def plot_empirical_freq_and_amplitude(u, t, I, w):
minima, maxima = minmax(t, u)
p = periods(maxima)
a = amplitudes(minima, maxima)
figure()
plot(range(len(p)), 2*pi/p, 'r-')
hold('on')
plot(range(len(a)), a, 'b-')
plot(range(len(p)), [w]*len(p), 'r--')
plot(range(len(a)), [I]*len(a), 'b--')
legend(['numerical frequency', 'numerical amplitude',
'analytical frequency', 'anaytical amplitude'],
loc='center right')
def plot_empirical_freq_and_amplitude(u, t, I, w):
minima, maxima = minmax(t, u)
p = periods(maxima)
a = amplitudes(minima, maxima)
figure()
plot(range(len(p)), 2*pi/p, 'r-')
hold('on')
plot(range(len(a)), a, 'b-')
plot(range(len(p)), [w]*len(p), 'r--')
plot(range(len(a)), [I]*len(a), 'b--')
legend(['numerical frequency', 'numerical amplitude',
'analytical frequency', 'anaytical amplitude'],
loc='center right')
def plot_empirical_freq_and_amplitude(u, t, I, w):
"""
Find the empirical angular frequency and amplitude of
simulations in u and t. u and t can be arrays or (in
the case of multiple simulations) multiple arrays.
One plot is made for the amplitude and one for the angular
frequency (just called frequency in the legends).
"""
from vib_empirical_analysis import minmax, periods, amplitudes
from math import pi
if not isinstance(u, (list, tuple)):
u = [u]
t = [t]
legends1 = []
legends2 = []
for i in range(len(u)):
minima, maxima = minmax(t[i], u[i])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(1)
plt.plot(range(len(p)), 2 * pi / p)
legends1.append("frequency, case%d" % (i + 1))
plt.hold("on")
plt.figure(2)
plt.plot(range(len(a)), a)
plt.hold("on")
legends2.append("amplitude, case%d" % (i + 1))
plt.figure(1)
plt.plot(range(len(p)), [w] * len(p), "k--")
legends1.append("exact frequency")
plt.legend(legends1, loc="lower left")
plt.axis([0, len(a) - 1, 0.8 * w, 1.2 * w])
plt.savefig("tmp1.png")
plt.savefig("tmp1.pdf")
plt.figure(2)
plt.plot(range(len(a)), [I] * len(a), "k--")
legends2.append("exact amplitude")
plt.legend(legends2, loc="lower left")
plt.axis([0, len(a) - 1, 0.8 * I, 1.2 * I])
plt.savefig("tmp2.png")
plt.savefig("tmp2.pdf")
plt.show()
def plot_empirical_freq_and_amplitude(u, t, I, w):
"""
Find the empirical angular frequency and amplitude of
simulations in u and t. u and t can be arrays or (in
the case of multiple simulations) multiple arrays.
One plot is made for the amplitude and one for the angular
frequency (just called frequency in the legends).
"""
from vib_empirical_analysis import minmax, periods, amplitudes
from math import pi
if not isinstance(u, (list, tuple)):
u = [u]
t = [t]
legends1 = []
legends2 = []
for i in range(len(u)):
minima, maxima = minmax(t[i], u[i])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(1)
plt.plot(range(len(p)), 2 * pi / p)
legends1.append('frequency, case%d' % (i + 1))
plt.hold('on')
plt.figure(2)
plt.plot(range(len(a)), a)
plt.hold('on')
legends2.append('amplitude, case%d' % (i + 1))
plt.figure(1)
plt.plot(range(len(p)), [w] * len(p), 'k--')
legends1.append('exact frequency')
plt.legend(legends1, loc='lower left')
plt.axis([0, len(a) - 1, 0.8 * w, 1.2 * w])
plt.savefig('tmp1.png')
plt.savefig('tmp1.pdf')
plt.figure(2)
plt.plot(range(len(a)), [I] * len(a), 'k--')
legends2.append('exact amplitude')
plt.legend(legends2, loc='lower left')
plt.axis([0, len(a) - 1, 0.8 * I, 1.2 * I])
plt.savefig('tmp2.png')
plt.savefig('tmp2.pdf')
plt.show()
def run_solvers_and_plot(solvers, timesteps_per_period=20,
num_periods=1, I=1, w=2*np.pi):
P = 2*np.pi/w # duration of one period
dt = P/timesteps_per_period
Nt = num_periods*timesteps_per_period
T = Nt*dt
t_mesh = np.linspace(0, T, Nt+1)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# Compute energy
dt = t[1] - t[0]
E = 0.5*((u[2:,0] - u[:-2,0])/(2*dt))**2 + 0.5*w**2*u[1:-1,0]**2
# Compute error in energy
E0 = 0.5*0**2 + 0.5*w**2*I**2
e_E = E - E0
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
print '*** Relative max error in energy for %s [0,%g] with dt=%g: %.3E' % (solver_name, t[-1], dt, np.abs(e_E).max()/E0)
# Make plots
if num_periods <= 80:
plt.figure(1)
if len(t_mesh) <= 50:
plt.plot(t, u[:,0]) # markers by default
else:
plt.plot(t, u[:,0], '-2') # no markers
plt.hold('on')
legends.append(solver.__class__.__name__)
