def comparison_plot(f, u, Omega, filename='tmp.pdf',
plot_title='', ymin=None, ymax=None,
u_legend='approximation'):
"""Compare f(x) and u(x) for x in Omega in a plot."""
x = sm.Symbol('x')
print 'f:', f
f = sm.lambdify([x], f, modules="numpy")
u = sm.lambdify([x], u, modules="numpy")
if len(Omega) != 2:
raise ValueError('Omega=%s must be an interval (2-list)' % str(Omega))
# 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
if not isinstance(Omega[0], (int,float)):
Omega[0] = float(Omega[0].evalf())
if not isinstance(Omega[1], (int,float)):
Omega[1] = float(Omega[1].evalf())
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
exact = f(xcoor)
approx = u(xcoor)
plot(xcoor, approx, '-')
hold('on')
plot(xcoor, exact, '-')
legend([u_legend, 'exact'])
title(plot_title)
xlabel('x')
if ymin is not None and ymax is not None:
axis([xcoor[0], xcoor[-1], ymin, ymax])
savefig(filename)
def run_solvers_and_check_amplitudes(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)
file_name = 'Amplitudes' # initialize filename for plot
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
solver_name = \
'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
file_name = file_name + '_' + solver_name
minima, maxima = minmax(t, u[:,0])
a = amplitudes(minima, maxima)
plt.plot(range(len(a)), a, '-', label=solver_name)
plt.hold('on')
plt.xlabel('Number of periods')
plt.ylabel('Amplitude (absolute value)')
plt.legend(loc='upper left')
plt.savefig(file_name + '.png')
plt.savefig(file_name + '.pdf')
plt.show()
def plot(self, include_exact=True, plt=None):
"""
Add solver.u curve to the plotting object plt,
and include the exact solution if include_exact is True.
This plot function can be called several times (if
the solver object has computed new solutions).
"""
if plt is None:
import scitools.std as plt # can use matplotlib as well
plt.plot(self.solver.t, self.solver.u, '--o')
plt.hold('on')
theta2name = {0: 'FE', 1: 'BE', 0.5: 'CN'}
name = theta2name.get(self.solver.theta, '')
legends = ['numerical %s' % name]
if include_exact:
t_e = linspace(0, self.problem.T, 1001)
u_e = self.problem.exact_solution(t_e)
plt.plot(t_e, u_e, 'b-')
legends.append('exact')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
plt.title('theta=%g, dt=%g' %
(self.solver.theta, self.solver.dt))
plt.savefig('%s_%g.png' % (name, self.solver.dt))
return plt
def comparison_plot(u, Omega, u_e=None, filename='tmp.eps',
plot_title='', ymin=None, ymax=None):
x = sp.Symbol('x')
u = sp.lambdify([x], u, modules="numpy")
if len(Omega) != 2:
raise ValueError('Omega=%s must be an interval (2-list)' % str(Omega))
# 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
if not isinstance(Omega[0], (int,float)):
Omega[0] = float(Omega[0].evalf())
if not isinstance(Omega[1], (int,float)):
Omega[1] = float(Omega[1].evalf())
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
approx = u(xcoor)
plot(xcoor, approx)
legends = ['approximation']
if u_e is not None:
exact = u_e(xcoor)
hold('on')
plot(xcoor, exact)
legends = ['exact']
legend(legends)
title(plot_title)
xlabel('x')
if ymin is not None and ymax is not None:
axis([xcoor[0], xcoor[-1], ymin, ymax])
savefig(filename)
def non_physical_behavior(I, a, T, dt, theta):
"""
Given lists/arrays a and dt, and numbers I, dt, and theta,
make a two-dimensional contour line B=0.5, where B=1>0.5
means oscillatory (unstable) solution, and B=0<0.5 means
monotone solution of u'=-au.
"""
a = np.asarray(a); dt = np.asarray(dt) # must be arrays
B = np.zeros((len(a), len(dt))) # results
for i in range(len(a)):
for j in range(len(dt)):
u, t = solver(I, a[i], T, dt[j], theta)
# Does u have the right monotone decay properties?
correct_qualitative_behavior = True
for n in range(1, len(u)):
if u[n] > u[n-1]: # Not decaying?
