A time-stepping L1 scheme for subdiffusion equation with a Riemann--Liouville time-fractional derivative is developed and analyzed. This is the first paper to show that the L1 scheme for the model problem under consideration is second-order accurate (sharp error estimate) over nonuniform time-steps. The established convergence analysis is novel, innovative and concise. For completeness, the L1 scheme is combined with the standard Galerkin finite elements for the spatial discretization, which will then define a fully-discrete numerical scheme. The error analysis for this scheme is also investigated. To support our theoretical contributions, some numerical tests are provided at the end. The considered (typical) numerical example suggests that the imposed time-graded meshes assumption can be further relaxed.

中文翻译：

---

分数反应扩散方程的 $L1$ 近似，时间分级网格上的二阶误差分析

开发并分析了具有 Riemann--Liouville 时间分数阶导数的子扩散方程的时间步长 L1 格式。这是第一篇论文，表明所考虑的模型问题的 L1 方案在非均匀时间步长上是二阶准确的（尖锐误差估计）。既定的收敛分析新颖、新颖、简洁。为完整起见，L1 方案与用于空间离散化的标准 Galerkin 有限元相结合，然后将定义一个完全离散的数值方案。还研究了该方案的误差分析。为了支持我们的理论贡献，最后提供了一些数值测试。所考虑的（典型）数值示例表明可以进一步放宽强加的时间分级网格假设。