Abstract
In the realm of ordinary differential equations, tackling stiff problems necessitates the application of robust numerical methods endowed with A-stability properties. To circumvent the constraints posed by the Dahlquist barrier theorem and mitigate errors arising from step-by-step implementation of linear multistep methods, block hyrid schemes have been introduced. This study focuses on the development of novel block schemes designed for the direct approximation of solutions to stiff initial value problems.
The methods proposed here leverage both interpolation and collocation, enhancing their consistency, convergence, and accuracy in solving initial value problems. The efficacy of the devised methods is demonstrated through a comprehensive analysis of stability regions for each class of the constructed symmetric block algorithms. Notably, these stability regions are proven to be unbounded for order p ≤ 15, This study establishes the suitability of the proposed block schemes for addressing both linear and non-linear stiff problems.
Furthermore, comparative assessments reveal their competitiveness with existing methods documented in the literature. In summary, this research introduces innovative approaches to address the challenges posed by stiff initial value problems, offering enhanced stability and accuracy in comparison to established methods.
