A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time meshes, and a fourth-order compact finite difference scheme is used for approximating the spatial variable coefficient differential operator. A full theoretical analysis of the stability and convergence of the method is given for the general case of the variable coefficients by developing an analysis technique different from the one for the constant coefficient problem. Taking the weak initial singularity of the solution into account, a sharp error estimate in the discrete \(L^{2}\)-norm is obtained. It is shown that the proposed method attains the temporal optimal second-order convergence provided a proper mesh parameter is employed. Numerical results demonstrate the sharpness of the theoretical error analysis result.

为了解决一类变系数反应-扩散问题，提出了一种基于非均匀时间网格的高阶紧致有限差分方法。通常，这种问题的解决方案在初始时具有典型的弱奇异性。将Caputo时间分数导数的均匀时间网格上的Alikhanov高阶近似推广到一类非均匀时间网格上，并使用四阶紧致有限差分方案近似空间变量系数微分算子。通过开发一种不同于常系数问题的分析技术，针对可变系数的一般情况，对该方法的稳定性和收敛性进行了完整的理论分析。考虑到解决方案的弱初始奇点，\（L ^ {2} \）-范数。结果表明，如果采用适当的网格参数，该方法可以获得时间最优的二阶收敛。数值结果证明了理论误差分析结果的准确性。