In this paper, an accurate numerical method is presented to solve a class of variable-order fractional diffusion problem. The problem first is discretized by a finite difference method in temporal direction, and then a local discontinuous Galerkin method in space. The stability and $L^2$ convergence of the proposed scheme are derived for all variable-order $\alpha \left(t\right)\in (0,1)$. We prove that the scheme is of accuracy-order $O(\tau +h^{k+1})$, where $\tau$, $h$ and $k$ are temporal step sizes, spatial step sizes and the degree of piecewise $P^k$ polynomials, respectively. Some numerical experiments are provided to verify the theoretical analysis and high-accuracy of the proposed method.

本文提出了一种精确的数值方法来解决一类变量的分数阶扩散问题。首先通过时间方向上的有限差分方法离散问题，然后通过空间中的局部不连续Galerkin方法离散化问题。稳定性和${\mathrm{大号}}^2$ 对所有可变阶均得出了所提方案的收敛性 $\alpha （Ť）\in （0，\mathrm{1个}）$。我们证明该方案是精度阶的$Ø（\tau +H^{ķ+\mathrm{1个}}）$，在哪里 $\tau$， $H$ 和 $ķ$ 是时间步长，空间步长和分段程度 $P^{ķ}$多项式分别。提供了一些数值实验，验证了所提方法的理论分析和高精度。