Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Duffing equation

With non-linearities, the frequency spectrum of an undamped Duffing oscillator should be composed of odd multiples of the driving frequency which can be interpreted as resonance driving terms. It is expected that the frequency spectrum of the corresponding numerical solution with high accurateness should contain nearly the same components. Hence, to contain these Fourier components and to calculate the amplitudes of these components in a more accurate and efficient way is the key to develop a new numerical method with high stability, accuracy and efficiency for the Duffing equation. To explore the possibility of using trigonometrically-fitting technique to build a numerical method with resonance spectrum, we design four types of Numerov methods, in which the first one is the traditional Numerov method, which contains no Fourier component, the second one contains only the first resonance term, the third one contains the first two resonance terms, and the last one contains the first three resonance terms, and apply them to the well-known undamped Duffing equation with Dooren's parameters. The numerical results demonstrate that the Numerov method fitted with the Fourier components is much more stable, accurate and efficient than the one with no Fourier component. The accuracy of the fitted method with the first three Fourier components can attain 10⁻⁹ for a remarkable range of step sizes, including nearly infinite, except individual small range of instability, which is much higher than the one of the traditional Numerov method, with eight orders for step size of π/2.011.