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 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 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 test_against_normal_solvers():
""" Test the HHT method on the damper system and compares the
result against a simulation using LSODAR
"""
end = 10.0
ode = flattened_2nd_order(3, 2, 3)
y0 = 3.0
v0 = 2.0
prob = Explicit_Problem(ode, [y0,v0])
exp_sim = LSODAR(prob)
t, y = exp_sim.simulate(end)
ode = mass_spring_damper(3, 2, 3)
prob = Explicit_Problem(ode, y0)
hht = HHT(prob, v0, alpha = -0.2)
t, y_2 = hht.simulate(end)
nose.tools.assert_almost_equal(y[-1][0]/y[-1][0], y_2[-1]/y[-1][0],places = 1)
def test_pend_newmark_against_normal():
""" Test out newmark solver againts the pendulum by comparing the result
of a simulation against one done by assimulo"""
import problems as p
from matplotlib.pylab import plot, show
pend = p.Pendulum2nd()
ode = pend.fcn
end = 15
y0 = pend.initial_condition()
prob = Explicit_Problem(ode, y0[0])
nmark = Newmark(prob, y0[1], beta = .7, gamma = .5)
t, y_2 = nmark.simulate(end)
ode = pend.fcn_1
prob = Explicit_Problem(ode, y0)
sim = LSODAR(prob)
t, y_3 = sim.simulate(end)
nose.tools.assert_almost_equal(y_3[-1][0]/y_3[-1][0], y_2[-1]/y_3[-1][0],places = 1)
def test_pend_hht_against_normal():
""" Test out hht solver againts the pendulum by comparing the result
of a simulation against one done by assimulo"""
import problems as p
from matplotlib.pylab import plot, show, figure
pend = p.Pendulum2nd()
ode = pend.fcn
end = 15
y0 = pend.initial_condition()
prob = Explicit_Problem(ode, y0[0])
hht = HHT(prob, y0[1], alpha=-0.2)
t, y_1 = hht.simulate(end)
ode = pend.fcn_1
prob = Explicit_Problem(ode, y0)
sim = LSODAR(prob)
t, y_3 = sim.simulate(end)
nose.tools.assert_almost_equal(y_3[-1][0]/y_3[-1][0], y_1[-1]/y_3[-1][0],places = 1)
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
class Test_LSODAR:
"""
Tests the LSODAR solver.
"""
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
@testattr(stddist=True)
def test_simulation(self):
"""
This tests the LSODAR with a simulation of the van der pol problem.
"""
self.sim.simulate(1.) #Simulate 2 seconds
nose.tools.assert_almost_equal(self.sim.y_sol[-1][0], -1.863646028, 4)
@testattr(stddist=True)
def test_setcoefficients(self):
elco, tesco = dcfode(1)
nose.tools.assert_almost_equal(elco[0, 2], 5. / 12., 9) # AM-2
nose.tools.assert_almost_equal(tesco[0, 2], 2., 9) # AM-2 error coeff
@testattr(stddist=True)
def test_readcommon(self):
"""
This tests the LSODAR's subroutine dsrcar (read functionality).
"""
self.sim.simulate(1.) #Simulate 2 seconds
r = N.ones((245, ), 'd')
i = N.ones((55, ), 'i')
dsrcar(r, i, 1)
nose.tools.assert_almost_equal(r[217], 2.22044605e-16, 20)
nose.tools.assert_equal(i[36], 3)
@testattr(stddist=True)
def test_writereadcommon(self):
"""
This tests the LSODAR's subroutine dsrcar (write and read functionality).