correct_qualitative_behavior = False
break # Jump out of loop
B[i,j] = float(correct_qualitative_behavior)
a_, dt_ = st.ndgrid(a, dt) # make mesh of a and dt values
st.contour(a_, dt_, B, 1)
st.grid('on')
st.title('theta=%g' % theta)
st.xlabel('a'); st.ylabel('dt')
st.savefig('osc_region_theta_%s.png' % theta)
st.savefig('osc_region_theta_%s.pdf' % theta)
def run_solvers_and_plot(solvers, rhs, T, dt, title=''):
Nt = int(round(T / float(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(rhs)
for solver in solvers:
solver.set_initial_condition([rhs.I, 0])
u, t = solver.solve(t_mesh)
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
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 RK4 on a much finer mesh
solver_exact.set_initial_condition([rhs.I, 0])
u_e, t_e = solver_exact.solve(t_fine)
plt.plot(t_e, u_e[:, 0], '-') # avoid markers by spec. line type
legends.append('exact (RK4, dt=%g)' % (t_fine[1] - t_fine[0]))
plt.legend(legends, loc='upper right')
plt.xlabel('t')
plt.ylabel('u')
plt.title(title)
plotfilestem = '_'.join(legends)
plt.savefig('tmp_%s.png' % plotfilestem)
plt.savefig('tmp_%s.pdf' % plotfilestem)
def run_solvers_and_check_amplitudes(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)
file_name = 'Amplitudes' # initialize filename for plot
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
solver_name = \
'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
file_name = file_name + '_' + solver_name
minima, maxima = minmax(t, u[:, 0])
a = amplitudes(minima, maxima)
plt.plot(range(len(a)), a, '-', label=solver_name)
plt.hold('on')
plt.xlabel('Number of periods')
plt.ylabel('Amplitude (absolute value)')
plt.legend(loc='upper left')
plt.savefig(file_name + '.png')
plt.savefig(file_name + '.pdf')
plt.show()
def non_physical_behavior(I, a, T, dt, theta):
"""
Given lists/arrays a and dt, and numbers I, dt, and theta,
make a two-dimensional contour line B=0.5, where B=1>0.5
means oscillatory (unstable) solution, and B=0<0.5 means
monotone solution of u'=-au.
"""
a = np.asarray(a)
dt = np.asarray(dt) # must be arrays
B = np.zeros((len(a), len(dt))) # results
for i in range(len(a)):
for j in range(len(dt)):
u, t = solver(I, a[i], T, dt[j], theta)
# Does u have the right monotone decay properties?
correct_qualitative_behavior = True
for n in range(1, len(u)):
if u[n] > u[n - 1]: # Not decaying?
correct_qualitative_behavior = False
break # Jump out of loop
B[i, j] = float(correct_qualitative_behavior)
a_, dt_ = st.ndgrid(a, dt) # make mesh of a and dt values
st.contour(a_, dt_, B, 1)
st.grid('on')
st.title('theta=%g' % theta)
st.xlabel('a')
st.ylabel('dt')
st.savefig('osc_region_theta_%s.png' % theta)
st.savefig('osc_region_theta_%s.eps' % theta)
def run_solvers_and_plot(solvers, rhs, T, dt, title=''):
Nt = int(round(T/float(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(rhs)
for solver in solvers:
solver.set_initial_condition([rhs.I, 0])
u, t = solver.solve(t_mesh)
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
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 RK4 on a much finer mesh
solver_exact.set_initial_condition([rhs.I, 0])
u_e, t_e = solver_exact.solve(t_fine)
plt.plot(t_e, u_e[:,0], '-') # avoid markers by spec. line type
legends.append('exact (RK4, dt=%g)' % (t_fine[1]-t_fine[0]))
plt.legend(legends, loc='upper right')
plt.xlabel('t'); plt.ylabel('u')
plt.title(title)
plotfilestem = '_'.join(legends)
plt.savefig('tmp_%s.png' % plotfilestem)
plt.savefig('tmp_%s.pdf' % plotfilestem)
def plot(self, include_exact=True, plt=None):
"""
Add solver.u curve to scitools plotting object plt,
and include the exact solution if include_exact is True.
This plot function can be called several times (if
the solver object has computed new solutions).
