def run_example(with_plots=True):
y0 = [2.0,-0.6] #Initial conditions
yd0 = [-.6,-200000.]
#Define an extended Assimulo problem
imp_mod = VanDerPolProblem(y0=y0,yd0=yd0)
#Define an explicit solver
imp_sim = Radau5DAE(imp_mod) #Create a Radau5 solver
#Simulate
t, y, yd = imp_sim.simulate(8.) #Simulate 8 seconds
#Plot
if with_plots:
import pylab as P
P.plot(t,y[:,0], marker='o')
P.xlabel('Time')
P.ylabel('State')
P.title(imp_mod.name)
P.show()
#Basic test
x1 = y[:,0]
assert N.abs(x1[-1]-1.14330840983) < 1e-3 #For test purpose
return imp_mod, imp_sim
def integrate(system, t0, tf):
# set up DAE
system.rhs = dae_rhs(system)
system.initial_conc_td = initial_derivative(system)
prob = ACAIE_Implicit_Problem(
system=system,
y0=system.initial_conc,
yd0=system.initial_conc_td,
t0=t0)
# set up solver
sol = Radau5DAE(prob)
sol.atol, sol.rtol = 1e-7, 1e-5
return sol.simulate(tfinal=tf)
# + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * u_perp res = np.zeros(6 * ct.nz) res[0:ct.nz] = np.real(dux + 1j * ct.omega * b_par + nabla_perp_u_perp) res[ct.nz:2*ct.nz] = np.imag(dux + 1j * ct.omega * b_par + nabla_perp_u_perp) res[2*ct.nz:3*ct.nz] = np.real(db_par + 1j / ct.omega * L_ux) res[3*ct.nz:4*ct.nz] = np.imag(db_par + 1j / ct.omega * L_ux) res[4*ct.nz:5*ct.nz] = np.real(L_u_perp - 1j * ct.omega * nabla_perp_b_par) res[5*ct.nz:] = np.imag(L_u_perp - 1j * ct.omega * nabla_perp_b_par) return res model = Implicit_Problem(residual, ct.U_x_min, ct.dU_x_min, ct.x_min) model.name = 'Resonant abosprotion' sim = Radau5DAE(model) sim.rtol = 1e-2 sim.atol = 1e-2 sim.verbosity = 10 x, U, dU = sim.simulate(ct.x_max, ct.nx-1) x = np.array(x) x0 = ct.x[ct.nx//2] z0 = ct.z[0] ux_r = U[:,0:ct.nz].T ux_i = U[:,ct.nz:2*ct.nz].T b_par_r = U[:,2*ct.nz:3*ct.nz].T b_par_i = U[:,3*ct.nz:4*ct.nz].T u_perp_r = U[:,4*ct.nz:5*ct.nz].T
def run_example(with_plots=True):
r"""
Example for the use of Radau5DAE 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)
imp_mod.name = 'Van der Pol (implicit)'
#Define an explicit solver
imp_sim = Radau5DAE(imp_mod) #Create a Radau5 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