In this paper, we construct and investigate an accurate numerical scheme for solving a class of variable-order (VO) fractional diffusion equation based on the Caputo–Fabrizio fractional derivative. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. For all variable-order $\alpha \left(t\right)\in (0,1)$, we derive the stability and $L^2$ convergence of proposed scheme and prove that the method 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. Several numerical tests are given to validate the theoretical analysis and efficiency of the proposed algorithm.

在本文中，我们构建并研究了一种基于Caputo–Fabrizio分数阶导数求解一类可变阶（VO）分数阶扩散方程的精确数值方案。该方案通过在时间变量上使用有限差分法和在空间上使用局部不连续Galerkin方法（LDG）提出。对于所有可变阶 $\alpha （Ť）\in （0，\mathrm{1个}）$，我们得出稳定性和 ${\mathrm{大号}}^{\mathrm{2个}}$ 所提方案的收敛性，证明了该方法的准确性 $Ø（\tau +H^{ķ+\mathrm{1个}}）$， 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。给出了若干数值测试，以验证所提出算法的理论分析和效率。