"""
if plt is None:
import scitools.std as plt
plt.plot(self.solver.t, self.solver.u, '--o')
plt.hold('on')
theta = self.solver.get('theta')
theta2name = {0: 'FE', 1: 'BE', 0.5: 'CN'}
name = theta2name.get(theta, '')
legends = ['numerical %s' % name]
if include_exact:
t_e = np.linspace(0, self.problem.get('T'), 1001)
u_e = self.problem.exact_solution(t_e)
plt.plot(t_e, u_e, 'b-')
legends.append('exact')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
dt = self.solver.get('dt')
plt.title('theta=%g, dt=%g' % (theta, dt))
plt.savefig('%s_%g.png' % (name, dt))
return plt
def visualize(u, t, title='', filename='tmp'):
plt.plot(t, u, 'b-')
plt.xlabel('t')
plt.ylabel('u')
dt = t[1] - t[0]
plt.title('dt=%g' % dt)
umin = 1.2*u.min(); umax = 1.2*u.max()
plt.axis([t[0], t[-1], umin, umax])
plt.title(title)
plt.savefig(filename + '.png')
plt.savefig(filename + '.pdf')
plt.show()
def plot(self, plt = None):
if plt is None:
import scitools.std as plt
plt.plot(self.solver.t, self.solver.v, 'b--o')
plt.hold("on")
plt.xlabel("t")
plt.ylabel("v")
plt.savefig("%g.png" % (self.solver.dt))
return plt
def visualize(list_of_curves, legends, title='', filename='tmp'):
"""Plot list of curves: (u, t)."""
for u, t in list_of_curves:
plt.plot(t, u)
plt.hold('on')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
plt.title(title)
plt.savefig(filename + '.png')
plt.savefig(filename + '.pdf')
plt.show()
def plot(self, plt=None):
if plt is None:
import scitools.std as plt
plt.plot(self.solver.t, self.solver.v, 'b--o')
plt.hold("on")
plt.xlabel("t")
plt.ylabel("v")
plt.savefig("%g.png" % (self.solver.dt))
return plt
def visualize(u, t, title='', filename='tmp'):
plt.plot(t, u, 'b-')
plt.xlabel('t')
plt.ylabel('u')
dt = t[1] - t[0]
plt.title('dt=%g' % dt)
umin = 1.2*u.min(); umax = 1.2*u.max()
plt.axis([t[0], t[-1], umin, umax])
plt.title(title)
plt.savefig(filename + '.png')
plt.savefig(filename + '.pdf')
plt.show()
def visualize(list_of_curves, legends, title='', filename='tmp'):
"""Plot list of curves: (u, t)."""
for u, t in list_of_curves:
plt.plot(t, u)
plt.hold('on')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
plt.title(title)
plt.savefig(filename + '.png')
plt.savefig(filename + '.pdf')
plt.show()
def visualize(u, t, title="", filename="tmp"):
plt.plot(t, u, "b-")
plt.xlabel("t")
plt.ylabel("u")
dt = t[1] - t[0]
plt.title("dt=%g" % dt)
umin = 1.2 * u.min()
umax = 1.2 * u.max()
plt.axis([t[0], t[-1], umin, umax])
plt.title(title)
plt.savefig(filename + ".png")
plt.savefig(filename + ".pdf")
plt.show()
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 run_solver_and_plot(solver, 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)
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
from vib_undamped import solver
u2, t2 = solver(I, w, dt, T)
plt.plot(t, u[:,1], 'r-', t2, u2, 'b-')
plt.legend(['Euler-Cromer', '2nd-order ODE'])
plt.xlabel('t'); plt.ylabel('u')
plt.savefig('tmp1.png'); plt.savefig('tmp1.pdf')
def logistic():
problem = Logistic(0.2, 1, 0.1)
T = 40
dt = 0.1
method = ForwardEuler(problem, dt)
method.set_initial_condition(problem.u0, 0)
u, t = method.solve(T)
# note that import * is not legal inside functions so we
# have to import each specific function:
from scitools.std import plot, hardcopy, xlabel, ylabel, title
plot(t, u)
xlabel('t'); ylabel('u')
title('Logistic growth: alpha=0.2, dt=%g, %d steps' \
% (dt, len(u)-1))
# compare with exponential growth:
#from scitools.std import hold, linspace, exp
#te = linspace(0, dt*N, N+1)
#ue = 0.1*exp(0.2*te)
#hold('on')
#plot(te, ue)
hardcopy('tmp.eps')
def logistic():
problem = Logistic(alpha=0.2, R=1, U0=0.1)
T = 40
solver = ForwardEuler(problem)
solver.set_initial_condition(problem.U0)
t = np.linspace(0, T, 401) # 400 intervals in [0,T]
u, t = solver.solve(t)
