Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S31 / 3

Put $y^{(0)}=y$ and $y^{(1)}=\dot{y}$. ThenThe second row is the original damped harmonic oscillator equation and the first row defines the auxiliary velocity. This converts a second-order ordinary differential equation into two first-order equations without changing the required two initial values.

A numerical evolution has numerical stability if small perturbations in initial data, forcing or arithmetic are not amplified without control over the time interval of interest. For this constant-coefficient decaying linear system, an absolute-stability test requires the homogeneous numerical amplification factors to have magnitude at most one, with no growing Jordan-block contribution on the unit circle. Stability is distinct from accuracy: a strongly overdamped numerical solution can be stable while representing the physical oscillation poorly.

The characteristic polynomial is ${\lambda }^2+2\gamma \lambda +k^2=0$. Since $0<\gamma <|k|$,Forward Euler method has amplification eigenvalues $g_{\pm }=1+h{\lambda }_{\pm }$. Their squared modulus isThusStrict inequality gives numerical damping; at the upper boundary the amplitude is constant despite physical decay, so that boundary is a poor accuracy choice. In the undamped limit $\gamma =0$, $|g{|}^2=1+k^2{h}^2>1$ for every $h>0$: forward Euler method is unstable for any nonzero step on an undamped harmonic oscillator.

Backward Euler method has update ${\mathbf{y}}^{n+1}=(I-hM{)}^{-1}{\mathbf{y}}^n$ and amplification eigenvalues $g_{\pm }=(1-h{\lambda }_{\pm }{)}^{-1}$. Hencefor every $h\ge 0$. It is therefore unconditionally stable for this linear system. Even with $\gamma =0$, it damps every nonzero numerical mode: stability is obtained at the cost of artificial damping and phase error. Large $h$ does not accurately resolve the oscillation period.

An explicit numerical method normally has a simple, inexpensive step and needs no simultaneous solve, but stability can force very small steps, especially for stiff differential equations. An implicit numerical method requires a linear or nonlinear solve, storage and suitable solver tolerances; its larger stability region can permit larger steps when stability is the main restriction. Accuracy, phase fidelity and preservation of physical structure still constrain the step. For this oscillator, backward Euler method is stable but dissipative, while forward Euler method is cheap but unstable without damping. Neither statement makes all implicit methods superior to all explicit ones; method order, problem structure and total cost determine the useful choice.