The Integration of Stiff Systems of Ordinary Differential Equations Using a Time-Stepping Algorithm with Off-Step Points
TL;DR
Numerical experiments demonstrate that the 2SBBDFO method is an effective and reliable tool for solving stiff ODEs, and achieves higher accuracy and competitive computational efficiency compared to existing block BDF methods, particularly for smaller step sizes.
Abstract
Stiff systems of ordinary differential equations (ODEs) arise frequently in science, engineering, and applied mathematics, and their numerical solutions require methods with strong stability properties to avoid severe step-size restrictions. This motivates the development of efficient and stable numerical schemes tailored for such problems. In this paper, a fully implicit hybrid block method, termed the Two-Point Superclass Block Backward Differentiation Formula with Off-step Points (2SBBDFO), is proposed for solving stiff ODE systems. The method extends the classical two-point block BDF by incorporating a non-zero coefficient $\beta_{k-1}$, enabling the simultaneous computation of two grid-point and two off-step solution values within each block. Theoretical analysis shows that the method is consistent, zero-stable, and convergent, with order-five accuracy. Stability analysis using the boundary-locus technique confirms that the method is nearly A-stable (A($\alpha$)-stable). The resulting nonlinear system is solved using Newton's iteration for efficient convergence. Numerical experiments on selected stiff initial value problems (IVPs) demonstrate that the method achieves higher accuracy and competitive computational efficiency compared to existing block BDF methods, particularly for smaller step sizes. These results indicate that the 2SBBDFO method is an effective and reliable tool for solving stiff ODEs.