def test_adaptive(ode, y0):
"""
Проверяем алгоритмы выбора шага
"""
t0, t1 = 0, 4 * np.pi
atol = 1e-6
rtol = 1e-3
tss = []
yss = []
methods = (
(ExplicitEulerMethod(), AdaptType.RUNGE),
(RungeKuttaMethod(coeffs.rk4_coeffs), AdaptType.RUNGE),
(EmbeddedRungeKuttaMethod(coeffs.dopri_coeffs), AdaptType.EMBEDDED),
)
for method, adapt_type in methods:
ode.clear_call_counter()
ts, ys = adaptive_step_integration(method=method,
ode=ode,
y_start=y0,
t_span=(t0, t1),
adapt_type=adapt_type,
atol=atol,
rtol=rtol)
print(f'{method.name} took {ode.get_call_counter()} function calls')
tss.append(np.array(ts))
yss.append(ys)
ts = np.array(sorted([t for ts in tss for t in ts]))
exact = ode[ts].T
y0 = np.array([y[0] for y in exact])
# plots
fig1, ax1 = plt.subplots(num='y(t)')
fig1.suptitle('test_adaptive: y(t)')
ax1.set_xlabel('t'), ax1.set_ylabel('y')
ax1.plot(ts, y0, 'ko-', label='exact')
fig2, ax2 = plt.subplots(num='dt(t)')
fig2.suptitle('test_adaptive: step sizes')
ax2.set_xlabel('t'), ax2.set_ylabel('dt')
fig3, ax3 = plt.subplots(num='dy(t)')
fig3.suptitle('test_adaptive: accuracies')
ax3.set_xlabel('t'), ax3.set_ylabel('accuracy')
for (m, _), ts, ys in zip(methods, tss, yss):
ax1.plot(ts, [y[0] for y in ys], '.', label=m.name)
ax2.plot(ts[:-1], ts[1:] - ts[:-1], '.-', label=m.name)
ax3.plot(ts, get_accuracy(ode[ts].T, ys), '.-', label=m.name)
ax1.legend()
ax2.legend()
ax3.legend()
plt.show()
def test_arenstorf():
"""
https://en.wikipedia.org/wiki/Richard_Arenstorf#The_Arenstorf_Orbit
https://commons.wikimedia.org/wiki/File:Arenstorf_Orbit.gif
Q: какие участки траектории наиболее быстрые?
"""
ode = Arenstorf()
t0, t1 = 0, 1 * ode.t_period
y0 = ode.y0
atol = 1e-6
rtol = 1e-3
tss = []
yss = []
methods = (
# (ExplicitEulerMethod(), AdaptType.RUNGE),
(RungeKuttaMethod(coeffs.rk4_coeffs), AdaptType.RUNGE),
(EmbeddedRungeKuttaMethod(coeffs.dopri_coeffs), AdaptType.EMBEDDED),
)
fig1, ax1 = plt.subplots(num='traj')
fig1.suptitle('Arenstorf orbit: trajectory')
ax1.set_xlabel('x1'), ax1.set_ylabel('x2')
fig2, ax2 = plt.subplots(num='dt(t)')
fig2.suptitle('Arenstorf orbit: step sizes')
ax2.set_xlabel('t'), ax2.set_ylabel('dt')
fig3, ax3 = plt.subplots(num='|f|')
fig3.suptitle('Arenstorf orbit: RHS analysis')
ax3.set_xlabel('t')
for method, adapt_type in methods:
ts, ys = adaptive_step_integration(method=method,
ode=ode,
y_start=y0,
t_span=(t0, t1),
adapt_type=adapt_type,
atol=atol,
rtol=rtol)
tss.append(np.array(ts))
yss.append(ys)
for (m, _), ts, ys in zip(methods, tss, yss):
ax1.plot([y[0] for y in ys], [y[1] for y in ys], ':', label=m.name)
ax2.plot(ts[:-1], ts[1:] - ts[:-1], '.-', label=m.name)
derivatives = [np.linalg.norm(ode(t, y)) for t, y in zip(ts, ys)]
# derivatives = [1/(np.linalg.norm(ode(t, y))) for t, y in zip(ts, ys)]
ax1.plot(0, 0, 'bo', label='Earth')
ax1.plot(1, 0, '.', color='grey', label='Moon')
ax3.plot(ts, derivatives, label='|f(t, y)|')
ax1.legend()
ax2.legend()
plt.show()
def test_stiff():
"""
Проверяем явные и неявные методы на жёсткой задаче
https://en.wikipedia.org/wiki/Van_der_Pol_oscillator
Q: почему даже метод Розенброка иногда уменьшает шаг почти до нуля?
