def run_example(with_plots=True):
r"""
This is the same example from the Sundials package (cvsRoberts_FSA_dns.c)
Its purpose is to demonstrate the use of parameters in the differential equation.
This simple example problem for CVode, due to Robertson
see http://www.dm.uniba.it/~testset/problems/rober.php,
is from chemical kinetics, and consists of the system:
.. math::
\dot y_1 &= -p_1 y_1 + p_2 y_2 y_3 \\
\dot y_2 &= p_1 y_1 - p_2 y_2 y_3 - p_3 y_2^2 \\
\dot y_3 &= p_3 y_ 2^2
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
def f(t, y, p):
yd_0 = -p[0]*y[0]+p[1]*y[1]*y[2]
yd_1 = p[0]*y[0]-p[1]*y[1]*y[2]-p[2]*y[1]**2
yd_2 = p[2]*y[1]**2
return N.array([yd_0,yd_1,yd_2])
def jac(t,y, p):
J = N.array([[-p[0], p[1]*y[2], p[1]*y[1]],
[p[0], -p[1]*y[2]-2*p[2]*y[1], -p[1]*y[1]],
[0.0, 2*p[2]*y[1],0.0]])
return J
def fsens(t, y, s, p):
J = N.array([[-p[0], p[1]*y[2], p[1]*y[1]],
[p[0], -p[1]*y[2]-2*p[2]*y[1], -p[1]*y[1]],
[0.0, 2*p[2]*y[1],0.0]])
P = N.array([[-y[0],y[1]*y[2],0],
[y[0], -y[1]*y[2], -y[1]**2],
[0,0,y[1]**2]])
return J.dot(s)+P
#The initial conditions
y0 = [1.0,0.0,0.0] #Initial conditions for y
#Create an Assimulo explicit problem
exp_mod = Explicit_Problem(f,y0, name='Robertson Chemical Kinetics Example')
exp_mod.rhs_sens = fsens
exp_mod.jac = jac
#Sets the options to the problem
exp_mod.p0 = [0.040, 1.0e4, 3.0e7] #Initial conditions for parameters
exp_mod.pbar = [0.040, 1.0e4, 3.0e7]
#Create an Assimulo explicit solver (CVode)
exp_sim = CVode(exp_mod)
#Sets the solver paramters
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.rtol = 1.e-4
exp_sim.atol = N.array([1.0e-8, 1.0e-14, 1.0e-6])
exp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
exp_sim.suppress_sens = False #Dont suppress the sensitivity variables in the error test.
exp_sim.report_continuously = True
#Simulate
t, y = exp_sim.simulate(4,400) #Simulate 4 seconds with 400 communication points
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 9.05518032e-01, 4)
nose.tools.assert_almost_equal(y[-1][1], 2.24046805e-05, 4)
nose.tools.assert_almost_equal(y[-1][2], 9.44595637e-02, 4)
nose.tools.assert_almost_equal(exp_sim.p_sol[0][-1][0], -1.8761, 2) #Values taken from the example in Sundials
nose.tools.assert_almost_equal(exp_sim.p_sol[1][-1][0], 2.9614e-06, 8)
nose.tools.assert_almost_equal(exp_sim.p_sol[2][-1][0], -4.9334e-10, 12)
#Plot
if with_plots:
P.plot(t, y)
P.title(exp_mod.name)
P.xlabel('Time')
P.ylabel('State')
P.show()
return exp_mod, exp_sim
