def setUp(self):
"""
This sets up the test case.
"""
#Define the residual
def f(t, y, yd):
eps = 1.e-6
my = 1. / eps
yd_0 = y[1]
yd_1 = my * ((1. - y[0]**2) * y[1] - y[0])
res_0 = yd[0] - yd_0
res_1 = yd[1] - yd_1
return N.array([res_0, res_1])
y0 = [2.0, -0.6] #Initial conditions
yd0 = [-.6, -200000.]
#Define an Assimulo problem
self.mod = Implicit_Problem(f, y0, yd0)
self.mod_t0 = Implicit_Problem(f, y0, yd0, 1.0)
#Define an explicit solver
self.sim = GLIMDA(self.mod) #Create a Radau5 solve
self.sim_t0 = GLIMDA(self.mod_t0)
#Sets the parameters
self.sim.atol = 1e-4 #Default 1e-6
self.sim.rtol = 1e-4 #Default 1e-6
self.sim.inith = 1.e-4 #Initial step-size
def test_simulate_explicit(self):
"""
Test a simulation of an explicit problem using GLIMDA.
"""
f = lambda t, y: N.array(-y)
y0 = [1.0]
problem = Explicit_Problem(f, y0)
simulator = GLIMDA(problem)
assert simulator.yd0[0] == -simulator.y0[0]
t, y = simulator.simulate(1.0)
nose.tools.assert_almost_equal(float(y[-1]), float(N.exp(-1.0)), 4)
def run_example(with_plots=True):
r"""
Example for the use of GLIMDA (general linear multistep 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= 10^6`.
on return:
- :dfn:`imp_mod` problem instance
- :dfn:`imp_sim` solver instance
"""
#Define the residual
def f(t, y, yd):
eps = 1.e-6
my = 1. / eps
yd_0 = y[1]
yd_1 = my * ((1. - y[0]**2) * y[1] - y[0])
res_0 = yd[0] - yd_0
res_1 = yd[1] - yd_1
return N.array([res_0, res_1])
y0 = [2.0, -0.6] #Initial conditions
yd0 = [-.6, -200000.]
#Define an Assimulo problem
imp_mod = Implicit_Problem(f,
y0,
yd0,
name='Glimbda Example: Van der Pol (implicit)')
#Define an explicit solver
imp_sim = GLIMDA(imp_mod) #Create a GLIMDA solver
#Sets the parameters
imp_sim.atol = 1e-4 #Default 1e-6
imp_sim.rtol = 1e-4 #Default 1e-6
imp_sim.inith = 1.e-4 #Initial step-size
#Simulate
t, y, yd = imp_sim.simulate(2.) #Simulate 2 seconds
#Plot
if with_plots:
P.subplot(211)
P.plot(t, y[:, 0]) #, marker='o')
P.xlabel('Time')
P.ylabel('State')
P.subplot(212)
P.plot(t, yd[:, 0] * 1.e-5) #, marker='o')
P.xlabel('Time')
P.ylabel('State derivatives scaled with $10^{-5}$')
P.suptitle(imp_mod.name)
P.show()
#Basic test
x1 = y[:, 0]
assert N.abs(x1[-1] - 1.706168035) < 1e-3 #For test purpose
return imp_mod, imp_sim