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).

The package is available on PyPI.

Requirements

• Python 3
• NumPy
• Matplotlib (for the example scripts)

Install the published package and its runtime dependency:

python -m pip install runge-kutta

For local development, install from a source checkout with 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 Derivative, State, rk4

def rhs(t: float, y: State) -> State:
    return -y

rhs: Derivative = rhs
t, y = rk4(rhs, t0=0.0, y0=np.array([1.0]), h=0.1)

Types

The module provides these aliases for annotating equations and states:

State = float | NDArray[np.float64]
Derivative = Callable[[float, State], State]

State represents either a scalar or a float64 NumPy array. A Derivative accepts a scalar time and a State, then returns the derivative as a State.

Solvers

f must be a Derivative.

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.