Runge-Kutta ODE Solvers
Small Python implementations of classical fourth-order Runge-Kutta (RK4) and Runge-Kutta-Fehlberg (RK45) methods for numerically solving ordinary differential equations (ODEs).
Requirements
- Python 3
- NumPy
- Matplotlib (for the example scripts)
Install the module and its dependencies:
python -m pip install .
Usage
Import a solver from runge_kutta. Each function advances the solution by
one integration step for an equation of the form dy/dt = f(t, y).
import numpy as np
from runge_kutta import rk4
def rhs(t, y):
return -y
t, y = rk4(rhs, t0=0.0, y0=np.array([1.0]), h=0.1)
Solvers
| Function | Description | Returns |
|---|---|---|
rk4(f, t0, y0, h) |
Performs one fixed-size classical fourth-order Runge-Kutta step. | (t, y) |
rk45(f, t0, y0, h) |
Performs one fixed-size Runge-Kutta-Fehlberg step using the fourth-order estimate. | (t, y, error) |
rk45_h_adj(f, t0, y0, h, epsilon) |
Performs an RK45 step and proposes a next step size from the embedded error estimate. | (t, y, error, h_adj) |
f must accept (t, y) and return the derivative at that point. y may be a
scalar or a NumPy array.
Examples
The included scripts compare RK4 and RK45 while plotting solution trajectories and RK45 error estimates:
python test1.py # Lorenz system
python test2.py # damped second-order system
Both scripts open Matplotlib windows. test1.py integrates the Lorenz system
for 500,000 steps and can take noticeably longer to run.
1 maintainer
Metadata
1 maintainer
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