A Nonstandard Finite Difference Schemes for the Damped Harmonic Oscillator with Trigonometric Denominators: Convergence Analysis, Systematic Damping-Regime Validation, and Energy Conservation

Abstract

This paper develops, analyses, and extends a class of nonstandard finite difference (NSFD) schemes for the damped harmonic oscillator equation $y'' + 2\varepsilon y' + y = 0$ using a hybrid interpolant that combines trigonometric and exponential basis functions within a dynamically renormalized denominator structure. We construct three new numerical schemes, New-h, New-Sin, and New-Exp distinguished by their choice of denominator function: $\psi = h$, $\psi = \sin(h)$, and $\psi = (e^{\gamma h}-1)/\gamma$, respectively. We provide complete proofs of consistency, local stability, and convergence for the proposed schemes and establish that all three schemes possess the same qualitative properties as the continuous model, including the preservation of monotone solution behaviour. New-Sin scheme is superior to the classical central difference method for undamped oscillators a comparative numerical study across four damping regimes and three step sizes, yielding maximum absolute errors up to two orders of magnitude smaller than the classical scheme.