4.3.1. ivp

This module defines the interface for solving initial-value problems for ordinary differential equations:

\[\frac{dy}{dt} = f(t, y), \quad y(t_0) = y_0.\]

4.3.1.1. Module Contents

type ivp.RhsFn = Callable[[float, np.ndarray, np.ndarray, object], int]

Signature of the right-hand side (RHS) function \(f(t, y)\).

The function accepts four arguments:

• t: current time,

• y: state vector at time \(t\),

• ydot: output array to which the result of function evalutation is stored,

• user_data: additional context (user-defined data) that must be passed to the function (e.g., parameters of the system).

class ivp.IVP(impl: str)

Interface for solving initial value problems.

This class serves as a gateway to the implementations of the solvers for initial-value problems for ordinary differential equations.

Parameters:

impl (str) – Name of the desired implementation.

Examples

Let’s solve the following initial value problem:

\[y'(t) = -y(t), \quad y(0) = 1.\]

First, import the necessary modules: >>> import numpy as np >>> from oif.interfaces.ivp import IVP

Define the right-hand side function:

>>> def rhs(t, y, ydot, user_data):
...     ydot[0] = -y[0]
...     return 0  # No errors, optional

Now define the initial condition:

>>> y0, t0 = np.array([1.0]), 0.0

Create an instance of the IVP solver using the implementation “jl_diffeq”, which is an adapter to the OrdinaryDiffeq.jl Julia package:

>>> s = IVP("jl_diffeq")

We set the initial value, the right-hand side function, and the tolerance:

>>> s.set_initial_value(y0, t0)
>>> s.set_rhs_fn(rhs)
>>> s.set_tolerances(1e-6, 1e-12)

Now we integrate to time t = 1.0 in a loop, outputting the current value of y with time step 0.1:

>>> t = t0
>>> times = np.linspace(t0, t0 + 1.0, num=11)
>>> for t in times[1:]:
...     s.integrate(t)
...     print(f"{t:.1f} {s.y[0]:.6f}")
0.1 0.904837
0.2 0.818731
0.3 0.740818
0.4 0.670320
0.5 0.606531
0.6 0.548812
0.7 0.496585
0.8 0.449329
0.9 0.406570
1.0 0.367879

y: numpy.ndarray

Current value of the state vector.

set_initial_value(y0: numpy.ndarray, t0: float)

Set initial value y(t0) = y0.

set_rhs_fn(rhs_fn: RhsFn)

Specify right-hand side function f.

set_tolerances(rtol: float, atol: float)

Specify relative and absolute tolerances, respectively.

set_user_data(user_data: object)

Specify additional data that will be used for right-hand side function.

set_integrator(integrator_name: str, integrator_params: dict = {})

Set integrator, if the name is recognizable.

integrate(t: float)

Integrate to time t and write solution to y.

print_stats()

Print integration statistics.