def demo():
"""
Demonstrate difference between Euler-Cromer and the
scheme for the corresponding 2nd-order ODE.
"""
I = 1.2
V = 0.2
m = 4
b = 0.2
s = lambda u: 2 * u
F = lambda t: 0
w = np.sqrt(2. / 4) # approx freq
dt = 0.9 * 2 / w # longest possible time step
w = 0.5
P = 2 * pi / w
T = 4 * P
from vib import solver as solver2
import scitools.std as plt
for k in range(4):
u2, t2 = solver2(I, V, m, b, s, F, dt, T, 'quadratic')
u, v, t = solver(I, V, m, b, s, F, dt, T, 'quadratic')
plt.figure()
plt.plot(t, u, 'r-', t2, u2, 'b-')
plt.legend(['Euler-Cromer', 'centered scheme'])
plt.title('dt=%.3g' % dt)
raw_input()
plt.savefig('tmp_%d' % k + '.png')
plt.savefig('tmp_%d' % k + '.pdf')
dt /= 2
def _test5():
problem = Problem3()
import argparse
parser = argparse.ArgumentParser(
description='Logistic ODE model')
parser = problem.define_command_line_arguments(parser)
# Try first with manual settings
problem.set(alpha=0.1, U0=2, T=130,
R=lambda t: 500 if t < 60 else 100)
solver = AutoSolver(problem, tol=1)
solver.solve(method=ODESolver.RungeKutta4)
print 'dt:', solver.dt
solver.plot()
# New example with data from the command line
# Try --alpha 0.11 --T 130 --U0 2 --R '500 if t < 60 else 300'
args = parser.parse_args()
problem.set(args=args)
print problem.alpha, problem.R, problem.U0, problem.T
solver = AutoSolver(problem, tol=1)
solver.solve(method=ODESolver.RungeKutta4)
print 'dt:', solver.dt
figure()
solver.plot()
def demo():
"""
Demonstrate difference between Euler-Cromer and the
scheme for the corresponding 2nd-order ODE.
"""
I = 1.2
w = 2.0
T = 5
dt = 2 / w # longest possible time step
from vib_undamped import solver as solver2 # 2nd-order ODE
import scitools.std as plt
for k in range(4):
dt /= 4
u2, t2 = solver2(I, w, dt, T)
u, v, t = solver(I, w, dt, T)
plt.figure()
plt.plot(t,
u,
t2,
u2,
legend=('Euler-Cromer', 'center scheme for $u'
'+u=0$'),
title='dt=%.3g' % dt)
raw_input()
plt.savefig('ECvs2nd_%d' % k + '.png')
plt.savefig('ECvs2nd_%d' % k + '.pdf')
def slides_waves():
L = 10.
a = 0.5
B0 = 1.
A = B0/5
t0 = 3.
v = 1.
x0 = lambda t: L/4 + v*t if t < t0 else L/4 + v*t0
sigma = 1.0
def bottom(x, t):
return (B(x, a, L, B0) + S(x, A, x0(t), sigma))
def depth(x, t):
return -bottom(x, t)
import scitools.std as plt
x = np.linspace(0, L, 101)
plt.plot(x, bottom(x, 0), x, depth(x, 0), legend=['bottom', 'depth'])
plt.show()
raw_input()
plt.figure()
dt_b = 0.01
solver(I=0, V=0,
f=lambda x, t: -(depth(x, t-dt_b) - 2*depth(x, t) + depth(x, t+dt_b))/dt_b**2,
c=lambda x, t: np.sqrt(np.abs(depth(x, t))),
U_0=None,
U_L=None,
L=L,
dt=0.1,
C=0.9,
T=20,
user_action=PlotSurfaceAndBottom(B, S, a, L, B0, A, x0, sigma, umin=-2, umax=2),
stability_safety_factor=0.9)
def demo5():
problem = Problem3()
import argparse
parser = argparse.ArgumentParser(
description='Logistic ODE model')
parser = problem.define_command_line_arguments(parser)
# Try first with manual settings
problem.set(alpha=0.1, U0=2, T=130,
R=lambda t: 500 if t < 60 else 100)
solver = AutoSolver(problem, tol=1)
solver.solve(method=ODESolver.RungeKutta4)
print 'dt:', solver.dt
solver.plot()
# New example with data from the command line
# Try --alpha 0.11 --T 130 --U0 2 --R '500 if t < 60 else 300'
args = parser.parse_args()
problem.set(args=args)
print problem.alpha, problem.R, problem.U0, problem.T
solver = AutoSolver(problem, tol=1)
solver.solve(method=ODESolver.RungeKutta4)
print 'dt:', solver.dt
figure()
solver.plot()
def comparison_plot(f, u, Omega, plotfile='tmp'):
"""Compare f(x,y) and u(x,y) for x,y in Omega in a plot."""
x, y = sm.symbols('x y')
f = sm.lambdify([x,y], f, modules="numpy")
u = sm.lambdify([x,y], u, modules="numpy")
# When doing symbolics, Omega can easily contain symbolic expressions,
# assume .evalf() will work in that case to obtain numerical
# expressions, which then must be converted to float before calling
# linspace below
for r in range(2):
for s in range(2):
if not isinstance(Omega[r][s], (int,float)):
Omega[r][s] = float(Omega[r][s].evalf())
resolution = 41 # no of points in plot
xcoor = linspace(Omega[0][0], Omega[0][1], resolution)
ycoor = linspace(Omega[1][0], Omega[1][1], resolution)
xv, yv = ndgrid(xcoor, ycoor)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
exact = f(xv, yv)
approx = u(xv, yv)
figure()
surfc(xv, yv, exact, title='f(x,y)',
colorbar=True, colormap=hot(), shading='flat')
if plotfile:
savefig('%s_f.pdf' % plotfile, color=True)
savefig('%s_f.png' % plotfile)
figure()
surfc(xv, yv, approx, title='f(x,y)',
colorbar=True, colormap=hot(), shading='flat')
if plotfile:
savefig('%s_u.pdf' % plotfile, color=True)
savefig('%s_u.png' % plotfile)
def demo():
"""
Demonstrate difference between Euler-Cromer and the
scheme for the corresponding 2nd-order ODE.
"""
I = 1.2
V = 0.2
m = 4
b = 0.2
s = lambda u: 2 * u
F = lambda t: 0
w = np.sqrt(2.0 / 4) # approx freq
dt = 0.9 * 2 / w # longest possible time step
w = 0.5
P = 2 * pi / w
T = 4 * P
from vib import solver as solver2
import scitools.std as plt
for k in range(4):
u2, t2 = solver2(I, V, m, b, s, F, dt, T, "quadratic")
u, v, t = solver(I, V, m, b, s, F, dt, T, "quadratic")
plt.figure()
plt.plot(t, u, "r-", t2, u2, "b-")
plt.legend(["Euler-Cromer", "centered scheme"])
plt.title("dt=%.3g" % dt)
raw_input()
plt.savefig("tmp_%d" % k + ".png")
plt.savefig("tmp_%d" % k + ".pdf")
dt /= 2
def visualize(self):
u = self.solver.u; t = self.solver.t # short forms
num_periods = vib_plot_empirical_freq_and_amplitude(u, t)
if num_periods <= 40:
plt.figure()
vib_visualize(u, t)
else:
vib_visualize_front(u, t, self.window_width, self.savefig)
vib_visualize_front_ascii(u, t)
plt.show()
def visualize(self):
u = self.solver.u; t = self.solver.t # short forms
num_periods = vib_plot_empirical_freq_and_amplitude(u, t)
if num_periods <= 40:
plt.figure()
vib_visualize(u, t)
else:
vib_visualize_front(u, t, self.window_width, self.savefig)
vib_visualize_front_ascii(u, t)
plt.show()
def plotting(xu, xeta, u1, eta1, bpu, bpeta):
sci.figure(1)
sci.plot(xeta,
eta1,
xeta,
-bpeta,
axes=(xu[0], xu[-1], -1.1 * max(bpeta), 2.0 * max(bpeta)))
sci.figure(2)
sci.plot(xu,
u1,
xu,
-bpu,
axes=(xu[0], xu[-1], -1.1 * max(bpu), 2.0 * max(bpu)))
def draw_basis_functions():
"""Illustrate P1, P2, and P3 basis functions on an element."""
