A Residual-Correction Derivation of the Second-Order Adams-Bashforth Method for Ordinary Differential Equations

Source abstract

The Adams-Bashforth family of explicit multistep methods is a staple of numerical ODE solvers. Yet, almost every textbook derives the second-order member via polynomial interpolation, a route that reveals little about the underlying error-correction mechanism. We offer an alternative: start from Euler's approximation, estimate its local truncation error from the slope difference between consecutive steps, and add that estimate back as a correction. The calculation yields exactly the familiar AB2 formula, with coefficients 3/2 -1/2 emerging naturally. This residual-correction perspective does more than reproduce known results—it links the method to defect correction, suggests straightforward error estimators, and provides a pedagogical bridge between one-step and multistep schemes. We analyse consistency, zero-stability, and second-order convergence, then validate the method on four benchmark problems (exponential decay, logistic growth, oscillatory dynamics, and a mildly stiff system). Numerical experiments confirm the theoretical order; experimental convergence rates consistently approach 2. The residual-correction framework reframes a classical technique, offering fresh insight for both teaching and extension.