Development of a Higher-Order 𝒜-Stable Block Approach with Symmetric Hybrid Points and an Adaptive Step-Size Strategy for Integrating Differential Systems Efficiently

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.