for ext in 'pdf', 'png':
for derivative in (0, 1):
for labels in True, False:
deriv = '' if derivative == 0 else 'd'
philab = '' if not labels else '_lab'
fe_basis_function_figure(
d=1, target_elm=1, N_e=4, derivative=derivative,
filename='fe_%sbasis_p1_4e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=2, target_elm=1, N_e=4, derivative=derivative,
filename='fe_%sbasis_p2_4e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=1, target_elm=1, N_e=5, derivative=derivative,
filename='fe_%sbasis_p1_5e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=3, target_elm=1, N_e=4, derivative=derivative,
filename='fe_%sbasis_p3_4e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=3, target_elm=2, N_e=5, derivative=derivative,
filename='fe_%sbasis_p3_5e%s.%s' % (deriv, philab, ext),
labels=labels)
def case(nperiods=4, showplot=False):
I = [1, 0] # Does not work well with float arrays and Cython
I = [1., 0.]
f = ode.Problem1()
t0 = time.clock()
time_points = np.linspace(0, nperiods*2*np.pi, nperiods*30+1)
u, t = ode.solver(f, I, time_points, ode.RK2)
t1 = time.clock()
if showplot and nperiods < 8:
from scitools.std import plot, figure
plot(t[-200:], u[-200:,0])
figure()
plot(u[-200:,0], u[-200:,1])
return t1-t0, time_points.size
def run_sines(help=False):
for N in (4, 12):
f = 10*(x-1)**2 - 1
psi = sines(x, N)
Omega = [0, 1]
if help: # u = 9 + sum
f0 = 9; f1 = -1
term = f0*(1-x) + x*f1
u = term + least_squares_orth(f-term, psi, Omega)
else:
u = least_squares_orth(f, psi, Omega)
figure()
comparison_plot(f, u, Omega, 'parabola_ls_sines%d%s.pdf' %
(N, '_wfterm' if help else ''))
def case(nperiods=4, showplot=False, ftype='class'):
I = [1, 0]
if ftype == 'class':
f = ProblemOpt()
else:
f = problem
t0 = time.clock()
u, t = RK2(f, I, t=np.linspace(0, nperiods * np.pi, nperiods * 30 + 1))
t1 = time.clock()
if showplot:
plot(t[-200:], u[-200:, 0])
figure()
plot(u[-200:, 0], u[-200:, 1])
return t1 - t0
def draw_basis_functions():
"""Illustrate P1, P2, and P3 basis functions on an element."""
for ext in 'pdf', 'png':
for derivative in (0, 1):
for labels in True, False:
deriv = '' if derivative == 0 else 'd'
philab = '' if not labels else '_lab'
fe_basis_function_figure(
d=1, target_elm=1, n_e=4, derivative=derivative,
filename='fe_%sbasis_p1_4e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=2, target_elm=1, n_e=4, derivative=derivative,
filename='fe_%sbasis_p2_4e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=1, target_elm=1, n_e=5, derivative=derivative,
filename='fe_%sbasis_p1_5e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=3, target_elm=1, n_e=4, derivative=derivative,
filename='fe_%sbasis_p3_4e%s.%s' % (deriv, philab, ext),
labels=labels)
plt.figure()
fe_basis_function_figure(
d=3, target_elm=2, n_e=5, derivative=derivative,
filename='fe_%sbasis_p3_5e%s.%s' % (deriv, philab, ext),
labels=labels)
def case(nperiods=4, showplot=False, ftype='class'):
I = [1, 0]
if ftype == 'class':
f = ProblemOpt()
else:
f = problem
t0 = time.clock()
u, t = RK2(f, I, t=np.linspace(0, nperiods*np.pi, nperiods*30+1))
t1 = time.clock()
if showplot:
plot(t[-200:], u[-200:,0])
figure()
plot(u[-200:,0], u[-200:,1])
return t1-t0
def plot_empirical_freq_and_amplitude(u, t):
minima, maxima = minmax(t, u)
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure()
from math import pi
w = 2*pi/p
plt.plot(range(len(p)), w, 'r-')
plt.hold('on')
plt.plot(range(len(a)), a, 'b-')
ymax = 1.1*max(w.max(), a.max())
ymin = 0.9*min(w.min(), a.min())
plt.axis([0, max(len(p), len(a)), ymin, ymax])
plt.legend(['estimated frequency', 'estimated amplitude'],
loc='upper right')
return len(maxima)
def plot_empirical_freq_and_amplitude(u, t):
minima, maxima = minmax(t, u)
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure()
from math import pi
w = 2 * pi / p
plt.plot(range(len(p)), w, 'r-')
plt.hold('on')
plt.plot(range(len(a)), a, 'b-')
ymax = 1.1 * max(w.max(), a.max())
ymin = 0.9 * min(w.min(), a.min())
plt.axis([0, max(len(p), len(a)), ymin, ymax])
plt.legend(['estimated frequency', 'estimated amplitude'],
loc='upper right')
return len(maxima)
def run_solvers_and_plot(solvers, timesteps_per_period=20,
num_periods=1, b=0):
w = 2*np.pi # frequency of undamped free oscillations
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)
t_fine = np.linspace(0, T, 8*Nt+1) # used for very accurate solution
legends = []
solver_exact = odespy.RK4(f)
for solver in solvers:
solver.set_initial_condition([solver.users_f.I, 0])
u, t = solver.solve(t_mesh)
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
# Make plots (plot last 10 periods????)
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)
# Compare with exact solution plotted on a very fine mesh
#t_fine = np.linspace(0, T, 10001)
#u_e = solver.users_f.exact(t_fine)
# Compare with RK4 on a much finer mesh
solver_exact.set_initial_condition([solver.users_f.I, 0])
u_e, t_e = solver_exact.solve(t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_e, u_e[:,0], '-') # avoid markers by spec. line type
legends.append('exact (RK4)')
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))
def run_solvers_and_plot(solvers, timesteps_per_period=20, num_periods=1, b=0):
w = 2 * np.pi # frequency of undamped free oscillations
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)
t_fine = np.linspace(0, T, 8 * Nt + 1) # used for very accurate solution
legends = []
solver_exact = odespy.RK4(f)
for solver in solvers:
solver.set_initial_condition([solver.users_f.I, 0])
u, t = solver.solve(t_mesh)