"""
r = N.ones((245, ), 'd')
i = N.ones((55, ), 'i')
dsrcar(r, i, 2)
r[0] = 100.
i[0] = 10
dsrcar(r, i, 1)
nose.tools.assert_almost_equal(r[0], 1., 4)
nose.tools.assert_equal(i[0], 1)
def test_rkstarter(self):
"""
This test checks the correctness of the Nordsieck array generated
from a RK starter
"""
pass
"""
A=N.array([[0.,1.],[-4.,0.]])
def f(t,x,sw0):
return N.dot(A,N.array(x))
H = 1.e-8
# Compute the exact solution at h=0,H/4,H/2,3H/4,H
T=N.array([0.,H/4.,H/2.,3./4.*H,H])
y0=N.array([1.,0.])
from scipy.linalg import expm
exact=N.array([N.dot(expm(A*t),y0) for t in T])
#polynomial interpolation
from scipy import polyfit
coeff = polyfit(T,exact,4)
d1coeff=N.array([4,3,2,1]).reshape(-1,1)*coeff[:-1,:]
d2coeff=N.array([3,2,1]).reshape(-1,1)*d1coeff[:-1,:]
d3coeff=N.array([2,1]).reshape(-1,1)*d2coeff[:-1,:]
d4coeff=N.array([1]).reshape(-1,1)*d3coeff[:-1,:]
h=H/4.
nordsieck_at_0=N.array([coeff[-1,:],h*d1coeff[-1,:],h**2/2.*d2coeff[-1,:],
h**3/6.*d3coeff[-1,:],h**4/24.*d4coeff[-1,:]])
rkNordsieck=odepack.RKStarterNordsieck(f,H)
computed=rkNordsieck(0,y0)
numpy.testing.assert_allclose(computed[1], nordsieck_at_0, atol=H/100., verbose=True)
"""
@testattr(stddist=True)
def test_interpol(self):
# a with interpolation and report_continuously
self.sim.report_continuously = True
t_sol, y_sol = self.sim.simulate(1., ncp_list=[0.5])
self.sim.reset()
t_sol1, y_sol1 = self.sim.simulate(0.5)
ind05 = N.nonzero(N.array(t_sol) == 0.5)[0][0]
#print y_sol[ind05],y_sol1[-1]
nose.tools.assert_almost_equal(y_sol[ind05, 0], y_sol1[-1, 0], 6)
def test_simulation_with_jac(self):
"""
This tests the LSODAR with a simulation of the van der pol problem.
"""
self.sim.usejac = True
self.sim.simulate(1.) #Simulate 2 seconds
nose.tools.assert_almost_equal(self.sim.y_sol[-1][0], -1.863646028, 4)
@testattr(stddist=True)
def test_simulation_ncp(self):
self.sim.simulate(1., 100) #Simulate 2 seconds
nose.tools.assert_almost_equal(self.sim.y_sol[-1][0], -1.863646028, 4)
@testattr(stddist=True)
def test_usejac_csc_matrix(self):
self.sim_sp.usejac = True
self.sim_sp.simulate(2.) #Simulate 2 seconds
assert self.sim_sp.statistics["nfcnjacs"] == 0
nose.tools.assert_almost_equal(self.sim_sp.y_sol[-1][0], 1.7061680350,
4)
@testattr(stddist=True)
def test_simulation_ncp_list(self):
self.sim.simulate(1., ncp_list=[0.5]) #Simulate 2 seconds
nose.tools.assert_almost_equal(self.sim.y_sol[-1][0], -1.863646028, 4)
@testattr(stddist=True)
def test_maxh(self):
self.sim.hmax = 1.0
assert self.sim.options["maxh"] == 1.0
assert self.sim.maxh == 1.0
self.sim.maxh = 2.0
assert self.sim.options["maxh"] == 2.0
assert self.sim.maxh == 2.0
@testattr(stddist=True)
def test_simulation_ncp_list_2(self):
self.sim.simulate(1., ncp_list=[0.5, 4]) #Simulate 2 seconds
nose.tools.assert_almost_equal(self.sim.y_sol[-1][0], -1.863646028, 4)
@testattr(stddist=True)
def test_simulation_ncp_with_jac(self):
"""
Test a simulation with ncp.
"""
self.sim.usejac = True
self.sim.simulate(1., 100) #Simulate 2 seconds
nose.tools.assert_almost_equal(self.sim.y_sol[-1][0], -1.863646028, 4)
def _default_solver_instance(self):
from assimulo.solvers import LSODAR
return LSODAR(self._assimulo_problem)