ODE solvers

Consider the following problem:

\[\begin{split}\frac{d x(t)}{dt} &=& rhs(t, x(t)) \\ x(t_0) &=& x_0\end{split}\]

with x and rhs some vector fields defined on a 1,2 or 3D domain.

The following time integrators are implemented in hysop:

  • forward Euler, Euler,

  • forward Runge-Kutta of order 2, 3 and 4 : RK2, RK3 and RK4

See runge_kutta for details, and ExplicitRungeKutta to build any Runge-Kutta scheme.

Usage

>>> import numpy as np
>>> from hysop.numerics.odesolvers.runge_kutta import Euler
>>> def f(out, X, t, step, steps):
...     for k in X.keys():
...         out[k][...] = X[k]
...
>>> y = np.linspace(0,1,11)
>>> v = {'y': y,}
>>> Euler(v, f, 0.1)
{'y': array([0.  , 0.11, 0.22, 0.33, 0.44, 0.55, 0.66, 0.77, 0.88, 0.99, 1.1 ])}

To build a time integrator, it is necessary to define a python function to compute the right-hand side. This function must be of the following form:

def some_func(out, X, t, step, steps, **kwds):
    out[name][...] = expr(X[name])

  • out and X are dictionaries with variables names as keys values are np.arrays contiaining datas

  • name is the name of the variable

  • t is the time

  • X must not be modified by the function

  • step equals the current step in RK algorithms with steps stages

  • Extra paramters can be forwarded throw extra keywords arguments