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
# Make plots (plot last 10 periods????)
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)
# Compare with exact solution plotted on a very fine mesh
#t_fine = np.linspace(0, T, 10001)
#u_e = solver.users_f.exact(t_fine)
# Compare with RK4 on a much finer mesh
solver_exact.set_initial_condition([solver.users_f.I, 0])
u_e, t_e = solver_exact.solve(t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_e, u_e[:, 0], '-') # avoid markers by spec. line type
legends.append('exact (RK4)')
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))
def main():
import argparse
parser = argparse.ArgumentParser()
parser.add_argument("--I", type=float, default=1.0)
parser.add_argument("--V", type=float, default=0.0)
parser.add_argument("--m", type=float, default=1.0)
parser.add_argument("--b", type=float, default=0.0)
parser.add_argument("--s", type=str, default="u")
parser.add_argument("--F", type=str, default="0")
parser.add_argument("--dt", type=float, default=0.05)
# parser.add_argument('--T', type=float, default=10)
parser.add_argument("--T", type=float, default=20)
parser.add_argument("--window_width", type=float, default=30.0, help="Number of periods in a window")
parser.add_argument("--damping", type=str, default="linear")
parser.add_argument("--savefig", action="store_true")
# Hack to allow --SCITOOLS options (scitools.std reads this argument
# at import)
parser.add_argument("--SCITOOLS_easyviz_backend", default="matplotlib")
a = parser.parse_args()
from scitools.std import StringFunction
s = StringFunction(a.s, independent_variable="u")
F = StringFunction(a.F, independent_variable="t")
I, V, m, b, dt, T, window_width, savefig, damping = (
a.I,
a.V,
a.m,
a.b,
a.dt,
a.T,
a.window_width,
a.savefig,
a.damping,
)
u, t = solver(I, V, m, b, s, F, dt, T, damping)
num_periods = plot_empirical_freq_and_amplitude(u, t)
if num_periods <= 40:
plt.figure()
visualize(u, t)
else:
visualize_front(u, t, window_width, savefig)
visualize_front_ascii(u, t)
plt.show()
def case(nperiods=4, showplot=False, ftype='class'):
I = [1, 0]
if ftype == 'class':
f = Problem1()
elif ftype == 'func2':
f = problem2
elif ftype == 'func3':
f = problem3
t0 = time.clock()
time_points = np.linspace(0, nperiods*2*np.pi, nperiods*30+1)
u, t = solver(f, I, time_points, RK2)
t1 = time.clock()
if showplot:
plot(t[-200:], u[-200:,0])
figure()
plot(u[-200:,0], u[-200:,1])
return t1-t0, time_points.size
def slides_waves():
L = 10.
a = 0.5
B0 = 1.
A = B0 / 5
t0 = 3.
v = 1.
x0 = lambda t: L / 4 + v * t if t < t0 else L / 4 + v * t0
sigma = 1.0
def bottom(x, t):
return (B(x, a, L, B0) + S(x, A, x0(t), sigma))
def depth(x, t):
return -bottom(x, t)
import scitools.std as plt
x = np.linspace(0, L, 101)
plt.plot(x, bottom(x, 0), x, depth(x, 0), legend=['bottom', 'depth'])
plt.show()
raw_input()
plt.figure()
dt_b = 0.01
solver(I=0,
V=0,
f=lambda x, t: -(depth(x, t - dt_b) - 2 * depth(x, t) + depth(
x, t + dt_b)) / dt_b**2,
c=lambda x, t: np.sqrt(np.abs(depth(x, t))),
U_0=None,
U_L=None,
L=L,
dt=0.1,
C=0.9,
T=20,
user_action=PlotSurfaceAndBottom(B,
S,
a,
L,
B0,
A,
x0,
sigma,
umin=-2,
umax=2),
stability_safety_factor=0.9)
def main():
import argparse
parser = argparse.ArgumentParser()
parser.add_argument('--I', type=float, default=1.0)
parser.add_argument('--V', type=float, default=0.0)
parser.add_argument('--m', type=float, default=1.0)
parser.add_argument('--b', type=float, default=0.0)
parser.add_argument('--s', type=str, default='4*pi**2*u')
parser.add_argument('--F', type=str, default='0')
parser.add_argument('--dt', type=float, default=0.05)
parser.add_argument('--T', type=float, default=20)
parser.add_argument('--window_width',
type=float,
default=30.,
help='Number of periods in a window')
parser.add_argument('--damping', type=str, default='linear')
parser.add_argument('--savefig', action='store_true')
# Hack to allow --SCITOOLS options
# (scitools.std reads this argument at import)
parser.add_argument('--SCITOOLS_easyviz_backend', default='matplotlib')
a = parser.parse_args()
from scitools.std import StringFunction
s = StringFunction(a.s, independent_variable='u')
F = StringFunction(a.F, independent_variable='t')
I, V, m, b, dt, T, window_width, savefig, damping = \
a.I, a.V, a.m, a.b, a.dt, a.T, a.window_width, a.savefig, \
a.damping
# compute u by both methods and then visualize the difference
u, t = solver2(I, V, m, b, s, F, dt, T, damping)
u_bw, _ = solver_bwdamping(I, V, m, b, s, F, dt, T, damping)
u_diff = u - u_bw
num_periods = plot_empirical_freq_and_amplitude(u_diff, t)
if num_periods <= 40:
plt.figure()
legends = [
'1st-2nd order method', '2nd order method', '1st order method'
]
visualize([(u_diff, t), (u, t), (u_bw, t)], legends)
else:
visualize_front(u_diff, t, window_width, savefig)
#visualize_front_ascii(u_diff, t)
plt.show()
def main():
import argparse
parser = argparse.ArgumentParser()
parser.add_argument('--I', type=float, default=1.0)
parser.add_argument('--V', type=float, default=0.0)
parser.add_argument('--m', type=float, default=1.0)
parser.add_argument('--b', type=float, default=0.0)
parser.add_argument('--s', type=str, default='4*pi**2*u')
parser.add_argument('--F', type=str, default='0')
parser.add_argument('--dt', type=float, default=0.05)
parser.add_argument('--T', type=float, default=20)
parser.add_argument('--window_width', type=float, default=30.,
help='Number of periods in a window')
parser.add_argument('--damping', type=str, default='linear')
parser.add_argument('--savefig', action='store_true')
# Hack to allow --SCITOOLS options
# (scitools.std reads this argument at import)
parser.add_argument('--SCITOOLS_easyviz_backend',
default='matplotlib')
a = parser.parse_args()
from scitools.std import StringFunction
s = StringFunction(a.s, independent_variable='u')
F = StringFunction(a.F, independent_variable='t')
I, V, m, b, dt, T, window_width, savefig, damping = \
a.I, a.V, a.m, a.b, a.dt, a.T, a.window_width, a.savefig, \
a.damping
# compute u by both methods and then visualize the difference
u, t = solver2(I, V, m, b, s, F, dt, T, damping)
u_bw, _ = solver_bwdamping(I, V, m, b, s, F, dt, T, damping)
u_diff = u - u_bw
num_periods = plot_empirical_freq_and_amplitude(u_diff, t)
if num_periods <= 40:
plt.figure()
legends = ['1st-2nd order method',
'2nd order method',
'1st order method']
visualize([(u_diff, t), (u, t), (u_bw, t)], legends)
else:
visualize_front(u_diff, t, window_width, savefig)
#visualize_front_ascii(u_diff, t)
plt.show()
def demo():
"""
Demonstrate difference between Euler-Cromer and the
scheme for the corresponding 2nd-order ODE.
