Block Hybrid Trapezoidal-type Methods for Solving Initial Value Problems in Ordinary Differential Equations

In solving 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 hybrid 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 herein 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 of the constructed block algorithms. Notably, these stability regions are proven to be unbounded for order $p\leq 15$. Comparative assessments reveal their competitiveness with existing methods. In fact, 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.