"""
t0 = 0
t1 = 800 * np.pi
mu = 1000
y0 = np.array([2., 0.])
f = VanDerPol(y0, mu)
fig1, ax1 = plt.subplots()
fig2, (ax21, ax22) = plt.subplots(1, 2)
fig3, ax3 = plt.subplots()
colors = 'rgbcmyk'
for i, (method, adapt_type) in enumerate([
(ExplicitEulerMethod(), AdaptType.RUNGE),
(ImplicitEulerMethod(), AdaptType.RUNGE),
(EmbeddedRosenbrockMethod(coeffs.rosenbrock23_coeffs),
AdaptType.EMBEDDED),
]):
f.clear_call_counter()
ts, ys = adaptive_step_integration(method,
f,
y0, (t0, t1),
adapt_type=adapt_type,
atol=1e-6,
rtol=1e-3)
print(
f'{method.name}: {len(ts)-1} steps, {f.get_call_counter()} RHS calls'
)
ax1.plot([y[0] for y in ys], [y[1] for y in ys],
f'{colors[i]}.--',
label=method.name)
ax21.plot(ts, [y[0] for y in ys], f'{colors[i]}.--', label=method.name)
ax22.plot(ts, [y[1] for y in ys], f'{colors[i]}.--', label=method.name)
ax3.plot(ts[:-1],
np.array(ts[1:]) - np.array(ts[:-1]),
f'{colors[i]}.--',
label=method.name)
ax1.set_xlabel('x'), ax1.set_ylabel('y'), ax1.legend()
fig1.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, y(x)')
ax21.set_xlabel('t'), ax21.set_ylabel('x'), ax21.legend()
ax22.set_xlabel('t'), ax22.set_ylabel('y'), ax22.legend()
fig2.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, x(t)')
ax3.set_xlabel('t'), ax3.set_ylabel('dt'), ax3.legend()
fig3.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, dt(t)')
plt.show()
def test_stiff():
"""
test explicit vs implicit methods on a stiff problem
"""
t0 = 0
t1 = 800*np.pi
mu = 1000
y0 = np.array([2., 0.])
f = VanDerPol(y0, mu)
fig1, ax1 = plt.subplots()
fig2, (ax21, ax22) = plt.subplots(1, 2)
fig3, ax3 = plt.subplots()
colors = 'rgbcmyk'
for i, (method, adapt_type) in enumerate(
[
(ExplicitEulerMethod(), AdaptType.RUNGE),
(ImplicitEulerMethod(), AdaptType.RUNGE),
(EmbeddedRosenbrockMethod(coeffs.rosenbrock23_coeffs), AdaptType.EMBEDDED),
]
):
f.clear_call_counter()
ts, ys = adaptive_step_integration(method, f, y0, (t0, t1),
adapt_type=adapt_type,
atol=1e-6, rtol=1e-3)
print(f'{method.name}: {len(ts)-1} steps, {f.get_call_counter()} RHS calls')
ax1.plot([y[0] for y in ys],
[y[1] for y in ys],
f'{colors[i]}.--', label=method.name)
ax21.plot(ts,
[y[0] for y in ys],
f'{colors[i]}.--', label=method.name)
ax22.plot(ts,
[y[1] for y in ys],
f'{colors[i]}.--', label=method.name)
ax3.plot(ts[:-1],
np.array(ts[1:]) - np.array(ts[:-1]),
f'{colors[i]}.--', label=method.name)
ax1.set_xlabel('x'), ax1.set_ylabel('y'), ax1.legend()
fig1.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, y(x)')
ax21.set_xlabel('t'), ax21.set_ylabel('x'), ax21.legend()
ax22.set_xlabel('t'), ax22.set_ylabel('y'), ax22.legend()
fig2.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, x(t)')
ax3.set_xlabel('t'), ax3.set_ylabel('dt'), ax3.legend()
fig3.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, dt(t)')
plt.show()
def test_stiff():
"""
Проверяем явные и неявные методы на жёсткой задаче
https://en.wikipedia.org/wiki/Van_der_Pol_oscillator
Q: почему даже метод Розенброка иногда уменьшает шаг почти до нуля?
"""
t0 = 0
t1 = 2500
mu = 1000
y0 = np.array([2., 0.])
ode = VanDerPol(y0, mu)
fig, axs = plt.subplots(2, 2, figsize=(10, 8))
colors = 'rgbcmyk'
for i, (method, adapt_type) in enumerate([
(ExplicitEulerMethod(), AdaptType.RUNGE),
(ImplicitEulerMethod(), AdaptType.RUNGE),
(EmbeddedRosenbrockMethod(coeffs.rosenbrock23_coeffs),
AdaptType.EMBEDDED),
]):
ode.clear_call_counter()
ts, ys = adaptive_step_integration(method,
ode,
y0, (t0, t1),
adapt_type=adapt_type,
atol=1e-6,
rtol=1e-3)
print(
f'{method.name}: {len(ts)-1} steps, {ode.get_call_counter()} RHS calls'
)
axs[0, 0].plot([y[0] for y in ys], [y[1] for y in ys],
f'{colors[i]}.--',
label=method.name)
axs[0, 1].plot(ts[:-1],
np.diff(ts),
f'{colors[i]}.--',
label=method.name)
axs[1, 0].plot(ts, [y[0] for y in ys],
f'{colors[i]}.--',
label=method.name)
axs[1, 1].plot(ts, [y[1] for y in ys],
f'{colors[i]}.--',
label=method.name)
axs[0, 0].legend(), axs[0, 0].set_title('y(x)')
axs[0, 1].legend(), axs[0, 1].set_title('dt(t)')
axs[1, 0].legend(), axs[1, 0].set_title('x(t)')
axs[1, 1].legend(), axs[1, 1].set_title('y(t)')
fig.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}')
fig.tight_layout()
plt.show()
def test_arenstorf():
"""
https://en.wikipedia.org/wiki/Richard_Arenstorf#The_Arenstorf_Orbit
https://commons.wikimedia.org/wiki/File:Arenstorf_Orbit.gif
Q: which parts of the orbit are fastest?