"""
I = 1.2; w = 2.0; T = 5
dt = 2/w # longest possible time step
from vib_undamped import solver as solver2 # 2nd-order ODE
from scitools.std import plot, figure, savefig
for k in range(4):
dt /= 4
u2, t2 = solver2(I, w, dt, T)
u, v, t = solver(I, w, dt, T)
figure()
plot(t, u, t2, u2,
legend=('Euler-Cromer', 'center scheme for $u''+u=0$'),
title='dt=%.3g' % dt)
raw_input()
savefig('ECvs2nd_%d' % k + '.png')
savefig('ECvs2nd_%d' % k + '.pdf')
def demo():
"""
Demonstrate difference between Euler-Cromer and the
scheme for the corresponding 2nd-order ODE.
"""
I = 1.2
w = 2.0
T = 5
dt = 2 / w # longest possible time step
from vib_undamped import solver as solver2 # 2nd-order ODE
import scitools.std as plt
for k in range(4):
dt /= 4
u2, t2 = solver2(I, w, dt, T)
u, v, t = solver(I, w, dt, T)
plt.figure()
plt.plot(t, u, t2, u2, legend=("Euler-Cromer", "center scheme for $u" "+u=0$"), title="dt=%.3g" % dt)
raw_input()
plt.savefig("ECvs2nd_%d" % k + ".png")
plt.savefig("ECvs2nd_%d" % k + ".pdf")
def run_Lagrange_interp_abs(N, ymin=None, ymax=None):
f = abs(1-2*x)
psi, points = Lagrange_polynomials_01(x, N)
u = interpolation(f, psi, points)
comparison_plot(f, u, Omega=[0, 1],
filename='Lagrange_interp_abs_%d.pdf' % (N+1),
plot_title='Interpolation by Lagrange polynomials '\
'of degree %d' % N, ymin=ymin, ymax=ymax)
# Make figures of Lagrange polynomials (psi)
figure()
xcoor = linspace(0, 1, 1001)
for i in (2, (N+1)/2+1):
fn = lambdify([x], psi[i])
ycoor = fn(xcoor)
plot(xcoor, ycoor)
legend(r'\psi_%d' % i)
hold('on')
plot(points, [fn(xc) for xc in points], 'ro')
#if ymin is not None and ymax is not None:
# axis([xcoor[0], xcoor[-1], ymin, ymax])
savefig('Lagrange_basis_%d.pdf' % (N+1))
def main():
import argparse
parser = argparse.ArgumentParser()
parser.add_argument('--I', type=float, default=1.0)
parser.add_argument('--V', type=float, default=0.0)
parser.add_argument('--m', type=float, default=1.0)
parser.add_argument('--b', type=float, default=0.0)
parser.add_argument('--s', type=str, default='u')
parser.add_argument('--F', type=str, default='0')
parser.add_argument('--dt', type=float, default=0.05)
#parser.add_argument('--T', type=float, default=10)
parser.add_argument('--T', type=float, default=20)
parser.add_argument('--window_width',
type=float,
default=30.,
help='Number of periods in a window')
parser.add_argument('--damping', type=str, default='linear')
parser.add_argument('--savefig', action='store_true')
# Hack to allow --SCITOOLS options (scitools.std reads this argument
# at import)
parser.add_argument('--SCITOOLS_easyviz_backend', default='matplotlib')
a = parser.parse_args()
from scitools.std import StringFunction
s = StringFunction(a.s, independent_variable='u')
F = StringFunction(a.F, independent_variable='t')
I, V, m, b, dt, T, window_width, savefig, damping = \
a.I, a.V, a.m, a.b, a.dt, a.T, a.window_width, a.savefig, \
a.damping
u, t = solver(I, V, m, b, s, F, dt, T, damping)
num_periods = plot_empirical_freq_and_amplitude(u, t)
if num_periods <= 40:
plt.figure()
visualize(u, t)
else:
visualize_front(u, t, window_width, savefig)
visualize_front_ascii(u, t)
plt.show()
def main():
import argparse
parser = argparse.ArgumentParser()
parser.add_argument('--I', type=float, default=1.0)
parser.add_argument('--V', type=float, default=0.0)
parser.add_argument('--m', type=float, default=1.0)
parser.add_argument('--b', type=float, default=0.0)
parser.add_argument('--s', type=str, default='u')
parser.add_argument('--F', type=str, default='0')
parser.add_argument('--dt', type=float, default=0.05)
#parser.add_argument('--T', type=float, default=10)
parser.add_argument('--T', type=float, default=20)
parser.add_argument('--window_width', type=float, default=30.,
help='Number of periods in a window')
parser.add_argument('--damping', type=str, default='linear')
parser.add_argument('--savefig', action='store_true')
# Hack to allow --SCITOOLS options (scitools.std reads this argument
# at import)
parser.add_argument('--SCITOOLS_easyviz_backend', default='matplotlib')
a = parser.parse_args()
from scitools.std import StringFunction
s = StringFunction(a.s, independent_variable='u')
F = StringFunction(a.F, independent_variable='t')
I, V, m, b, dt, T, window_width, savefig, damping = \
a.I, a.V, a.m, a.b, a.dt, a.T, a.window_width, a.savefig, \
a.damping
u, t = solver(I, V, m, b, s, F, dt, T, damping)
num_periods = plot_empirical_freq_and_amplitude(u, t)
if num_periods <= 40:
plt.figure()
visualize(u, t)
else:
visualize_front(u, t, window_width, savefig)
visualize_front_ascii(u, t)
plt.show()
def amplification_factor(names):
# Use SciTools since it adds markers to colored lines
from scitools.std import (plot, title, xlabel, ylabel, hold, savefig,
axis, legend, grid, show, figure)
figure()
curves = {}
p = linspace(0, 3, 99)
curves['exact'] = A_exact(p)
plot(p, curves['exact'])
hold('on')
name2theta = dict(FE=0, BE=1, CN=0.5)
for name in names:
curves[name] = A(p, name2theta[name])
plot(p, curves[name])
axis([p[0], p[-1], -20, 20])
#semilogy(p, curves[name])
plot([p[0], p[-1]], [0, 0], '--') # A=0 line
title('Amplification factors')
grid('on')
legend(['exact'] + names, loc='lower left', fancybox=True)
xlabel(r'$p=-a\cdot dt$')
ylabel('Amplification factor')
savefig('A_growth.png')
savefig('A_growth.pdf')
def amplification_factor(names):
# Use SciTools since it adds markers to colored lines
from scitools.std import (
plot, title, xlabel, ylabel, hold, savefig,
axis, legend, grid, show, figure)
figure()
curves = {}
p = linspace(0, 3, 99)
curves['exact'] = A_exact(p)
plot(p, curves['exact'])
hold('on')
name2theta = dict(FE=0, BE=1, CN=0.5)
for name in names:
curves[name] = A(p, name2theta[name])
plot(p, curves[name])
axis([p[0], p[-1], -20, 20])
#semilogy(p, curves[name])
plot([p[0], p[-1]], [0, 0], '--') # A=0 line
title('Amplification factors')
grid('on')
legend(['exact'] + names, loc='lower left', fancybox=True)
xlabel(r'$p=-a\cdot dt$')
ylabel('Amplification factor')
savefig('A_growth.png'); savefig('A_growth.pdf')
def solve3(self, exact=None, s=0, ax=[], navn="tmp"):
if self.dim > 2:
return None
problem, dt, N = self.problem, self.dt, self.N
f_text = problem.f
alpha = problem.alpha
rho = problem.rho
I = problem.I
nx, ny = self.nx, self.ny
V, mesh = self.make_mesh()
u = TrialFunction(V)
v = TestFunction(V)
I = Expression(I, s=s) # can take one parameter with the name s
u_k = interpolate(I, V)
a = (u * v + dt / rho * alpha(u_k) * inner(nabla_grad(u), nabla_grad(v))) * dx
u = Function(V)
if self.dim == 2:
mlab.figure(1)
self.plot2D(u_k, nx, ny, navn, -1)
if self.dim == 1:
sci.figure(1)
self.plot1D(u_k, nx, ny, exact, mesh, ax)
for n in range(N):
time.sleep(0.2)
t = (n + 1) * dt
f = Expression(f_text, rho=rho, t=t, dt=dt)
L = (u_k * v + dt * f * v / rho) * dx
solve(a == L, u)
numVec = self.make_numpy(u, nx, ny)
if self.dim == 2:
mlab.figure(1)
self.plot2D(u, nx, ny, navn, n)
if self.dim == 1:
sci.figure(1)
self.plot1D(u, nx, ny, exact, mesh, ax, t=t, n=n + 1)
if exact != None:
if ny == 0:
sci.figure(2)
else:
mlab.figure(2)
self.plotError(u, exact, nx, ny, mesh, t, navn, n=n)
u_k.assign(u)
if self.dim == 2:
self.makemovie(navn)
r = raw_input("Done solving!")
