ode_solve

ode_solve(
    f,
    initial_value,
    start_time,
    end_time=None,
    time_steps=None,
    dt=None,
    dt_min=None,
    method='rk4',
    tolerance=None,
)

Solve a user-written ordinary differential equation.

Solves dy/dt = f(t, y) forward from start_time with a ported Numerics Runge-Kutta method (P2 “math extras”): fixed-step second- or fourth-order Runge-Kutta over an equally-spaced grid (method="rk2"/"rk4" with end_time/time_steps), the fourth-order method’s single-step form (method="rk4" with dt alone), or one of the two adaptive-step-size methods, Runge-Kutta-Fehlberg or Runge-Kutta-Cash-Karp (method="rkf"/"cash_karp" with dt/dt_min).

Parameters

Name	Type	Description	Default
f	callable	A function taking two numbers (t, y) and returning dy/dt, one number.	required
initial_value	float	The value of y at start_time.	required
start_time	float	The time to start integrating from.	required
end_time	(float, int)	The end time and the number of equally-spaced points between start_time and end_time (inclusive of both ends). Required together for method="rk2"; for method="rk4", supplying them selects this array form over the single-step form (see dt below) – their PRESENCE, not a separate flag.	None
time_steps	(float, int)	The end time and the number of equally-spaced points between start_time and end_time (inclusive of both ends). Required together for method="rk2"; for method="rk4", supplying them selects this array form over the single-step form (see dt below) – their PRESENCE, not a separate flag.	None
dt	float	The step size. For method="rk4" without end_time/time_steps, the single step to advance by (the ODE is solved once, at start_time + dt). For method="rkf"/"cash_karp", the maximum internal step size, required together with dt_min.	None
dt_min	float	The minimum internal step size for method="rkf"/"cash_karp", required together with dt.	None
method	str	One of "rk4" (the default), "rk2", "rkf", or "cash_karp".	"rk4"
tolerance	float	The absolute error tolerance for method="rkf"/"cash_karp" alone. Left unset, the ported routine’s own default (1e-3) applies.	None

Returns

Name	Type	Description
	float or numpy.ndarray	For method="rk2", or "rk4" with end_time/time_steps: an array of length time_steps, the solution at each grid point (y[0] is initial_value). For every other case: a single number, the solution at start_time + dt ("rk4"’s single-step form) or under the adaptive step control ("rkf"/"cash_karp").

Examples

>>> import corehydropy as ch
>>> y = ch.ode_solve(lambda t, y: y, initial_value=1, start_time=0, end_time=1,
...                   time_steps=100)
>>> round(y[-1], 5)
2.71828