In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in $(0,1)$. If its solution $u(x,t)$ has strong regularity, for example $u(\cdot ,t)\in C^2[0,T]$ for a given time $T$, then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution $u(x,t)$ likely behaves a certain regularity at the starting time, i.e., $\frac{\partial u}{\partial t}$ and $\frac{{\partial }^{2}u}{{\partial }^{2}t}$ can blow up as $t\to 0^{+}$ albeit $u(\cdot ,t)\in C[0,T]$ for a given time $T$. In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis.

在本文中，我们研究了广义时间分数Burgers方程，其中时间分数导数在Caputo的意义上具有导数阶 $（0，\mathrm{1个}）$。如果它的解决方案$ü（X，Ť）$ 有很强的规律性，例如 $ü（\cdot ，Ť）\in C^{\mathrm{2个}}[0，Ť]$ 在给定的时间内 $Ť$，然后在均匀网格上使用L1方案近似Caputo时间分数导数，并使用局部不连续伽勒金（LDG）方法逼近空间导数。但是，解决方案$ü（X，Ť）$ 在开始时可能表现出一定规律性，即 $\frac{\partial ü}{\partial Ť}$ 和 $\frac{{\partial }^{\mathrm{2个}}ü}{{\partial }^{\mathrm{2个}}Ť}$ 可以炸毁为 $Ť\to 0^{+}$ 尽管 $ü（\cdot ，Ť）\in C[0，Ť]$ 在给定的时间内 $Ť$。在这种情况下，我们在非均匀网格上使用L1方案来近似Caputo时间分数导数，并使用LDG方法来离散化空间导数。建立并分析了两种情况的完全离散方案。结果表明，所推导的方案在数值上是稳定的和收敛的。最后，提供了一些数值实验，这些实验支持理论分析。