Abstract
This paper examines the consistency, convergence, error propagation and absolute stability of a step-doubling adaptive modified Euler method for nonlinear initial value problems. The underlying Heun update has a second-order Taylor expansion and a cubic local defect. Two half steps retain second-order global convergence and, when paired with a full step, provide an asymptotic local-error estimate for the accepted fine solution. A discrete error recursion supplies a conditional O(hmax²) global bound on finite intervals under appropriate smoothness and Lipschitz assumptions, provided the accepted mesh becomes uniformly fine. On the Dahlquist test equation, the accepted fine update has stability function [1+z/2+z²/8]²; the real negative stability interval in terms of the total attempted step extends to ?4 < z < 0, compared with ?2 < z < 0 for a single full Heun step. This extension costs two propagation half steps and does not imply A-stability or solve stiffness. Numerical checks on nonlinear logistic and Riccati equations illustrate the distinction between local tolerance, endpoint error and stability. The results establish the method’s mathematical scope and practical limits.
Introduction
A numerical method for nonlinear ordinary differential equations is useful only when its errors can be understood and its calculations remain stable in the intended regime. Variable time steps complicate that assessment. A successful local error test does not prove convergence by itself, nor does a small reported defect guarantee that an explicit discretisation can handle stiff modes. Analysis must separate truncation, propagation, step-size selection and the stability of the underlying advance.
This paper addresses that separation for the adaptive modified Euler construction described in the accompanying methodological study. Its core is the explicit trapezoidal Heun rule, combined with a coarse-full-step versus two-half-step comparison. The technique responds to the doctoral synopsis’s call for theoretical work on error, convergence and stability. It builds on standard numerical analysis principles (Gear, 1971; Hairer, Nørsett and Wanner, 1993) and adaptive-control literature (Hall and Higham, 1988; Söderlind, 2002).
The terminology matters. “Stable” can refer to propagation of perturbations on a bounded interval, or to absolute stability of a method on a decaying linear test problem; these are related but not identical concepts. Similarly, consistency is an asymptotic statement about a vanishing time step, while tolerance compliance is an algorithmic decision made from an estimated local error. The purpose here is to derive valid conclusions for each notion, state necessary assumptions and avoid interpreting step-size adaptation as an unconditional stability guarantee.
Conclusion
The accepted two-half-step modified Euler map is consistent and converges with global order two on bounded finite intervals under standard smoothness, Lipschitz and mesh-refinement assumptions. Its coarse/fine discrepancy supplies a leading-order estimate of accepted local error, not a certified error bound. Its negative-real absolute stability interval in the total-step variable is (?4,0), compared with (?2,0) for a single Heun step; this reflects substepping at additional computational cost and does not confer A-stability. These results form a rigorous baseline for the adaptive nonlinear ODE programme and identify the further algorithmic and numerical evidence required for stronger performance claims.
References
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