def run_example(with_plots=True):
r"""
Example to demonstrate the use of a preconditioner
.. math::
\dot y_1 & = 2 t \sin y_1 + t \sin y_2 \\
\dot y_2 & = 3 t \sin y_1 + 2 t \sin y_2
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
#Define the rhs
def rhs(t, y):
A = np.array([[2.0, 1.0], [3.0, 2.0]])
yd = np.dot(A * t, np.sin(y))
return yd
#Define the preconditioner setup function
def prec_setup(t, y, fy, jok, gamma, data):
A = np.array([[2.0, 1.0], [3.0, 2.0]])
#If jok is false the jacobian data needs to be recomputed
if jok == False:
#Extract the diagonal of the jacobian to form a Jacobi preconditioner
a0 = A[0, 0] * t * np.cos(y[0])
a1 = A[1, 1] * t * np.cos(y[1])
a = np.array([(1. - gamma * a0), (1. - gamma * a1)])
#Return true (jacobian data was recomputed) and the new data
return [True, a]
#If jok is true the existing jacobian data can be reused
if jok == True:
#Return false (jacobian data was reused) and the old data
return [False, data]
#Define the preconditioner solve function
def prec_solve(t, y, fy, r, gamma, delta, data):
#Solve the system Pz = r
z0 = r[0] / data[0]
z1 = r[1] / data[1]
z = np.array([z0, z1])
return z
#Initial conditions
y0 = [1.0, 2.0]
#Define an Assimulo problem
exp_mod = Explicit_Problem(
rhs, y0, name="Example of using a preconditioner in SUNDIALS")
#Set the preconditioner setup and solve function for the problem
exp_mod.prec_setup = prec_setup
exp_mod.prec_solve = prec_solve
#Create a CVode solver
exp_sim = CVode(exp_mod)
#Set the parameters for the solver
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.atol = 1e-5
exp_sim.rtol = 1e-5
exp_sim.linear_solver = 'SPGMR'
exp_sim.precond = "PREC_RIGHT" #Set the desired type of preconditioning
#Simulate
t, y = exp_sim.simulate(5)
if with_plots:
exp_sim.plot()
#Basic verification
nose.tools.assert_almost_equal(y[-1, 0], 3.11178295, 4)
nose.tools.assert_almost_equal(y[-1, 1], 3.19318992, 4)
return exp_mod, exp_sim
