def plot_pvt():
astate = analytic_spring_ode(times, state[-1])
dstate = list(state)
for t in times[1:]:
dstate.append(rk4(discrete_spring_ode, t, dt, dstate[-1]))
dstate = array(dstate)
agreement = astate[:,0] - dstate[:,0]
fig, ax = subplots()
agmark, = ax.plot(times, agreement, 'k--', lw=1, alpha=.7)
amark, = ax.plot(times, astate[:,0], 'b-', lw=6)
dmark, = ax.plot(times, dstate[:,0], 'r-', lw=2)
ax.legend( [amark, dmark, agmark],
[r'Analytic $\left(\mathcal{A}_t\right)$', r'Discrete $\left(\mathcal{D}_t\right)$', r'Agreement $\left(\mathcal{A}_t-\mathcal{D}_t\right)$'],
numpoints=1,
loc="lower right")
configure( ax=ax,
title=r"$k=1.0$ $\frac{1}{seconds^2}$; $m=1.0$ $gram(s)$; $v_0=0.0$ $\frac{meter(s)}{second}$",
xlabel="Time (seconds)\n$\\Delta t=" + str(dt) + "$",
ylabel="Spatial Displacement (meters)\n$x_0="+str(x0)+"$",
xbounds=(t0, tf), ybounds=(-1.,1.))
fig.suptitle("Simple Harmonic Oscillator", size=30)
fig.subplots_adjust(left=0.05, right=0.95, top=0.9, bottom=0.08)
def plot_samples_and_models(times, pos, vel, acc):
F1 = [array([pos[0], vel[0]])]
F2 = [array([pos[0], vel[0]])]
for t,dt,p,v in zip(times,dts,pos,vel)[1:]:
F1.append(rk4(falld1, t, dt, F1[-1]))
F2.append(rk4(falld2, t, dt, F2[-1]))
F1 = array(F1)
F2 = array(F2)
fig, axes = subplots(2)
pmark, = axes[0].plot(times, pos, 'b--o')
plmark, = axes[0].plot(times[:-2], F1[:,0], 'g--o')
pqmark, = axes[0].plot(times[:-2], F2[:,0], 'r--o')
axes[0].legend( [pmark, plmark, pqmark],
[r'$\left(t,p\right)$', r'Linear', r'Quadratic'],
numpoints=1,
loc="upper right")
configure( ax=axes[0],
title=r'Position vs Time $\left(v_t='+ str(vT)+ r'\right)$',
xlabel=r"Time $\left(seconds\right)$",
ylabel=r"Position $\left(meters\right)$",
xbounds=None, ybounds=(0,.5))
vmark, = axes[1].plot(times[:-2], vel, 'b--o')
vlmark, = axes[1].plot(times[:-2], F1[:,1], 'g--o')
vqmark, = axes[1].plot(times[:-2], F2[:,1], 'r--o')
axes[1].legend( [vmark, vlmark, vqmark],
[r'$\left(t,v\right)$', r'Linear', r'Quadratic'],
numpoints=1,
loc="upper right")
configure( ax=axes[1],
title=r'Velocity vs Time $\left(v_t='+str(vT)+r'\right)$',
xlabel=r"Time $\left(seconds\right)$",
ylabel=r"Velocity $\left(\frac{meters}{second}\right)$",
xbounds=None, ybounds=None)
fig.suptitle("Falling Coffee Filter", size=30)
fig.subplots_adjust(left=0.05, right=0.95, top=0.9, bottom=0.08)
show()
def plot_evt(dt, integral, suptitle='Simple Harmonic Oscillator'):
# Do this once.
a_state = analytic_spring_ode(times, state[-1])
a_energy = spring_energy(times, a_state)
d_state = list(state)
if integral != predict_correct:
for t in times[1:]:
d_state.append(integral(discrete_spring_ode, t, dt, d_state[-1]))
else:
d_times, d_state = predict_correct(discrete_spring_ode, list(times), dt, d_state)
d_state = array(d_state)
d_energy = spring_energy(times, d_state)
fig, ax = subplots()
amark, = ax.plot(times, a_energy, 'b-', lw=6)
if integral == predict_correct:
print len(times); print len(d_energy)
dmark, = ax.plot(d_times, d_energy[:-2], 'r-', lw=2)
else:
print len(times); print len(d_energy)
dmark, = ax.plot(times, d_energy, 'r-', lw=2)
ax.legend( [amark, dmark],
[r'Analytic $\left(\mathcal{A}_t\right)$', r'Discrete $\left(\mathcal{D}_t\right)$'],
numpoints=1,
loc="lower right")
configure( ax=ax,
title=r"$k=1.0$ $\frac{1}{seconds^2}$; $m=1.0$ $gram(s)$; $v_0=0.0$ $\frac{meter(s)}{second}$",
xlabel="Time (seconds)\n$\\Delta t=" + str(dt) + "$",
ylabel="Energy (joules)\n$E_0="+str(E0)+"$",
xbounds=None, ybounds=(0.49,0.51))
error = abs(E0 - d_energy[-1]) / E0
print error
fig.suptitle(suptitle, size=30)
