NEW HYBRID IMPLICIT QUARTER-STEP BLOCK METHOD FORTHE SOLUTION OF FIRST ORDER ORDINARY DIFFERENTIALEQUATION

Abstract

This paper introduces an innovative hybrid implicit quarter-step block method designed for solving first-order ordinary differential equations (ODEs) numerically. Traditional linear multistep methods often necessitate the creation of separate predictors, which can ramp up both computational costs and complexity. To tackle this issue, we propose a continuous implicit hybrid method that utilizes interpolation and collocation of a power series approximation at both grid and off-grid points within a single integration interval. By evaluating the continuous formulation at quarter-step points, we derive discrete block formulas that come together to create a unified, self-starting block method. This approach allows for the simultaneous generation of numerical approximations at four consecutive points, eliminating the need for a separate predictor. We’ve established that this method is zero-stable, consistent, and therefore convergent. To evaluate the effectiveness of our proposed method, we conducted numerical experiments across a variety of initial value problems. The findings reveal that our new quarter-step block method not only achieves greater accuracy but also outperforms existing methods of similar order, highlighting its potential as a powerful and efficient computational tool for integrating first-order ODEs.