def simulate_n_paths(n, N, x0, p0, M, m):
xm = np.zeros(N + 1) # mean of x
pm = np.zeros(N + 1) # mean of p
xs = np.zeros(N + 1) # standard deviation of x
ps = np.zeros(N + 1) # standard deviation of p
for i in range(n):
x, p = simulate_one_path(N, x0, p0, M, m)
# Accumulate paths
xm += x
pm += p
xs += x**2
ps += p**2
# Show 5 random sample paths
if i % (n / 5) == 0:
figure(1)
plot(x, title='sample paths of investment')
hold('on')
figure(2)
plot(p, title='sample paths of interest rate')
hold('on')
figure(1)
savefig('tmp_sample_paths_investment.pdf')
figure(2)
savefig('tmp_sample_paths_interestrate.pdf')
# Compute average
xm /= float(n)
pm /= float(n)
# Compute standard deviation
xs = xs / float(n) - xm * xm # variance
ps = ps / float(n) - pm * pm # variance
# Remove small negative numbers (round off errors)
print 'min variance:', xs.min(), ps.min()
xs[xs < 0] = 0
ps[ps < 0] = 0
xs = np.sqrt(xs)
ps = np.sqrt(ps)
return xm, xs, pm, ps
def simulate_n_paths(n, N, x0, p0, M, m):
xm = np.zeros(N+1) # mean of x
pm = np.zeros(N+1) # mean of p
xs = np.zeros(N+1) # standard deviation of x
ps = np.zeros(N+1) # standard deviation of p
for i in range(n):
x, p = simulate_one_path(N, x0, p0, M, m)
# Accumulate paths
xm += x
pm += p
xs += x**2
ps += p**2
# Show 5 random sample paths
if i % (n/5) == 0:
figure(1)
plot(x, title='sample paths of investment')
hold('on')
figure(2)
plot(p, title='sample paths of interest rate')
hold('on')
figure(1); savefig('tmp_sample_paths_investment.eps')
figure(2); savefig('tmp_sample_paths_interestrate.eps')
# Compute average
xm /= float(n)
pm /= float(n)
# Compute standard deviation
xs = xs/float(n) - xm*xm # variance
ps = ps/float(n) - pm*pm # variance
# Remove small negative numbers (round off errors)
print 'min variance:', xs.min(), ps.min()
xs[xs < 0] = 0
ps[ps < 0] = 0
xs = np.sqrt(xs)
ps = np.sqrt(ps)
return xm, xs, pm, ps
def run_solvers_and_plot(solvers,
timesteps_per_period=20,
num_periods=1,
I=1,
w=2 * np.pi):
P = 2 * np.pi / w # one period
dt = P / timesteps_per_period
N = num_periods * timesteps_per_period
T = N * dt
t_mesh = np.linspace(0, T, N + 1)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# 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('vb_%d_%d_u.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_u.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%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('vb_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%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('vb_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%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('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
grad = []
h = alpha[1] - alpha[0]
for i in range(1, len(alpha)-1):
deriv = 1/(2*h)*(-E[i-1] + E[i+1])
grad.append(deriv);
return grad
aGrad = get_grad_1d(E, alpha)
if two:
aGrad2 = get_grad_1d(E2, alpha2)
figure(1)
plot(alpha, E, xlabel="alpha", legend="IS", hold="on")
#figure(2)
#plot(alpha, E, xlabel="alpha", legend="IS", hold="on")
#figure(3)
#plot(alpha, E, xlabel="alpha", legend="IS", hold="on")
figure(4)
plot(alpha[1:-1], aGrad, legend="IS", hold="on")
if two:
figure(1)
plot(alpha2, E2, xlabel="alpha", ylabel="E", title="2p Qdot noInteraction",\
legend="BF", hardcopy=path+"full.eps")
#figure(2)
#plot(alpha2, E2, xlabel="alpha", ylabel="E", title="2p Qdot noInteraction",\
# legend="BF", axis=[1,1.5, 1,3], hardcopy=path+"closeToOne.eps")
p[mu-1][i+1] = eodsq - u[i+1]*r2d
p[mu+1][nps-3], p[mu][nps-2] = eodsq + u[nps-2]*r2d, diag
return p
u0 = np.zeros(9, float)
u0[1], u0[2:5], u0[5] = .5, 1., .5
t0, tn, n_points, ml, mu = 0., .4, 5, 1, 1
time_points = np.linspace(t0, tn, n_points)
atol, rtol = 1e-3, 1e-3
exact_final = [1.07001457e-1, 2.77432492e-1, 5.02444616e-1, 7.21037157e-1,
9.01670441e-1, 8.88832048e-1, 4.96572850e-1, 9.46924362e-2,
-6.90855199e-3]
st.figure()
method = Lsodi
# Test case 1: Lsodi, with res, adda_full, & jac_full
m = method(res=res, rtol=rtol, atol=atol,
adda_lsodi=adda_full, jac_lsodi=jac_full)
m.set_initial_condition(u0)
u,t = m.solve(time_points)
st.plot(t, u[:,0], 'r-', title="Lsodi with Python functions",
legend="with res, adda_full & jac_full", hold="on")
print('Max error with test case 1 is %g' % max(u[-1] - exact_final))
# Test case 2: Lsodi, with res & adda_full
m = method(res=res, rtol=rtol, atol=atol, adda_lsodi=adda_full)
m.set_initial_condition(u0)
import scitools.std as plt
import sys
import numpy as np
import matplotlib.pyplot as plt
x, y = np.loadtxt('volt.txt', delimiter=',', unpack=True)
plt.figure(1)
plt.plot(x, y, '*', linewidth=1) # avoid markers by spec. line type
#plt.xlim([0.0, 10])
#plt.ylim([0.0, 2])
plt.legend(['Force-- Tip'],
loc='upper right',
prop={"family": "Times New Roman"})
plt.xlabel('Sampled time')
plt.ylabel('$Force /m$')
plt.savefig('volt1v.png')
plt.savefig('volt1v.pdf')
plt.hold(True)
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 simulate(
beta=0.9,
Theta=30,
epsilon=0,
num_periods=6,
time_steps_per_period=60,
plot=True,
):
from math import sin, cos, pi
Theta = Theta*np.pi/180 #convert to radians
#Initial position and velocity
#(we order the equations such that Euler-Cromer in odespy
#can be used, i.e., vx, x, vy, y)
ic = [0, #x'=vx
(1 + epsilon)*sin(Theta), #x
0, #y'=vy
1 - (1 + epsilon)*cos(Theta), #y
]
def f(u, t, beta):
vx, x, vy, y = u
L = np.sqrt(x**2 + (y-1)**2)
h = beta/(1-beta)*(1- beta/L) #help factor
return [-h*x, vx, -h*(y-1) - beta, vy]
#Non-elastic pendulum (scaled similarly in the limit beta=1)
#Solution Theta*cos(t)
P = 2*pi
dt = P/time_steps_per_period
T = num_periods*P