plt.figure(2)
if len(t_mesh) <= 50:
plt.plot(u[:,0], u[:,1]) # markers by default
else:
plt.plot(u[:,0], u[:,1], '-2') # no markers
plt.hold('on')
if num_periods > 20:
minima, maxima = minmax(t, u[:,0])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(3)
plt.plot(range(len(p)), 2*np.pi/p, '-')
plt.hold('on')
plt.figure(4)
plt.plot(range(len(a)), a, '-')
plt.hold('on')
# Compare with exact solution plotted on a very fine mesh
t_fine = np.linspace(0, T, 10001)
u_e = I*np.cos(w*t_fine)
v_e = -w*I*np.sin(w*t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_fine, u_e, '-') # avoid markers by spec. line type
legends.append('exact')
plt.legend(legends, loc='upper left')
plt.xlabel('t'); plt.ylabel('u')
plt.title('Time step: %g' % dt)
plt.savefig('vib_%d_%d_u.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_u.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_u.eps' % (timesteps_per_period, num_periods))
plt.figure(2)
plt.plot(u_e, v_e, '-') # avoid markers by spec. line type
plt.legend(legends, loc='lower right')
plt.xlabel('u(t)'); plt.ylabel('v(t)')
plt.title('Time step: %g' % dt)
plt.savefig('vib_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_pp.eps' % (timesteps_per_period, num_periods))
del legends[-1] # fig 3 and 4 does not have exact value
if num_periods > 20:
plt.figure(3)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated periods')
plt.savefig('vib_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.figure(4)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated amplitudes')
plt.savefig('vib_%d_%d_a.eps' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_a.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_a.eps' % (timesteps_per_period, num_periods))
def run_solvers_and_plot(solvers,
timesteps_per_period=20,
num_periods=1,
I=1,
w=2 * np.pi):
P = 2 * np.pi / w # duration of one period
dt = P / timesteps_per_period
Nt = num_periods * timesteps_per_period
T = Nt * dt
t_mesh = np.linspace(0, T, Nt + 1)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# Compute energy
dt = t[1] - t[0]
E = 0.5 * ((u[2:, 0] - u[:-2, 0]) /
(2 * dt))**2 + 0.5 * w**2 * u[1:-1, 0]**2
# Compute error in energy
E0 = 0.5 * 0**2 + 0.5 * w**2 * I**2
e_E = E - E0
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
print '*** Relative max error in energy for %s [0,%g] with dt=%g: %.3E' % (
solver_name, t[-1], dt, np.abs(e_E).max() / E0)
# Make plots
if num_periods <= 80:
plt.figure(1)
if len(t_mesh) <= 50:
plt.plot(t, u[:, 0]) # markers by default
else:
plt.plot(t, u[:, 0], '-2') # no markers
plt.hold('on')
legends.append(solver_name)
plt.figure(2)
if len(t_mesh) <= 50:
plt.plot(u[:, 0], u[:, 1]) # markers by default
else:
plt.plot(u[:, 0], u[:, 1], '-2') # no markers
plt.hold('on')
if num_periods > 20:
minima, maxima = minmax(t, u[:, 0])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(3)
plt.plot(range(len(p)), 2 * np.pi / p, '-')
plt.hold('on')
plt.figure(4)
plt.plot(range(len(a)), a, '-')
plt.hold('on')
# Compare with exact solution plotted on a very fine mesh
t_fine = np.linspace(0, T, 10001)
u_e = I * np.cos(w * t_fine)
v_e = -w * I * np.sin(w * t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_fine, u_e, '-') # avoid markers by spec. line type
legends.append('exact')
plt.legend(legends, loc='upper left')
plt.xlabel('t')
plt.ylabel('u')
plt.title('Time step: %g' % dt)
plt.savefig('vib_%d_%d_u.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_u.pdf' % (timesteps_per_period, num_periods))
plt.figure(2)
plt.plot(u_e, v_e, '-') # avoid markers by spec. line type
plt.legend(legends, loc='lower right')
plt.xlabel('u(t)')
plt.ylabel('v(t)')
plt.title('Time step: %g' % dt)
plt.savefig('vib_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
del legends[-1] # fig 3 and 4 does not have exact value
if num_periods > 20:
plt.figure(3)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated periods')
plt.savefig('vib_%d_%d_p.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.figure(4)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated amplitudes')
plt.savefig('vib_%d_%d_a.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_a.png' % (timesteps_per_period, num_periods))
