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Python implementation of solvers for differential algebraic equation's (DAE's) that should be added to scipy one day.

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scipy_dae - solving differential algebraic equations (DAE's) and implicit differential equations (IDE's) in Python

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Python implementation of solvers for differential algebraic equations (DAE's) and implicit differential equations (IDE's) that should be added to scipy one day.

Currently, two different methods are implemented.

  • Implicit Radau IIA methods of order 2s - 1 with arbitrary number of odd stages.
  • Implicit backward differentiation formula (BDF) of variable order with quasi-constant step-size and stability/ accuracy enhancement using numerical differentiation formula (NDF).

More information about both methods are given in the specific class documentation.

To pique your curiosity

The Kármán vortex street solved by a finite element discretization of the weak form of the incompressible Navier-Stokes equations using FEniCS and the three stage Radau IIA method.

Karman

Basic usage

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=np.common_type(y, yp))
    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.plot(t, y[0], label="y1")
ax.plot(t, y[1] * 1e4, label="y2 * 1e4")
ax.plot(t, y[2], label="y3")
ax.set_xlabel("t")
ax.set_xscale("log")
ax.legend()
ax.grid()
plt.show()

Robertson

Advanced usage

More examples are given in the examples directory, which includes

Work-precision

In order to investigate the work precision of the implemented solvers, we use different DAE examples with differentiation index 1, 2 and 3 as well as IDE examples.

Index 1 DAE - Brenan

Brenan's index 1 problem is described by the system of differential algebraic equations

$$ \begin{aligned} \dot{y}_1 - t \dot{y}_2 &= y_1 - (1 + t) y_2 \\ 0 &= y_2 - \sin(t) . \end{aligned} $$

For the consistent initial conditions $t_0 = 0$, $y_1(t_0) = 1$, $y_2(t_0) = 0$, $\dot{y}_1 = -1$ and $\dot{y}_2 = 1$, the analytical solution is given by $y_1(t) = e^{-t} + t \sin(t)$ and $y_2(t) = \sin(t)$.

This problem is solved for $atol = rtol = 10^{-(1 + m / 4)}$, where $m = 0, \dots, 45$. The resulting error at $t_1 = 10$ is compared with the elapsed time of the used solvers in the figure below. For reference, the work-precision diagram of sundials IDA solver is also added. Note that the elapsed time is scaled by a factor of 100 since the sundials C-code is way faster.

Brenan_work_precision

Clearly, the family of Radau IIA methods outplay the BDF/NDF methods for low tolerances. For medium to high tolerances, both methods are appropriate.

Index 2 DAE - knife edge

The knife edge index 2 problem is a simple mechanical example with nonholonomic constraint. It is described by the system of differential algebraic equations

$$ \begin{aligned} \dot{x} &= u \\ \dot{y} &= v \\ \dot{\varphi} &= \omega \\ m \dot{u} &= m g \sin\alpha + \sin\varphi \lambda \\ m \dot{v} &= -\cos\varphi \lambda \\ J \dot{\omega} &= 0 \\ 0 &= u \sin\varphi - v \cos\varphi . \end{aligned} $$

Since the implemented solvers are designed for index 1 DAE's we have to perform some sort of index reduction. Therefore, we transform the semi-explicit form into a general form as proposed by Gear. The resulting index 1 system is given as

$$ \begin{aligned} \dot{x} &= u \\ \dot{y} &= v \\ \dot{\varphi} &= \omega \\ m \dot{u} &= m g \sin\alpha + \sin\varphi \dot{\Lambda} \\ m \dot{v} &= -\cos\varphi \dot{\Lambda} \\ J \dot{\omega} &= 0 \\ 0 &= u \sin\varphi - v \cos\varphi . \end{aligned} $$

For the initial conditions $t_0 = 0$, $x(t_0) = \dot{x}(t_0) = y(t_0) = \dot{y}(t_0) = \varphi(t_0) = 0$ and $\dot{\varphi}(t_0) = \Omega$, a closed form solution is given by

$$ \begin{aligned} x(t) &= \frac{g \sin\alpha}{2 \Omega^2} \sin^2(\Omega t) \\ y(t) &= \frac{g \sin\alpha}{2 \Omega^2} \left(\Omega t - \frac{1}{2}\sin(2 \Omega t)\right) \\ \varphi(t) &= \Omega t \\ u(t) &= \frac{g \sin\alpha}{\Omega} \sin(\Omega t) \cos(\Omega t) \\ v(t) &= \frac{g \sin\alpha}{2 \Omega} \left(1 - \cos(2 \Omega t)\right) = \frac{g \sin\alpha}{\Omega} \sin^2(\Omega t) \\ \omega(t) &= \Omega \\ \Lambda(t) &= \frac{2g \sin\alpha}{\Omega} (\cos(\Omega t) - 1) , % (2 * m * g * salpha / Omega) * (np.cos(Omega * t) - 1) \end{aligned} $$

with the Lagrange multiplier $\dot{\Lambda}(t) = - 2g \sin\alpha \sin(\Omega t)$.

