Abstract

This study introduces a novel single-step hybrid block method with four intra-step points that attains six-order accuracy, ensures A-stability, consistency, and provides an efficient, accurate, and computationally economical tool for solving ordinary differential equations. The scheme incorporates intra-step points, which provide richer information within each integration step and significantly improve both precision and stability. When function values are not naturally defined at the chosen nodes, suitable interpolation techniques are introduced to approximate the missing terms without compromising accuracy. A detailed theoretical framework is established, including the analysis of convergence behavior and the derivation of local truncation error expressions. The stability of the method is further examined by identifying its stability regions and proving zero-stability under practical constraints on the step size. These theoretical guarantees ensure that the scheme is not only accurate but also reliable for long-time numerical integration. To complement the analysis, a series of comprehensive numerical experiments are conducted on benchmark problems frequently used in literature. The experimental results consistently demonstrate the superiority of the proposed method over existing approaches in terms of accuracy, efficiency, and overall robustness.

Keywords

References

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