omega = 2*pi/P
time_points = np.linspace(
0, T, num_periods*time_steps_per_period+1)
solver = odespy.EulerCromer(f, f_args=(beta,))
solver.set_initial_condition(ic)
u, t = solver.solve(time_points)
x = u[:,1]
y = u[:,3]
theta = np.arctan(x/(1-y))
if plot:
plt.figure()
plt.plot(x, y, 'b-', title='Pendulum motion',
daspect=[1,1,1], daspectmode='equal',
axis=[x.min(), x.max(), 1.3*y.min(), 1])
plt.savefig('tmp_xy.png')
plt.savefig('tmp_xy.pdf')
#plot theta in degrees
plt.figure()
plt.plot(t, theta*180/np.pi, 'b-',
title='Angular displacement in degrees')
plt.savefig('tmp_theta.png')
plt.savefig('tmp_theta.pdf')
if abs(Theta) < 10*pi/180:
#compare theta and theta_e for small angles (< 10 degrees)
theta_e = Theta*np.cos(omega*t) #non-elastic scaled sol.
plt.figure()
plt.plot(t, theta, t, theta_e,
legend=['theta elastic', 'theta non-elastic'],
title='Elastic vs non-elastic pendulum, '\
'beta=%g' % beta)
plt.savefig('tmp_compare.png')
plt.savefig('tmp_compare.pdf')
#Plot y vs x (the real physical motion)
return x, y, theta, t
"""
Exercise 5.25: Investigate the behaviour of Langrange's interpolating polynomials
Author: Weiyun Lu
"""
import Lagrange_poly2
from scitools.std import hold, figure
figure()
for n in [2,4,6,10]:
Lagrange_poly2.graph(abs, n, -2, 2)
hold('on')
hold('off')
figure()
for n in [13,20]:
Lagrange_poly2.graph(abs, n, -2, 2)
hold('on')
hold('off')
def run(gamma, beta=10, delta=40, scaling=1, animate=False):
"""Run the scaled model for welding."""
gamma = float(gamma) # avoid integer division
if scaling == 'a':
v = gamma
a = 1
L = 1.0
b = 0.5 * beta**2
elif scaling == 'b':
v = 1
a = 1.0 / gamma
L = 1.0
b = 0.5 * beta**2
elif scaling == 'c':
v = 1
a = beta / gamma
L = beta
b = 0.5
elif scaling == 'd':
# PDE: u_t = gamma**(-1)u_xx + gamma**(-1)*delta*f
v = 1
a = 1.0 / gamma
L = 1.0
b = 0.5 * beta**2
delta *= 1.0 / gamma
ymin = 0
# Need global ymax to be able change ymax in closure process_u
global ymax
ymax = 1.2
I = lambda x: 0
f = lambda x, t: delta * np.exp(-b * (x - v * t)**2)
import time
import scitools.std as plt
plot_arrays = []
if scaling == 'c':
plot_times = [0.2 * beta, 0.5 * beta]
else:
plot_times = [0.2, 0.5]
def process_u(u, x, t, n):
"""
Animate u, and store arrays in plot_arrays if
t coincides with chosen times for plotting (plot_times).
"""
global ymax
if animate:
plt.plot(x,
u,
'r-',
x,
f(x, t[n]) / delta,
'b-',
axis=[0, L, ymin, ymax],
title='t=%f' % t[n],
xlabel='x',
ylabel='u and f/%g' % delta)
if t[n] == 0:
time.sleep(1)
plot_arrays.append(x)
dt = t[1] - t[0]
tol = dt / 10.0
if abs(t[n] - plot_times[0]) < tol or \
abs(t[n] - plot_times[1]) < tol:
plot_arrays.append((u.copy(), f(x, t[n]) / delta))
if u.max() > ymax:
ymax = u.max()
Nx = 100
D = 10
if scaling == 'c':
T = 0.5 * beta
else:
T = 0.5
u_L = u_R = 0
theta = 1.0
cpu = solver(I, a, f, L, Nx, D, T, theta, u_L, u_R, user_action=process_u)
x = plot_arrays[0]
plt.figure()
for u, f in plot_arrays[1:]:
plt.plot(x,
u,
'r-',
x,
f,
'b--',
axis=[x[0], x[-1], 0, ymax],
xlabel='$x$',
ylabel=r'$u, \ f/%g$' % delta)
plt.hold('on')
plt.legend([
'$u,\\ t=%g$' % plot_times[0],
'$f/%g,\\ t=%g$' % (delta, plot_times[0]),
'$u,\\ t=%g$' % plot_times[1],
'$f/%g,\\ t=%g$' % (delta, plot_times[1])
])
filename = 'tmp1_gamma%g_%s' % (gamma, scaling)
plt.title(r'$\beta = %g,\ \gamma = %g,\ $' % (beta, gamma) +
'scaling=%s' % scaling)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return cpu
D = 500
dt = dx**2*D
dt = 1.25
D = dt/dx**2
T = 2.5
umin = u_R
umax = u_L
a_consts = [[0, 1]]
a_consts = [[0, 1], [0.5, 8]]
a_consts = [[0, 1], [0.5, 8], [0.75, 0.1]]
a = fill_a(a_consts, L, Nx)
#a = random.uniform(0, 10, Nx+1)
from scitools.std import plot, hold, subplot, figure, show
figure()
subplot(2,1,1)
u, x, cpu = viz(I, a, L, Nx, D, T, umin, umax, theta, u_L, u_R)
v = u_exact_stationary(x, a, u_L, u_R)
print 'v', v
print 'u', u
hold('on')
symbol = 'bo' if Nx < 32 else 'b-'
plot(x, v, symbol, legend='exact stationary')
subplot(2,1,2)
plot(x, a, legend='a')
show()
def simulate(
beta=0.9, # dimensionless parameter
Theta=30, # initial angle in degrees
epsilon=0, # initial stretch of wire
num_periods=6, # simulate for num_periods
time_steps_per_period=60, # time step resolution
plot=True, # make plots or not
):
from math import sin, cos, pi
Theta = Theta*np.pi/180 # convert to radians
# Initial position and velocity
# (we order the equations such that Euler-Cromer in odespy
# can be used, i.e., vx, x, vy, y)
ic = [0, # x'=vx
(1 + epsilon)*sin(Theta), # x
0, # y'=vy
1 - (1 + epsilon)*cos(Theta), # y
]
def f(u, t, beta):
vx, x, vy, y = u
L = np.sqrt(x**2 + (y-1)**2)
h = beta/(1-beta)*(1 - beta/L) # help factor
return [-h*x, vx, -h*(y-1) - beta, vy]
# Non-elastic pendulum (scaled similarly in the limit beta=1)
# solution Theta*cos(t)
P = 2*pi
dt = P/time_steps_per_period
T = num_periods*P
omega = 2*pi/P
time_points = np.linspace(
0, T, num_periods*time_steps_per_period+1)
solver = odespy.EulerCromer(f, f_args=(beta,))
solver.set_initial_condition(ic)
u, t = solver.solve(time_points)
x = u[:,1]
y = u[:,3]
theta = np.arctan(x/(1-y))
if plot:
plt.figure()
plt.plot(x, y, 'b-', title='Pendulum motion',
daspect=[1,1,1], daspectmode='equal',
axis=[x.min(), x.max(), 1.3*y.min(), 1])
plt.savefig('tmp_xy.png')
plt.savefig('tmp_xy.pdf')
# Plot theta in degrees
plt.figure()
plt.plot(t, theta*180/np.pi, 'b-',
title='Angular displacement in degrees')
plt.savefig('tmp_theta.png')
plt.savefig('tmp_theta.pdf')
if abs(Theta) < 10*pi/180:
# Compare theta and theta_e for small angles (<10 degrees)
theta_e = Theta*np.cos(omega*t) # non-elastic scaled sol.
