We noticed recently that Forward Euler is pretty bad for simulating the *pendulum* equation. We feel that the pendulum example is too small of an example for the results to be so poor.
Explicit integration schemes suffer from a small region of stability, as was mentioned before. In order to be stable they need a very small step-size and thus if you want to solve the ODEs over a longer timespan you need many steps. This usually leads to a large accumulated error.
Thus unless you want to solve only for a very short time period I would switch to an implicit scheme, e.g. Implicit runge-kutta or multi-step methods.
Implicit is expensive/complicated.
Adams-Bashforth and variants is not stable for the pendulum.
3 or 4 step explicit Runge-Kutta methods are stable for the pendulum
But these RK methods will damp the pendulum motion (with less damping as the timestep gets smaller).
For the pendulum, Stormer-Verlet is simple, explicit, and will not damp the oscillations.
For Hamiltonian systems (like the pendulum) symplectic methods are clearly the best choice.
Gauss methods are implicit and therefore by the definition of simple - not simple.
Use Verlet:
x = position (or angle)
v = velocity (or angular velocity)
a(x) = acceleration (or angular acceleration)
(1) v_(n+1/2) = v_n + (dt/2)*a(x_n) explicit
(2) x_(n+1) = x_n + dt*v_(n+1/2) explicit
(3) v_(n+1) = v_(n+1/2) + (dt/2)*a(x_(n+1)) explicit
Simple
Stable for any timestep up to the period/pi (which is huge).
Conserves mass, momentum, angular momentum, energy within a small bound.
2nd order accurate
Works for any complicated coupled nonlinear dynamics you can dream up.
You can't do better than this given your criteria.
- Badges
- Science topic
- Similar topics
- Mathematics