ODE Integration

The Apogee simulator solves initial value problems (IVPs) of the form:

\[ \frac{d\mathbf{y}}{dt} = \mathbf{f}(\mathbf{y}, t), \quad \mathbf{y}(t_0) = \mathbf{y}_0 \]

Solver Choice¶

We use the Tsitouras 5(4) explicit Runge-Kutta method from Diffrax.

Why Tsit5?¶

Feature	Benefit
Explicit	No linear system solves
5th order	High accuracy
Embedded 4th order	Error estimation for adaptivity
FSAL	Efficient step reuse

Configuration¶

solver = diffrax.Tsit5()
stepsize_controller = diffrax.PIDController(rtol=rtol, atol=atol)

Topics¶

Runge-Kutta Methods - Theory and derivation
Adaptive Stepping - Error control

Implementation¶

def _solve_segment(*, term, solver, t0, t1, dt0, y0, args, event, rtol, atol, max_steps):
    """Solve ODE segment with event detection."""
    return diffrax.diffeqsolve(
        term,
        solver,
        t0=t0,
        t1=t1,
        dt0=dt0,
        y0=y0,
        args=args,
        saveat=diffrax.SaveAt(t0=True, steps=True),
        stepsize_controller=diffrax.PIDController(rtol=rtol, atol=atol),
        event=event,
        max_steps=max_steps,
    )

References¶

Hairer, E., Nørsett, S.P., & Wanner, G. (1993). Solving Ordinary Differential Equations I: Nonstiff Problems. Springer.
Tsitouras, Ch. (2011). "Runge-Kutta pairs of order 5(4) satisfying only the first column simplifying assumption." Computers & Mathematics with Applications, 62, 770-775.