提出一种带有 Caputo 导数的时间分数阶变系数扩散方程的数值解法. 方程的解在初 始时刻附近通常具有弱正则性, 采用非一致网格上的 L1 公式离散时间分数阶导数, 并使用局 部间断 Galerkin (local discontinuous Galerkin, LDG) 方法离散空间导数, 给出方程的全离 散格式. 基于离散的分数阶 Gronwall 不等式, 证明了格式的数值稳定性和收敛性, 且所得结 果关于 α 是鲁棒的, 即当 α → 1⁻ 时不会发生爆破. 最后, 通过数值算例验证理论分析的结果.

We present an efficient method for seeking the numerical solution of a Caputo- type diffusion equation with a variable coefficient. Since the solution of such an equation is likely to have a weak singularity near the initial time, the time-fractional derivative is discretized using the L1 formula on nonuniform meshes. For spatial derivative, we employ the local discontinuous Galerkin method to derive a fully discrete scheme. Based on a dis- crete fractional Gronwall inequality, the numerical stability and convergence of the derived scheme are proven which are both α-robust, that is, the bounds obtained do not blow up as α → 1⁻. Finally, numerical experiments are displayed to confirm the theoretical results.