This problem is solved for $atol = rtol = 10^{-(1 + m / 4)}$, where $m = 0, \dots, 32$. The resulting error at $t_1 = 2 \pi / \Omega$ is compared with the elapsed time of the used solvers in the figure below.

knife_edge_work_precision

Index 3 DAE - Arevalo

Arevalo's index 3 problem describes the motion of a particle on a circular track. It is described by the system of differential algebraic equations

$$ \begin{aligned} \dot{x} &= u \\ \dot{y} &= v \\ \dot{u} &= 2 y + x \lambda \\ \dot{v} &= -2 x + y \lambda \\ 0 &= x^2 + y^2 - 1 . \end{aligned} $$

Since the implemented solvers are designed for index 1 DAE's we have to perform some sort of index reduction. Therefore, we use the stabilized index 1 formulation of Hiller and Anantharaman. The resulting index 1 system is given as

$$ \begin{aligned} \dot{x} &= u + x \dot{\Gamma} \\ \dot{y} &= v + y \dot{\Gamma} \\ \dot{u} &= 2 y + x \dot{\Lambda} \\ \dot{v} &= -2 x + y \dot{\Lambda} \\ 0 &= x u + y v \\ 0 &= x^2 + y^2 - 1 . \end{aligned} $$

The analytical solution to this problem is given by

$$ \begin{aligned} x(t) &= \sin(t^2) \\ y(t) &= \cos(t^2) \\ u(t) &= 2 t \cos(t^2) \\ v(t) &= -2 t \sin(t^2) \\ \Lambda(t) &= -\frac{4}{3} t^3 \\ \Gamma(t) &= 0 , \end{aligned} $$

with the Lagrange multipliers $\dot{\Lambda} = -4t^2$ and $\dot{\Gamma} = 0$.

This problem is solved for $atol = rtol = 10^{-(3 + m / 4)}$, where $m = 0, \dots, 24$. The resulting error at $t_1 = 5$ is compared with the elapsed time of the used solvers in the figure below.

Arevalo_work_precision

IDE - Weissinger

A simple example of an implicit differential equations is called Weissinger's equation

$$ t y^2 (\dot{y})^3 - y^3 (\dot{y}^2) + t (t^2 + 1) \dot{y} - t^2 y = 0 . $$

It has the analytical solution $y(t) = \sqrt{t^2 + \frac{1}{2}}$ and $\dot{y}(t) = \frac{t}{\sqrt{t^2 + \frac{1}{2}}}$.

Starting at $t_0 = \sqrt{1 / 2}$, this problem is solved for $atol = rtol = 10^{-(4 + m / 4)}$, where $m = 0, \dots, 28$. The resulting error at $t_1 = 10$ is compared with the elapsed time of the used solvers in the figure below.

Weissinger_work_precision

Nonlinear index 1 DAE - Kvaernø

In a final example, an nonlinear index 1 DAE is investigated as proposed by Kvaernø. The system is given by

$$ \begin{aligned} (\sin^2(\dot{y}_1) + \sin^2(y_2)) (\dot{y}_2)^2 - (t - 6)^2 (t - 2)^2 y_1 e^{-t} &= 0 \\ (4 - t) (y_2 + y_1)^3 - 64 t^2 e^{-t} y_1 y_2 &= 0 , \end{aligned} $$

It has the analytical solution $y_1(t) = t^4 e^{-t}$, $y_2(t) = t^3 e^{-t} (4 - t)$ and $\dot{y}_1(t) = (4 t^3 - t^4) e^{-t}$, $y_2(t) = (-t^3 + (4 - t) 3 t^2 - (4 - t) t^3) e^{-t}$.

Starting at $t_0 = 0.5$, this problem is solved for $atol = rtol = 10^{-(4 + m / 4)}$, where $m = 0, \dots, 32$. The resulting error at $t_1 = 1$ is compared with the elapsed time of the used solvers in the figure below.

Kvaerno_work_precision

Install

An editable developer mode can be installed via

python -m pip install -e .[dev]

The tests can be started using

python -m pytest --cov