We are interested in the numerical solution of the DIB (Dual-Ion Batteries) model for the formation of spatio-temporal patterns in battery electrodeposition [1], a 2D reaction-diffusion Partial Differential Equation (PDE) system, which couples an equation for the morphological dynamics with one for the surface chemical dynamics. The numerical treatment of this problem requires the employment of non-trivial techniques, since a related spatial discretization performed e.g. via finite differences, finite elements, or spectral methods, leads to a large and highly stiff initial value problem. In this talk we propose numerical schemes which are able to provide accurate and stable solutions for these types of problems, also exploiting a-priori known properties of the model. In particular, we derive new classes of efficient linearly implicit and multivalue methods [2, 3]. The first are obtained through a stabilization of explicit numerical schemes using the so-called Time-Accurate and highly-Stable Explicit preconditioners [2, 3]. Numerical experiments show that the new methods efficiently solve the DIB model, employing reasonable computing times, and reproducing the expected Turing patterns. The good performance of the new methods will also be confirmed through tests on other problems and on a model for vegetation growth in arid environments characterized by low rainfall [4]. This talk is part of a research activity within the project PRIN PNRR 2022 P20228C2PP (CUP: F53D23010020001) BAT-MEN (BATtery Modeling, Experiments & Numerics). [1] B. Bozzini, D. Lacitignola, and I. Sgura., J Solid State Electrochem., 17, 467–479, 2013. [2] D. Conte, J. Martin-Vaquero, G. Pagano, and B. Paternoster, SIAM J. Sci. Comput., accepted for publication. [3] D. Conte, G. Pagano, and B. Paternoster, Commun. Nonlinear Sci. Numer. Simul., 119, 20, 2023.. [4] D. Conte, G. Pagano, and B. Paternoster, J. Comput. Appl. Math., 419:Paper No. 114790, 17, 2023. [5] R. D'Ambrosio, B. Paternoster, J. Comput. Appl. Math. 387, article number 112515 (2021).