fig.subplots_adjust(left=0.05, right=0.95, top=0.9, bottom=0.08)
def plot_pc():
a_state = analytic_spring_ode(times, state[-1])
pc_times, pc_state = predict_correct(discrete_spring_ode,times,dt,state)
fig, ax = subplots()
amark, = ax.plot(times, a_state[:,0], 'b-', lw=6)
pcmark, = ax.plot(pc_times, pc_state[1:-1,0], 'r-', lw=2)
ax.legend( [amark, pcmark],
[r'Analytic $\left(\mathcal{A}_t\right)$', r'Predictor-Corrector $\left(\mathcal{D}_t\right)$'],
numpoints=1,
loc="lower right")
configure( ax=ax,
title=r"$k=1.0$ $\frac{1}{seconds^2}$; $m=1.0$ $gram(s)$; $v_0=0.0$ $\frac{meter(s)}{second}$",
xlabel="Time (seconds)\n$\\Delta t=" + str(dt) + "$",
ylabel="Position (meters)\n$x_0="+str(x0)+"$",
xbounds=(t0,tf), ybounds=None)
fig.suptitle("Predictor Corrector", size=30)
fig.subplots_adjust(left=0.05, right=0.95, top=0.9, bottom=0.08)
def plot_acc_vs_vel(vel,acc):
fig, ax = plt.subplots()
avmark, = ax.plot(vel, acc, 'b--o')
start_avmark, = ax.plot(vel[0], acc[0], 'go', ms=10)
end_avmark, = ax.plot(vel[-1], acc[-1], 'ro', ms=10)
ax.legend( [avmark, start_avmark, end_avmark],
[r'$\left(v,a\right)$', r'First', r'Last'],
numpoints=1,
loc="lower right")
configure( ax=ax,
title="Acceleration vs Velocity",
xlabel=r"Velocity $\left(\frac{meters}{second}\right)$",
ylabel=r"Acceleration $\left(\frac{meters}{seconds^2}\right)$",
xbounds=None, ybounds=(-10,10))
fig.suptitle("Falling Coffee Filter", size=30)
fig.subplots_adjust(left=0.05, right=0.95, top=0.9, bottom=0.08)
show()
def plot_samples(times, pos, vel, acc):
fig, axes = subplots(2, 2)
ptmark, = axes[0,0].plot(times, pos, 'b--o')
start_ptmark, = axes[0,0].plot(times[0], pos[0], 'go', ms=10)
end_ptmark, = axes[0,0].plot(times[-1], pos[-1], 'ro', ms=10)
axes[0,0].legend( [ptmark, start_ptmark, end_ptmark],
[r'$\left(t,p\right)$', r'First', r'Last'],
numpoints=1,
loc="upper right")
configure( ax=axes[0,0],
title="Position vs Time",
xlabel=r"Time $\left(seconds\right)$",
ylabel=r"Position $\left(meters\right)$",
xbounds=None, ybounds=None)
vtmark, = axes[1,0].plot(times[1:-1], vel, 'b--o')
start_vtmark, = axes[1,0].plot(times[1], vel[0], 'go', ms=10)
end_vtmark, = axes[1,0].plot(times[-2], vel[-1], 'ro', ms=10)
axes[1,0].legend( [vtmark, start_vtmark, end_vtmark],
[r'$\left(t,v\right)$', r'First', r'Last'],
numpoints=1,
loc="upper right")
configure( ax=axes[1,0],
title="Velocity vs Time",
xlabel=r"Time $\left(seconds\right)$",
ylabel=r"Velocity $\left(\frac{meters}{second}\right)$",
xbounds=None, ybounds=None)
atmark, = axes[0,1].plot(times[1:-1], acc, 'b--o')
start_atmark, = axes[0,1].plot(times[1], acc[0], 'go', ms=10)
end_atmark, = axes[0,1].plot(times[-2], acc[-1], 'ro', ms=10)
axes[0,1].legend( [atmark, start_atmark, end_atmark],
[r'$\left(t,a\right)$', r'First', r'Last'],
numpoints=1,
loc="lower right")
configure( ax=axes[0,1],
title="Acceleration vs Time",
xlabel=r"Time $\left(seconds\right)$",
ylabel=r"Acceleration $\left(\frac{meters}{seconds^2}\right)$",
xbounds=None, ybounds=(-10,10))
avmark, = axes[1,1].plot(vel, acc, 'b--o')
start_avmark, = axes[1,1].plot(vel[0], acc[0], 'go', ms=10)
end_avmark, = axes[1,1].plot(vel[-1], acc[-1], 'ro', ms=10)
axes[1,1].legend( [avmark, start_avmark, end_avmark],
[r'$\left(v,a\right)$', r'First', r'Last'],
numpoints=1,
loc="lower right")
configure( ax=axes[1,1],
title="Acceleration vs Velocity",
xlabel=r"Velocity $\left(\frac{meters}{second}\right)$",
ylabel=r"Acceleration $\left(\frac{meters}{seconds^2}\right)$",
xbounds=None, ybounds=(-10,10))
fig.suptitle("Falling Coffee Filter", size=30)
fig.subplots_adjust(left=0.05, right=0.95, top=0.9, bottom=0.08)
show()
