An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term.

The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices.

Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper.

为时间分数扩散反应问题的数值解设计了一种有效的策略。最初应用空间导数的标准有限差分离散化，这导致线性刚性项。然后，在隐式-显式 (IMEX) 框架中实施矩形积积分 (PI) 规则，以便隐式处理此线性刚性项并以显式方式处理非线性且通常为非刚性项。

内核压缩方案（KCS）被相继采用，以减少持久内存项的计算和存储需求的过载。为了减少计算工作量，半离散化问题以矩阵形式描述，以便只需要用小矩阵来求解 Sylvester 方程。

通过在二维域中进行的数值实验，推导出并验证了精度的理论结果以及整个方法主要参数的最佳选择策略。还显示了相对于其他方法的计算优势，并在论文末尾说明了检测模式形成的一些应用。