Difference Method for a Variable-Order, Variable-Coefficient Time-Fractional Damped Wave Equation

摘要: 本文研究一维变阶变系数时间分数阶阻尼波动方程的有限差分求解。引入速度变量，在首个半时间层采用中心离散，在后续时间层采用加权三点公式，并以变阶$L2-1\sigma$ 型公式逼近Caputo导数，空间上采用二阶中心差分。证明了离散解存在唯一，利用离散能量方法建立稳定性估计，并给出后向欧拉-L₁低阶格式作为对照。数值算例表明，主格式在时间和空间方向均表现出二阶收敛精度，且计算效率优于低阶格式。

Abstract: This paper is concerned with a finite difference approximation for a one-dimensional variable-order time-fractional damped wave equation with variable coefficients. By introducing an auxiliary velocity variable, a central difference formula is employed at the first half-time level, whereas a weighted three-level discretization is applied at subsequent time levels. The variable-order Caputo fractional derivative is approximated by a corresponding variable-order formula, while the spatial derivatives are discretized by the standard second-order central difference method. The unique solvability of the resulting fully discrete scheme is established, and a stability estimate is derived through a discrete energy argument. For comparison, a low-order scheme based on the backward Euler discretization is also considered. Numerical results confirm that the proposed scheme attains second-order accuracy in both temporal and spatial directions and, meanwhile, achieves higher computational efficiency than the low-order method.