plt.figure()
plt.plot(t, theta, t, theta_e,
legend=['theta elastic', 'theta non-elastic'],
title='Elastic vs non-elastic pendulum, '\
'beta=%g' % beta)
plt.savefig('tmp_compare.png')
plt.savefig('tmp_compare.pdf')
# Plot y vs x (the real physical motion)
return x, y, theta, t
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))
from __future__ import print_function
# Stiff ODE: Van der Pol oscillator
# u'' = 3*(1 - u**2)*u' - u
from odespy import *
import scitools.std as st
solver_no = 1 # number of solvers in figure
st.figure()
exceptions=['RKC', 'Lsodes', 'Leapfrog', 'Lsodi', 'Lsodis', 'Lsoibt',
'MyRungeKutta', 'MySolver', 'AdaptiveResidual', 'EulerCromer']
solvers = [solver for solver in list_available_solvers() if solver not in exceptions]
f = lambda (u00,u11),t: [u11, 3.*(1 - u00**2)*u11 - u00]
jac = lambda (u00,u11),t: \
[[0., 1.], [-6.*u00*u11 - 1., 3.*(1. - u00**2)]]
u0, time_points = [2., 0.], np.linspace(0., 10., 100)
print("""Van der Pol oscillator problem:
u'' = 3*(1 - u**2)*u' - u""")
# Loop for all possible solvers
for solver in solvers:
try:
method = eval(solver)(f)
method.set_initial_condition(u0)
u,t = method.solve(time_points)
if solver_no == 8: # More than 8 solvers in current figure?
st.figure() # Initialize new figure.
solver_no = 1
# Stiff ODE: Van der Pol oscillator
# u'' = 3*(1 - u**2)*u' - u
from odespy import *
import scitools.std as st
solver_no = 1 # number of solvers in figure
st.figure()
exceptions=['RKC', 'Lsodes', 'Leapfrog', 'Lsodi', 'Lsodis', 'Lsoibt',
'MyRungeKutta', 'MySolver', 'AdaptiveResidual', 'EulerCromer']
solvers = [solver for solver in list_available_solvers() if solver not in exceptions]
f = lambda (u00,u11),t: [u11, 3.*(1 - u00**2)*u11 - u00]
jac = lambda (u00,u11),t: \
[[0., 1.], [-6.*u00*u11 - 1., 3.*(1. - u00**2)]]
u0, time_points = [2., 0.], np.linspace(0., 10., 100)
print """Van der Pol oscillator problem:
u'' = 3*(1 - u**2)*u' - u"""
# Loop for all possible solvers
for solver in solvers:
try:
method = eval(solver)(f)
method.set_initial_condition(u0)
u,t = method.solve(time_points)
if solver_no == 8: # More than 8 solvers in current figure?
st.figure() # Initialize new figure.
solver_no = 1
st.plot(t, u[:,0], hold="on", legend=solver, axis=[0.,10.,-4.,4.])
"""
import sys
try:
n_points = int(sys.argv[1]) # Read from input
except:
n_points = 10 # default number of time-steps
# Compile these Fortran subroutines
f_f77, jac_f77 = compile_f77(f_str, jac_full_str)
t0, tn, u0 = 0., 10., [2., 0.]
time_points = np.linspace(t0, tn, n_points)
atol, rtol = 1e-4, 1e-4
st.figure()
method = Radau5Explicit
# Test case 1: Radau5, with f & jac
m = method(None, f_f77=f_f77, rtol=rtol, atol=atol, jac_f77_radau5=jac_f77)
m.set_initial_condition(u0)
u, t = m.solve(time_points)
st.plot(t,
u[:, 0],
title="Van der Pol oscillator, with Radau5 & Lsoda",
legend="Radau5 with f & jac",
hold="on")
# Test case 2: Radau5, with f
m = method(None, f_f77=f_f77, rtol=rtol, atol=atol)
m.set_initial_condition(u0)
def viz(
I,
V,
f,
c,
L,
dt,
C,
T,
ymax, # y axis: [-ymax, ymax]
u_exact, # u_exact(x, t)
animate='u and u_exact', # or 'error'
movie_filename='movie',
):
"""Run solver and visualize u at each time level."""
import scitools.std as plt
import time, glob, os
class Plot:
def __init__(self, ymax, frame_name='frame'):
self.max_error = [] # hold max amplitude errors
self.max_error_t = [] # time points corresp. to max_error
self.frame_name = frame_name
self.ymax = ymax
def __call__(self, u, x, t, n):
"""user_action function for solver."""
if animate == 'u and u_exact':
plt.plot(x,
u,
'r-',
x,
u_exact(x, t[n]),
'b--',
xlabel='x',
ylabel='u',
axis=[0, L, -self.ymax, self.ymax],
title='t=%f' % t[n],
show=True)
else:
error = u_exact(x, t[n]) - u
local_max_error = np.abs(error).max()
# self.max_error holds the increasing amplitude error
if self.max_error == [] or \
local_max_error > max(self.max_error):
self.max_error.append(local_max_error)
self.max_error_t.append(t[n])
# Use user's ymax until the error exceeds that value.
# This gives a growing max value of the yaxis (but
# never shrinking)
self.ymax = max(self.ymax, max(self.max_error))
plt.plot(x,
error,
'r-',
xlabel='x',
ylabel='error',
axis=[0, L, -self.ymax, self.ymax],
title='t=%f' % t[n],
show=True)
plt.savefig('%s_%04d.png' % (self.frame_name, n))
# Clean up old movie frames
for filename in glob.glob('frame_*.png'):
os.remove(filename)
plot = Plot(ymax)
u, x, t, cpu = solver(I, V, f, c, L, dt, C, T, plot)
# Make plot of max error versus time
plt.figure()
plt.plot(plot.max_error_t, plot.max_error)
plt.xlabel('time')
plt.ylabel('max abs(error)')
plt.savefig('error.png')
plt.savefig('error.pdf')
# Make .flv movie file
fps = 4 # Frames per second
codec2ext = dict(flv='flv') #, libx64='mp4',
#libvpx='webm', libtheora='ogg')
filespec = 'frame_%04d.png'
movie_program = 'avconv' # or 'ffmpeg'
for codec in codec2ext:
ext = codec2ext[codec]
cmd = '%(movie_program)s -r %(fps)d -i %(filespec)s '\
'-vcodec %(codec)s %(movie_filename)s.%(ext)s' % vars()
os.system(cmd)
def estimate(truncation_error, T, N_0, m, makeplot=True):
"""
Compute the truncation error in a problem with one independent
variable, using m meshes, and estimate the convergence
rate of the truncation error.
