It is well known that the efficiency of a battery strongly depends on the charging ability and that today it is crucial to develop highly sustainable processes to control and reduce the distribution of material inside it. Towards this aim, in the last decade we have proposed a morphochemical reaction-diffusion PDE system, called DIB model, to describe the distribution of metal growth at the battery electrodes in terms of Turing pattern formation, see e.g. [1].\\ Numerical approximation of morphological features in Turing patterns is a challenging task because both very fine space discretization and longtime integration are needed. We show that, after semi-discretization in space, the Kronecker structure of the diffusion matrix can be exploited to build matrix-oriented (MO) versions of some classical time integrators. In particular, we consider finite differences on square domains and classical Lagrangian FEM on more general domains and special surfaces. In the first case, the fully discrete problem is reformulated as a sequence of \emph{Sylvester matrix equations}, that we solve by the \emph{reduced approach} in the associated spectral space [2]. In the second case, at each time step we solve a \emph{multiterm Sylvester matrix equation} by the matrix form of the Preconditioned Conjugate Gradient (MO-PCG) [3]. We present encouraging results wrt the classical vector approach (solving large sparse linear systems) for Turing pattern approximation in terms of computational execution times and memory storage. Joint work with V. Simoncini (University of Bologna, Italy).\\ Research supported by – NextGeneration EU– PRIN2022 PNRR Project Title “BAT-MEN” (BATtery Modeling, Experiments & Numerics)}, Project code P20228C2PP_001, CUP F53D23010020001 BIBLIOGRAPHY [1] D. Lacitignola, B. Bozzini, M. Frittelli, I. Sgura, Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition, Comm Nonlin Sci Numer Simul, 48 (2017) 484–508, doi.org/10.1016/j.cnsns.2017.01.008 [2] M.C. D'Autilia, I. Sgura, V. Simoncini, Matrix-oriented discretization methods for reaction-diffusion PDEs: comparisons and applications, CAMWA 79, (2020) 2067--2085, doi.org/10.1016/j.camwa.2019.10.020 [3] M. Frittelli, I. Sgura - Matrix-or-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domains}, Applied Numerical Mathematics (APNUM) 200 (2024) 286--308, doi.org/10.1016/j.apnum.2023.07.010