Several evolutionary processes arising in sustainability-related applications, such as the analysis of spatial disease spread for public health management, electrodeposition [3], or combustion processes [6], can be described by multidimensional reaction–diffusion PDEs, typically equipped with Neumann boundary conditions. The numerical simulation of these problems is particularly demanding and often requires long-time integration. Therefore, the use of efficient and stable time integrators combined with suitable spatial discretizations is required. The main aim of this talk is to show how to enhance the efficiency of existing time integrators for reaction-diffusion problems by means of splitting and matrix-oriented techniques. These approaches exploit the structure of the matrices arising from the spatial discretization of the diffusion operator, leading to a significant reduction in computational cost. We introduce a class of matrix-oriented W-methods of order up to four [1, 6] and compare their performance with that of the corresponding AMF (Approximate Matrix Factorization) splitting schemes [2]. Moreover, we consider the AMF-W framework also in a more general setting involving time-dependent boundary conditions. Inspired by recent techniques proposed for Dirichlet problems [5], we present a new boundary correction technique for the case of time-dependent Neumann boundary conditions, that prevents the reduction of the order of accuracy and preserves the theoretical convergence properties of the used methods [4]. References: [1] D. Conte, S. González-Pinto, D. Hernández-Abreu, G. Pagano. J. Sci. Comput. 100(2), 34. 2024. [2] D. Conte, S. Iscaro, G. Pagano. On Matrix-Oriented and AMF-W Methods for advection-reaction diffusion Partial Differential Equations. In preparation. [3] M. C. D’Autilia, I. Sgura, V. Simoncini. Comput. Math. Appl. 79.7, pp. 2067–2085. 2020. [4] S. González-Pinto, D. Hernández-Abreu, S. Iscaro. On the treatment of boundary conditions for AMF-W methods in diffusion-reaction PDES. In preparation. [5] González-Pinto, S., Hernández-Abreu, D. Appl. Numer. Math., 210: 95-112. 2025. [6] W. Hundsdorfer, J. G. Verwer. Vol. 33. Springer Science and Business Media., 2003.