The user-supplied function truncation_error(dt, N) computes
the truncation error on a uniform mesh with N intervals of
length dt::
R, t, R_a = truncation_error(dt, N)
where R holds the truncation error at points in the array t,
and R_a are the corresponding theoretical truncation error
values (None if not available).
The truncation_error function is run on a series of meshes
with 2**i*N_0 intervals, i=0,1,...,m-1.
The values of R and R_a are restricted to the coarsest mesh.
and based on these data, the convergence rate of R (pointwise)
and time-integrated R can be estimated empirically.
"""
N = [2**i*N_0 for i in range(m)]
R_I = np.zeros(m) # time-integrated R values on various meshes
R = [None]*m # time series of R restricted to coarsest mesh
R_a = [None]*m # time series of R_a restricted to coarsest mesh
dt = np.zeros(m)
legends_R = []; legends_R_a = [] # all legends of curves
for i in range(m):
dt[i] = T/float(N[i])
R[i], t, R_a[i] = truncation_error(dt[i], N[i])
R_I[i] = np.sqrt(dt[i]*np.sum(R[i]**2))
if i == 0:
t_coarse = t # the coarsest mesh
stride = N[i]/N_0
R[i] = R[i][::stride] # restrict to coarsest mesh
R_a[i] = R_a[i][::stride]
if makeplot:
plt.figure(1)
plt.plot(t_coarse, R[i], log='y')
legends_R.append('N=%d' % N[i])
plt.hold('on')
plt.figure(2)
plt.plot(t_coarse, R_a[i] - R[i], log='y')
plt.hold('on')
legends_R_a.append('N=%d' % N[i])
if makeplot:
plt.figure(1)
plt.xlabel('time')
plt.ylabel('pointwise truncation error')
plt.legend(legends_R)
plt.savefig('R_series.png')
plt.savefig('R_series.pdf')
plt.figure(2)
plt.xlabel('time')
plt.ylabel('pointwise error in estimated truncation error')
plt.legend(legends_R_a)
plt.savefig('R_error.png')
plt.savefig('R_error.pdf')
# Convergence rates
r_R_I = convergence_rates(dt, R_I)
print 'R integrated in time; r:',
print ' '.join(['%.1f' % r for r in r_R_I])
R = np.array(R) # two-dim. numpy array
r_R = [convergence_rates(dt, R[:,n])[-1]
for n in range(len(t_coarse))]
# Plot convergence rates
if makeplot:
plt.figure()
plt.plot(t_coarse, r_R)
plt.xlabel('time')
plt.ylabel('r')
plt.axis([t_coarse[0], t_coarse[-1], 0, 2.5])
plt.title('Pointwise rate $r$ in truncation error $\sim\Delta t^r$')
plt.savefig('R_rate_series.png')
plt.savefig('R_rate_series.pdf')
from scitools.std import plot, figure, hold, savefig
random.seed(1)
x0 = 1 # initial investment
p0 = 5 # initial interest rate
N = 10*12 # number of months
M = 3 # p changes (on average) every M months
n = 1000 # number of simulations
m = 0.5 # adjustment of p
# Verify
x, p = simulate_one_path(3, 1, 10, M, 0)
print '3 months with p=10:', x
xm, xs, pm, ps = simulate_n_paths(n, N, x0, p0, M, m)
figure(3)
months = range(len(xm)) # indices along the x axis
plot(months, xm, 'r',
months, xm-xs, 'y',
months, xm+xs, 'y',
title='Mean +/- 1 st.dev. of investment',
savefig='tmp_mean_investment.eps')
figure(4)
plot(months, pm, 'r',
months, pm-ps, 'y',
months, pm+ps, 'y',
title='Mean +/- 1 st.dev. of annual interest rate',
savefig='tmp_mean_interestrate.eps')
def run(gamma, beta=10, delta=40, scaling=1, animate=False):
"""Run the scaled model for welding."""
if scaling == 1:
v = gamma
a = 1
elif scaling == 2:
v = 1
a = 1.0 / gamma
b = 0.5 * beta**2
L = 1.0
ymin = 0
# Need gloal to be able change ymax in closure process_u
global ymax
ymax = 1.2
I = lambda x: 0
f = lambda x, t: delta * np.exp(-b * (x - v * t)**2)
import time
import scitools.std as plt
plot_arrays = []
def process_u(u, x, t, n):
global ymax
if animate:
plt.plot(x,
u,
'r-',
x,
f(x, t[n]) / delta,
'b-',
axis=[0, L, ymin, ymax],
title='t=%f' % t[n],
xlabel='x',
ylabel='u and f/%g' % delta)
if t[n] == 0:
time.sleep(1)
plot_arrays.append(x)
dt = t[1] - t[0]
tol = dt / 10.0
if abs(t[n] - 0.2) < tol or abs(t[n] - 0.5) < tol:
plot_arrays.append((u.copy(), f(x, t[n]) / delta))
if u.max() > ymax:
ymax = u.max()
Nx = 100
D = 10
T = 0.5
u_L = u_R = 0
theta = 1.0
cpu = solver(I, a, f, L, Nx, D, T, theta, u_L, u_R, user_action=process_u)
x = plot_arrays[0]
plt.figure()
for u, f in plot_arrays[1:]:
plt.plot(x,
u,
'r-',
x,
f,
'b--',
axis=[x[0], x[-1], 0, ymax],
xlabel='$x$',
ylabel=r'$u, \ f/%g$' % delta)
plt.hold('on')
plt.legend([
'$u,\\ t=0.2$',
'$f/%g,\\ t=0.2$' % delta, '$u,\\ t=0.5$',
'$f/%g,\\ t=0.5$' % delta
])
filename = 'tmp1_gamma%g_s%d' % (gamma, scaling)
s = 'diffusion' if scaling == 1 else 'source'
plt.title(r'$\beta = %g,\ \gamma = %g,\ $' % (beta, gamma) +
'scaling=%s' % s)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return cpu
def run_solvers_and_plot(solvers, timesteps_per_period=20,
num_periods=1, I=1, w=2*np.pi):
P = 2*np.pi/w # one period
dt = P/timesteps_per_period
N = num_periods*timesteps_per_period
T = N*dt
t_mesh = np.linspace(0, T, N+1)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# 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('vb_%d_%d_u.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_u.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%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('vb_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%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('vb_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%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('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
