A Taylor-Augmented Simpson Method for Highly Oscillatory ODEs: Consistency Order, Interval of Periodicity and Computational Cost

The numerical integration of ordinary differential equations exhibiting highly oscillatory dynamics requires a balance between computational efficiency and accuracy that classical fixed-step solvers struggle to achieve. We examine a Taylor-augmented variant of the Simpson 1/3 corrector, in which an explicit second-derivative term evaluated on a three-node central stencil is appended to the classical two-step formula. A formal local truncation error analysis shows the augmented scheme to be consistent of order two, compared with the fourth-order accuracy of the unaugmented Milne–Simpson corrector; no claim of higher-order convergence is made, and the measured convergence rate confirms this value. A linear multistep analysis yields the interval of periodicity in closed form, extending to a digital frequency Ω≈1.7968 and a maximum functional step h≈0.286T, marginally wider than the classical 3 bound. Using a linearized harmonic oscillator at 1 kHz, 100 kHz and 1 MHz, the scheme is benchmarked against Runge–Kutta (RK4), Dormand–Prince, Rosenbrock and Milne–Simpson baselines in terms of maximum relative error and execution time. Under coarse discretization it loses tracking, exhibiting the phase-alternation signature of the parasitic root of this method family. At fine step sizes, it attains accuracy comparable to RK4 at roughly one-seventh of the execution time, positioning it as a low-cost fixed-step option where per-step expense dominates and asymptotic order is not the binding constraint.