def run_example(with_plots=True):
r"""
An example for IDA with scaled preconditioned GMRES method
as a special linear solver.
Note, how the operation Jacobian times vector is provided.
ODE:
.. math::
\dot y_1 - y_2 &= 0\\
\dot y_2 -9.82 &= 0
on return:
- :dfn:`imp_mod` problem instance
- :dfn:`imp_sim` solver instance
"""
#Defines the residual
def res(t, y, yd):
res_0 = yd[0] - y[1]
res_1 = yd[1] + 9.82
return N.array([res_0, res_1])
#Defines the Jacobian*vector product
def jacv(t, y, yd, res, v, c):
jy = N.array([[0, -1.], [0, 0]])
jyd = N.array([[1, 0.], [0, 1]])
j = jy + c * jyd
return N.dot(j, v)
#Initial conditions
y0 = [1.0, 0.0]
yd0 = [0.0, -9.82]
#Defines an Assimulo implicit problem
imp_mod = Implicit_Problem(
res, y0, yd0, name='Example using the Jacobian Vector product')
imp_mod.jacv = jacv #Sets the jacobian
imp_sim = IDA(imp_mod) #Create an IDA solver instance
#Set the parameters
imp_sim.atol = 1e-5 #Default 1e-6
imp_sim.rtol = 1e-5 #Default 1e-6
imp_sim.linear_solver = 'SPGMR' #Change linear solver
#imp_sim.options["usejac"] = False
#Simulate
t, y, yd = imp_sim.simulate(
5, 1000) #Simulate 5 seconds with 1000 communication points
#Basic tests
nose.tools.assert_almost_equal(y[-1][0], -121.75000000, 4)
nose.tools.assert_almost_equal(y[-1][1], -49.100000000)
#Plot
if with_plots:
P.plot(t, y)
P.xlabel('Time')
P.ylabel('State')
P.title(imp_mod.name)
P.show()
return imp_mod, imp_sim
