Source code for assimulo.examples.euler_vanderpol
#!/usr/bin/env python
# -*- coding: utf-8 -*-
# Copyright (C) 2010 Modelon AB
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, version 3 of the License.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
import numpy as N
import pylab as P
import nose
from assimulo.solvers import ImplicitEuler
from assimulo.problem import Explicit_Problem
[docs]def run_example(with_plots=True):
r"""
Example for the use of the implicit Euler method to solve
Van der Pol's equation
.. math::
\dot y_1 &= y_2 \\
\dot y_2 &= \mu ((1.-y_1^2) y_2-y_1)
with :math:`\mu=\frac{1}{5} 10^3`.
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
eps = 5.e-3
my = 1./eps
#Define the rhs
def f(t,y):
yd_0 = y[1]
yd_1 = my*((1.-y[0]**2)*y[1]-y[0])
return N.array([yd_0,yd_1])
#Define the Jacobian
def jac(t,y):
jd_00 = 0.0
jd_01 = 1.0
jd_10 = -1.0*my-2*y[0]*y[1]*my
jd_11 = my*(1.-y[0]**2)
return N.array([[jd_00,jd_01],[jd_10,jd_11]])
y0 = [2.0,-0.6] #Initial conditions
#Define an Assimulo problem
exp_mod = Explicit_Problem(f,y0,
name = "ImplicitEuler: Van der Pol's equation (as explicit problem) ")
exp_mod.jac = jac
#Define an explicit solver
exp_sim = ImplicitEuler(exp_mod) #Create a ImplicitEuler solver
#Sets the parameters
exp_sim.h = 1e-4 #Stepsize
exp_sim.usejac = True #If the user defined jacobian should be used or not
#Simulate
t, y = exp_sim.simulate(2.0) #Simulate 2 seconds
#Plot
if with_plots:
P.plot(t,y[:,0], marker='o')
P.title(exp_mod.name)
P.ylabel("State: $y_1$")
P.xlabel("Time")
P.show()
#Basic test
x1 = y[:,0]
assert N.abs(x1[-1]-1.8601438) < 1e-1 #For test purpose
return exp_mod, exp_sim
if __name__=='__main__':
mod,sim = run_example()