Simple pendulum Runge-Kutta-Fehlberg solver with error analysis

This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the problem of the simple pendulum:

\begin{aligned} \dfrac{d^2 \theta}{dt^2} = -\dfrac{g}{l} \cos{\theta} \end{aligned}

where $g$ is the acceleration due to gravity in metres per second squared, $l$ is the length of the pendulum in metres, $\theta$ is the angle from the positive $x$ axis (in radians) and $t$ is the time in seconds. Below you can specify the various parameters for the problem we will solve.

The error in $\dot{\theta}$ mentioned in the table below is approximated using this formula derived by integrating the above equation

\begin{aligned} \dot{\theta} = \pm \sqrt{\dot{\theta}_0^2 + \dfrac{2g}{l}\left(\sin{\theta_0} - \sin{\theta}\right)}. \end{aligned}

Our $\theta$ approximations are substituted in, and our $\dot{\theta}$ RKF45 approximation is subtracted from this value. From this equation, the period $T$ of the problem is approximated when the conditions for periodicity are satisfied, namely using the equation

\begin{aligned} T &= 2 \left|\int_{\theta_\mathrm{min}}^{\theta_\mathrm{max}} \dfrac{d\theta}{\sqrt{\dot{\theta}_0^2 + \dfrac{2g}{l} (\sin{\theta_0} - \sin{\theta})}}\right| \end{aligned}

where $\theta_\mathrm{min}$ and $\theta_\mathrm{max}$ are the two closest values for which $\dot{\theta} = 0$ . $T$ is calculated using Chebyshev-Gauss quadrature (Simpson's rule could not be used as there are unremovable singularities at the endpoints which makes Simpson's rule markedly less accurate).

Simulation parameter form.
Parameter	Value	Explanation
$g$ :		Acceleration due to gravity.
$l$ :		Length of pendulum in metres.
$t_f$ :		End time for the simulation in seconds. Default is four times the period of the problem.
$\theta_{0}$ :		Initial angle relative to the positive $x$ -axis.
$\dot{\theta}_{0}$ :		Initial rate of change of the aforementioned angle.
$N$ :		Number of quadrature nodes used to approximate period
$\epsilon$ :		Error tolerance.
$\mathrm{Tol}_{\mathrm{type}}$ :		Tolerance type, can be either absolute (0) or relative (1).
$h_{\mathrm{Initial}}$ :		Initial step size.
$h_{\mathrm{Min}}$ :		Minimum allowed step size.
$\Delta t$ :		Time increment for skipping ahead in animation.
$t_1$ :		Time you want to skip ahead to in animation when you press the skip button.
$\mathrm{Width}$ :		Width (in px) of Plotly windows used for plotting and animation below.
$\mathrm{Height}$ :		Height (in px) of Plotly windows used for plotting and animation below.
$t_\mathrm{Scale}$ :		Proportion of animation time passed per real time. $t_{\mathrm{Scale}}=1.0$ means animation and real time match. $t_{\mathrm{Scale}}<1.0$ means the animation is going more slowly than real time. $t_{\mathrm{Scale}}>1.0$ means it is going more rapidly.

Information	Data	Notes
$T$ :		Period of the problem, auto-calculated.
$\theta_\mathrm{min}$		The value of $\theta_\mathrm{min}$ used in the above period equation.
$\theta_\mathrm{max}$		The value of $\theta_\mathrm{max}$ used in the above period equation.
RKF45 step number		Number of steps used in our Runge-Kutta-Fehlberg approximation of the solution.

© Brenton Horne. I do not claim to be infallible, so feel free to open issues or pull requests on GitHub if you notice flaws or things that can be improved.
I favour Oxford spelling, so if you notice I generally use different spelling to Australian spelling, that is probably why.
Homepage. Source code: https://github.com/fusion809/fusion809.github.io.
Website built with Franklin.jl.

Simple pendulum Runge-Kutta-Fehlberg solver with error analysis

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