ODE Solvers Comparison

A from-scratch implementation and comparison of six numerical methods for solving ordinary differential equations:

• Euler method — explicit forward Euler (1st order)
• Modified Euler method — Heun's predictor-corrector (2nd order)
• RK2 — midpoint method (2nd order)
• RK4 — classic 4-stage Runge-Kutta (4th order)
• RKF45 — Runge-Kutta-Fehlberg, embedded 4th/5th order with adaptive step size
• DOP853 — Dormand-Prince 8th order, adaptive (via scipy.integrate.solve_ivp, since its ~13-stage Butcher tableau isn't practical to hand-code)

All six solvers are run on the same test problem, and their accuracy is compared against the known closed-form solution.

Output

Left: the six numerical solutions vs. the exact solution. Right: each method's error over time on a log scale — DOP853 stays near machine precision (~10⁻¹⁴), while Euler's error grows to nearly 1.

Features

• Each method implemented as its own function with a shared signature (solver(f, t0, y0, t_end, h)), so any of them can be dropped into a different ODE or system of ODEs
• Works for both scalar and vector-valued y, since state is handled as a NumPy array internally
• Adaptive step-size control for RKF45, implemented from the classic Fehlberg coefficients (Butcher tableau)
• Side-by-side accuracy comparison: a results table (final value, error, steps used) plus solution and error-vs-time plots

Requirements

• Python 3.9+
• See requirements.txt

Setup

git clone https://github.com/linkedaven/Differential-Equation-Solver.git
cd Differential-Equation-Solver
pip install -r requirements.txt

Usage

python ode_solvers.py

A window opens showing the six numerical solutions against the exact solution (left) and each method's error on a log scale (right). The four fixed-step methods (Euler, Modified Euler, RK2, RK4) use a step size of h = 0.2; RKF45 runs adaptively with tol = 1e-6; DOP853 runs with rtol = 1e-10 and atol = 1e-12.

Test problem

By default, the script solves:

y' = y - t² + 1,    y(0) = 0.5,    t ∈ [0, 2]

whose closed-form solution is:

y(t) = (t + 1)² - 0.5·eᵗ

To try a different problem, edit f(t, y), y_exact(t), and the values of t0, y0, and t_end near the top of ode_solvers.py. Since every solver shares the same signature, the rest of the script works unchanged. To change the step size of the fixed-step methods, edit h in the comparison section at the bottom of the script.

License

MIT — see LICENSE.