# Note that import * is not legal inside functions so we
# have to import each specific function.
from scitools.std import plot, hardcopy, xlabel, ylabel, title
plot(t, u)
xlabel('t'); ylabel('u')
title('Logistic growth: alpha=%s, R=%g, dt=%g' \
% (problem.alpha, problem.R, t[1]-t[0]))
# Compare with exponential growth
#from scitools.std import hold, linspace, exp
#te = linspace(0, dt*N, N+1)
#ue = 0.1*exp(0.2*te)
#hold('on')
#plot(te, ue)
hardcopy('tmp.eps')
def logistic():
problem = Logistic(alpha=0.2, R=1, U0=0.1)
T = 40
method = ForwardEuler(problem)
method.set_initial_condition(problem.U0)
t = np.linspace(0, T, 401) # 400 intervals in [0,T]
u, t = method.solve(t)
# Note that import * is not legal inside functions so we
# have to import each specific function.
from scitools.std import plot, hardcopy, xlabel, ylabel, title
plot(t, u)
xlabel('t'); ylabel('u')
title('Logistic growth: alpha=%s, R=%g, dt=%g' \
% (problem.alpha, problem.R, t[1]-t[0]))
# Compare with exponential growth
#from scitools.std import hold, linspace, exp
#te = linspace(0, dt*N, N+1)
#ue = 0.1*exp(0.2*te)
#hold('on')
#plot(te, ue)
hardcopy('tmp.eps')
def run_solver_and_plot(solver,
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)
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
from vib_undamped import solver
u2, t2 = solver(I, w, dt, T)
plt.plot(t, u[:, 1], 'r-', t2, u2, 'b-')
plt.legend(['Euler-Cromer', '2nd-order ODE'])
plt.xlabel('t')
plt.ylabel('u')
plt.savefig('tmp1.png')
plt.savefig('tmp1.pdf')
def viz(I, V, f, c, L, Nx, 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 corresponding 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, Nx, 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')
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))
# -*- coding: utf-8 -*-
"""
Created on Mon Oct 19 21:13:30 2015
@author: antalcides
"""
from ForwardEuleruni import *
from LogisticUni import *
problem = Logistic(0.2, 1, 0.1)
T = 40
dt = 0.1
method = ForwardEuler(problem, dt)
method.set_initial_condition(problem.u0, 0)
u, t = method.solve(T)
from scitools.std import plot, hardcopy, xlabel, ylabel, title
plot(t, u)
xlabel('t')
ylabel('u')
title('Logistic growth: alpha=0.2, dt=%g, %d steps' \
% (dt, len(u)-1))
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))
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 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))
T = num_periods * P
t = np.linspace(0, T, time_steps_per_period * num_periods + 1)
import odespy
def f(u, t, alpha):
# Note the sequence of unknowns: v, u (v=du/dt)
v, u = u
return [-alpha * np.sign(v) - u, v]
solver = odespy.RK4(f, f_args=[alpha])
solver.set_initial_condition([beta, 1]) # sequence must match f
uv, t = solver.solve(t)
u = uv[:, 1] # recall sequence in f: v, u
v = uv[:, 0]
return u, t
if __name__ == "__main__":
alpha_values = [0, 0.05, 0.1]
for alpha in alpha_values:
u, t = simulate(alpha, 0, 6, 60)
plt.plot(t, u)
plt.hold("on")
plt.legend([r"$\alpha=%g$" % alpha for alpha in alpha_values])
plt.xlabel(r"$\bar t$")
plt.ylabel(r"$\bar u$")
plt.savefig("tmp.png")
plt.savefig("tmp.pdf")
plt.show()
raw_input() # for scitools' matplotlib engine
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)
T = num_periods * P
t = np.linspace(0, T, time_steps_per_period * num_periods + 1)
import odespy
def f(u, t, alpha):
# Note the sequence of unknowns: v, u (v=du/dt)
v, u = u
return [-alpha * np.sign(v) - u, v]
solver = odespy.RK4(f, f_args=[alpha])
solver.set_initial_condition([beta, 1]) # sequence must match f
uv, t = solver.solve(t)
u = uv[:, 1] # recall sequence in f: v, u
v = uv[:, 0]
return u, t
if __name__ == '__main__':
alpha_values = [0, 0.05, 0.1]
for alpha in alpha_values:
u, t = simulate(alpha, 0, 6, 60)
plt.plot(t, u)
plt.hold('on')
plt.legend([r'$\alpha=%g$' % alpha for alpha in alpha_values])
plt.xlabel(r'$\bar t$')
plt.ylabel(r'$\bar u$')
plt.savefig('tmp.png')
plt.savefig('tmp.pdf')
plt.show()
raw_input() # for scitools' matplotlib engine
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))