scipy_dae - solving differential algebraic equations (DAE's) and implicit differential equations (IDE's) in Python
Python implementation of solvers for differential algebraic equations (DAE's) and implicit differential equations (IDE's) that should be added to scipy one day. See the GitHub repository.
Currently, two different methods are implemented.
- Implicit backward differentiation formula (BDF) of variable order with quasi-constant step-size and stability/ accuracy enhancement using numerical differentiation formula (NDF).
- Implicit Radau IIA methods of order 2s - 1 with arbitrary number of odd stages.
More information about both methods are given in the specific class documentation.
The Robertson problem of semi-stable chemical reaction is a simple system of differential algebraic equations of index 1. It demonstrates the basic usage of the package.
import numpy as np
import matplotlib.pyplot as plt
from scipy_dae.integrate import solve_dae
def F(t, y, yp):
"""Define implicit system of differential algebraic equations."""
y1, y2, y3 = y
y1p, y2p, y3p = yp
F = np.zeros(3, dtype=y.dtype)
F[0] = y1p - (-0.04 * y1 + 1e4 * y2 * y3)
F[1] = y2p - (0.04 * y1 - 1e4 * y2 * y3 - 3e7 * y2**2)
F[2] = y1 + y2 + y3 - 1 # algebraic equation
return F
# time span
t0 = 0
t1 = 1e7
t_span = (t0, t1)
t_eval = np.logspace(-6, 7, num=1000)
# initial conditions
y0 = np.array([1, 0, 0], dtype=float)
yp0 = np.array([-0.04, 0.04, 0], dtype=float)
# solver options
method = "Radau"
# method = "BDF" # alternative solver
atol = rtol = 1e-6
# solve DAE system
sol = solve_dae(F, t_span, y0, yp0, atol=atol, rtol=rtol, method=method, t_eval=t_eval)
t = sol.t
y = sol.y
# visualization
fig, ax = plt.subplots()
ax.set_xlabel("t")
ax.plot(t, y[0], label="y1")
ax.plot(t, y[1] * 1e4, label="y2 * 1e4")
ax.plot(t, y[2], label="y3")
ax.set_xscale("log")
ax.legend()
ax.grid()
plt.show()
More examples are given in the examples directory, which includes
- ordinary differential equations (ODE's)
- differential algebraic equations (DAE's)
- implicit differential equations (IDE's)
An editable developer mode can be installed via
python -m pip install -e .[dev]The tests can be started using
python -m pytest --cov