A Modified Third-Order Block Numerical Integration Technique for First Order Stiff and Oscillatory Differential Equations

Abstract

This paper proposes a modified two-point block backward differentiation formula (MBBDF) for the numerical solution of highly stiff and oscillatory ordinary differential equations. The method achieves third-order accuracy with a reduced error constant and computes two solution values simultaneously at each step. For the selected parameter value $\rho=-\dfrac{7}{8}$, stability analysis shows that the resulting scheme is A-stable, making it suitable for the numerical integration of stiff initial value problems. The method is also consistent and zero-stable, and hence convergent. Numerical experiments on benchmark stiff problems show that the relative accuracy of the considered methods depends on the problem and step size. While 2SBBDF generally produces the smallest maximum errors, the proposed MBBDF method maintains competitive accuracy and often requires substantially less computational time than 2SBBDF and 2ISBBDF. The results indicate that MBBDF provides a favorable compromise between numerical accuracy, stability, and computational efficiency for the solution of stiff and oscillatory ordinary differential equations.