ODEs-and-solvers-in-Python

This repository is a Python learning project focusing on simulating systems of ordinary differential equations (ODEs) using hand-written numerical solvers and a modular model interface. It is the Python twin of the original MATLAB project ODEs-and-solvers.

Contents:

solvers.py:

• ee — Explicit (forward) Euler

• ie — Implicit (backward) Euler

• trap — Implicit trapezoidal rule (Crank–Nicolson)

• rk4 — Classic 4th-order Runge–Kutta

• rk45 — Runge–Kutta–Fehlberg 4(5) with adaptive steps
  $~~~~~~~~~~~~~~~~~~~~~~~~~$ Sharp tolerances: per-component absolute and relative tests must both pass.
  $~~~~~~~~~~~~~~~~~~~~~~~~~$ Propagate either the 5th-order state (standard) or the 4th-order state (often a bit more robust on
  $~~~~~~~~~~~~~~~~~~~~~~~~~$ tricky trajectories).

• newton_it — Newton-Rhapson iteration for implicit methods

• num_jacobian — Numerical Jacobian for the Newton-Rhapson iteration

All solvers share the same solve(f, (t0, tf), y0, h) -> (t, Y) signature.

models.py:

• vdp — van der Pol oscillator

• lv — Lotka–Volterra predator–prey model

• rayleigh — Rayleigh oscillator

• cr3bp — Planar circular restricted three-body problem (Earth–Moon parameters)

• coupled_pendulums — Coupled pendulums on a cart with damping, exhibiting spontaneous synchronization

• sir — Simple Susceptible-Infected-Recovered compartmental epidemic model

Each model implements f(t: float, y: ndarray) -> ndarray, parameters can be overridden via params = dict(...) in main.py.

Quik Start

• Clone this repo.
• Run pip install numpy matplotlib.
• Run main.py, select system, solver and simulation settings at the top of the main() funtion.

Contributing

This project was meant as a learning playground, but improvements, bug fixes, and new systems/solvers are welcome.

License

MIT License: feel free to use it for education, research, and fun.