Citation: Singla, R.; Singh, G.; Ramos, H.; Kanwar, V. Development of a Higher-Order A-Stable Block Approach with Symmetric Hybrid Points and an Adaptive Step-Size Strategy for Integrating Differential Systems Efficiently. Symmetry 2023, 15, 1635. https://doi.org/10.3390/ sym15091635 Academic Editor: Serkan Araci Received: 27 July 2023 Revised: 18 August 2023 Accepted: 21 August 2023 Published: 24 August 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Development of a Higher-Order A-Stable Block Approach with Symmetric Hybrid Points and an Adaptive Step-Size Strategy for Integrating Differential Systems Efficiently Rajat Singla 1,2,†,‡ , Gurjinder Singh 1,‡ , Higinio Ramos 3,4, * and Vinay Kanwar 5 1 Department of Mathematics, I. K. Gujral Punjab Technical University Jalandhar, Main Campus, Kapurthala 144603, Punjab, India; rajatmath1310@gmail.com (R.S.); gurjinder11@gmail.com (G.S.) 2 Department of Mathematics, Akal University, Raman Road, Talwandi Sabo 151302, Punjab, India 3 Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain 4 Department of Mathematics, Escuela Politécnica Superior de Zamora, Campus Viriato, 49022 Zamora, Spain 5 University Institute of Engineering and Technology, Panjab University, Sector-25, Chandigarh 160025, Chandigarh, India; vmithil@yahoo.co.in * Correspondence: higra@usal.es † Current address: Department of Engineering, Plaksha University, Alpha City, Sector-101, Mohali 140306, Punjab, India. ‡ These authors contributed equally to this work. Abstract: This article introduces a computational hybrid one-step technique designed for solving initial value differential systems of a first order, which utilizes second derivative function evaluations. The method incorporates three intra-step symmetric points that are calculated to provide an optimum version of the suggested scheme. By combining the hybrid and block methodologies, an efficient numerical method is achieved. The hybrid nature of the algorithm determines that the first Dahlquist barrier is overcome, ensuring its effectiveness. The proposed technique exhibits an eighth order of convergence and demonstrates A-stability characteristics, making it particularly well suited for handling stiff problems. Additionally, an adjustable step size variant of the algorithm is developed using an embedded-type technique. Through numerical experiments, it is shown that the suggested approach outperforms some other well-known methods with similar properties when applied to initial-value ordinary differential problems. Keywords: ODEs; initial-value problems; hybrid methods; adaptive step size; A-stability; optimization strategy MSC: 65LXX; 65L04; 65L05; 65L06; 65L20 1. Introduction In this article, our aim is to construct an efficient algorithm for integrating initial-value differential problems given by z ′ ( x)= f( x, z); z( x 0 )= z 0 , (1) with x ∈ [ x 0 , x N ], z : [ x 0 , x N ] → R m , f : [ x 0 , x N ] × R m → R m , assuming that all prerequisites for the existence of a unique solution are fulfilled. Differential equations are used to model continuous phenomena that frequently occur in real-world situations. Unfortunately, very few of such equations can be tackled analyt- ically. In this scenario, usually the problem of interest is dealt with numerically, that is, an approximate solution is obtained on a discrete set of points. The classes of Runge–Kutta and linear multi-steps techniques have usually been to obtain reliable approximations of the true solution of (1). For more details, one can see the monographs written by Butcher [1], Symmetry 2023, 15, 1635. https://doi.org/10.3390/sym15091635 https://www.mdpi.com/journal/symmetry