def setUp(self):
"""
This sets up the test case.
"""
def f(t, y):
eps = 1.e-6
my = 1. / eps
yd_0 = y[1]
yd_1 = my * ((1. - y[0]**2) * y[1] - y[0])
return N.array([yd_0, yd_1])
def jac(t, y):
eps = 1.e-6
my = 1. / eps
J = N.zeros([2, 2])
J[0, 0] = 0.
J[0, 1] = 1.
J[1, 0] = my * (-2. * y[0] * y[1] - 1.)
J[1, 1] = my * (1. - y[0]**2)
return J
def jac_sparse(t, y):
eps = 1.e-6
my = 1. / eps
J = N.zeros([2, 2])
J[0, 0] = 0.
J[0, 1] = 1.
J[1, 0] = my * (-2. * y[0] * y[1] - 1.)
J[1, 1] = my * (1. - y[0]**2)
return sp.csc_matrix(J)
#Define an Assimulo problem
y0 = [2.0, -0.6] #Initial conditions
exp_mod = Explicit_Problem(f, y0)
exp_mod_t0 = Explicit_Problem(f, y0, 1.0)
exp_mod_sp = Explicit_Problem(f, y0)
exp_mod.jac = jac
exp_mod_sp.jac = jac_sparse
self.mod = exp_mod
#Define an explicit solver
self.sim = LSODAR(exp_mod) #Create a LSODAR solve
self.sim_sp = LSODAR(exp_mod_sp)
#Sets the parameters
self.sim.atol = 1e-6 #Default 1e-6
self.sim.rtol = 1e-6 #Default 1e-6
self.sim.usejac = False
def run_example(with_plots=True):
r"""
Example for the use of LSODAR 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
"""
#Define the rhs
def f(t,y):
eps = 1.e-6
my = 1./eps
yd_0 = y[1]
yd_1 = my*((1.-y[0]**2)*y[1]-y[0])
return N.array([yd_0,yd_1])
y0 = [2.0,-0.6] #Initial conditions
#Define an Assimulo problem
exp_mod = Explicit_Problem(f,y0, name = "LSODAR: Van der Pol's equation")
#Define an explicit solver
exp_sim = LSODAR(exp_mod) #Create a Radau5 solver
#Sets the parameters
exp_sim.atol = 1e-4 #Default 1e-6
exp_sim.rtol = 1e-4 #Default 1e-6
#Simulate
t, y = exp_sim.simulate(2.) #Simulate 2 seconds
#Plot
if with_plots:
import pylab as P
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]-1.706168035) < 1e-3 #For test purpose
return exp_mod, exp_sim
def test_event_localizer(self):
exp_mod = Extended_Problem() #Create the problem
exp_sim = LSODAR(exp_mod) #Create the solver
exp_sim.verbosity = 0
exp_sim.report_continuously = True
#Simulate
t, y = exp_sim.simulate(
10.0, 1000) #Simulate 10 seconds with 1000 communications points
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 8.0)
nose.tools.assert_almost_equal(y[-1][1], 3.0)
nose.tools.assert_almost_equal(y[-1][2], 2.0)
def run_example(with_plots=True):
r"""
Example of the use of Euler's method for a differential equation
with a discontinuity (state event) and the need for an event iteration.
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
#Create an instance of the problem
exp_mod = Extended_Problem() #Create the problem
exp_sim = LSODAR(exp_mod) #Create the solver
exp_sim.verbosity = 0
exp_sim.continuous_output = True
#Simulate
t, y = exp_sim.simulate(
10.0, 100) #Simulate 10 seconds with 1000 communications points
#Plot
if with_plots:
import pylab as P
P.plot(t, y)
P.title("Solution of a differential equation with discontinuities")
P.ylabel('States')
P.xlabel('Time')
P.show()
return exp_mod, exp_sim
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 8.0)
nose.tools.assert_almost_equal(y[-1][1], 3.0)
nose.tools.assert_almost_equal(y[-1][2], 2.0)
def run_example(with_plots=True):
"""
Bouncing ball example to demonstrate LSODAR's
discontinuity handling.
Also a way to use :program:`problem.initialize` and :program:`problem.handle_result`
in order to provide extra information is demonstrated.
The governing differential equation is
.. math::
\\dot y_1 &= y_2\\\\
\\dot y_2 &= -9.81
and the switching functions are
.. math::
\\mathrm{event}_0 &= y_1 \\;\\;\\;\\text{ if } \\;\\;\\;\\mathrm{sw}_0 = 1\\\\
\\mathrm{event}_1 &= y_2 \\;\\;\\;\\text{ if }\\;\\;\\; \\mathrm{sw}_1 = 1
otherwise the events are deactivated by setting the respective value to something different from 0.
The event handling sets
:math:`y_1 = - 0.88 y_1` and :math:`\\mathrm{sw}_1 = 1` if the first event triggers
and :math:`\\mathrm{sw}_1 = 0` if the second event triggers.
"""
#Create an instance of the problem
exp_mod = Extended_Problem() #Create the problem
exp_sim = LSODAR(exp_mod) #Create the solver
exp_sim.atol = 1.e-8
exp_sim.report_continuously = True
exp_sim.verbosity = 30
#Simulate
t, y = exp_sim.simulate(10.0) #Simulate 10 seconds
#Plot
if with_plots:
P.subplot(221)
P.plot(t, y)
P.title('LSODAR Bouncing ball (one step mode)')
P.ylabel('States: $y$ and $\dot y$')
P.subplot(223)
P.plot(exp_sim.t_sol, exp_sim.h_sol)
P.title('LSODAR step size plot')
P.xlabel('Time')
P.ylabel('Sepsize')
P.subplot(224)
P.plot(exp_sim.t_sol, exp_sim.nq_sol)
P.title('LSODAR order plot')
P.xlabel('Time')
P.ylabel('Order')
P.suptitle(exp_mod.name)
P.show()
return exp_mod, exp_sim
def _default_solver_instance(self):
from assimulo.solvers import LSODAR
return LSODAR(self._assimulo_problem)