** Convergence Tests of Variable-Step Extrapolated Runge-Kutta Scheme**

In the homework a fourth-order Runge-Kutta scheme was applied to the Duffing oscillator. The initial conditions were chosen such that the particle barely made it across the barrier in the center, which makes the dynamics very sensitive to the accurace of the code.

Solution *u* for *tol=0.025*

Solution *u* for *tol=1e-11*

Time dependence of the total energy.

Time dependence of the time steps.

Difference between approximations to *u* for *delta t* and *delta t/2*.

Dependence of change in total energy in the oscillator on the tolerance applied in the choice of time step. The energy change and the tolerance are linearly related.

The difference in the value of the dependent variable *u* between two approximations
using timesteps *delta t* and *delta t/2* as a function of the tolerance. The accuracy
of the solution and the tolerance are linearly related.

The dependence of the energy change on the time step shows the 5th-order accuracy.

The computational effort grows with the fifth-power of the accurace (change in energy).