"""
problem = Arenstorf()
t0, t1 = 0, 1 * problem.t_period
y0 = problem.y0
atol = 1e-6
rtol = 1e-3
tss = []
yss = []
methods = (
# (ExplicitEulerMethod(), AdaptType.RUNGE),
(RungeKuttaMethod(coeffs=collection.rk4_coeffs), AdaptType.RUNGE),
(EmbeddedRungeKuttaMethod(coeffs=collection.dopri_coeffs),
AdaptType.EMBEDDED),
)
plt.figure('traj'), plt.suptitle(
'Arenstorf orbit: trajectory'), plt.xlabel('x1'), plt.ylabel('x2')
plt.figure('dt(t)'), plt.suptitle(
'Arenstorf orbit: step sizes'), plt.xlabel('t'), plt.ylabel('dt')
for method, adapt_type in methods:
ts, ys = adaptive_step_integration(method=method,
func=problem,
y_start=y0,
t_span=(t0, t1),
adapt_type=adapt_type,
atol=atol,
rtol=rtol)
tss.append(np.array(ts))
yss.append(ys)
for (m, _), ts, ys in zip(methods, tss, yss):
plt.figure('traj'), plt.plot([y[0] for y in ys], [y[1] for y in ys],
':',
label=m.name)
plt.figure('dt(t)'), plt.plot(ts[:-1],
ts[1:] - ts[:-1],
'.-',
label=m.name)
plt.figure('traj')
plt.plot(0, 0, 'bo', label='Earth')
plt.plot(1, 0, '.', color='grey', label='Moon')
plt.legend()
plt.figure('dt(t)'), plt.legend()
plt.show()
def test_adaptive_order():
"""
Проверяем сходимость
Q: почему наклон линии (число в скобках) соответствует порядку метода?
"""
t0, t1 = 0, 2 * np.pi
y0 = np.array([1., 1.])
ode = Harmonic(y0, 1, 1)
methods = (
(ExplicitEulerMethod(), AdaptType.RUNGE),
(RungeKuttaMethod(coeffs.rk4_coeffs), AdaptType.RUNGE),
(RungeKuttaMethod(coeffs.dopri_coeffs), AdaptType.RUNGE),
(EmbeddedRungeKuttaMethod(coeffs.dopri_coeffs), AdaptType.EMBEDDED),
)
tols = 10.**-np.arange(3, 9)
plt.figure(figsize=(9, 6))
for i, (method, adapt_type) in enumerate(methods):
print(method.name)
fcs = []
errs = []
for tol in tols:
ode.clear_call_counter()
ts, ys = adaptive_step_integration(method=method,
ode=ode,
y_start=y0,
t_span=(t0, t1),
adapt_type=adapt_type,
atol=tol,
rtol=tol * 1e3)
err = np.linalg.norm(ys[-1] - ode[t1])
fc = ode.get_call_counter()
print(
f'{method.name} with {adapt_type.name}: {fc} RHS calls, err = {err:.5f}'
)
errs.append(err)
fcs.append(fc)
x = np.log10(fcs)
y = -np.log10(errs)
k, b = np.polyfit(x, y, 1)
plt.plot(x, k * x + b, 'k:')
plt.plot(x, y, 'p', label=f'{method.name} {adapt_type} ({k:.2f})')
plt.title('test_adaptive_order: check RHS evals')
plt.xlabel('log10(function_calls)')
plt.ylabel('accuracy')
plt.legend()
plt.show()
def test_stiff():
"""
test explicit vs implicit methods on a stiff problem
"""
t0 = 0
t1 = 8*np.pi
mu = 0
y0 = np.array([2., 0.])
f = VanDerPol(y0, mu)
colors = 'rgbcmyk'
for i, (method, adapt_type) in enumerate(
[
(ExplicitEulerMethod(), AdaptType.RUNGE),
(ImplicitEulerMethod(), AdaptType.RUNGE),
(RungeKuttaMethod(collection.rk4_coeffs), AdaptType.RUNGE),
(EmbeddedRosenbrockMethod(collection.rosenbrock23_coeffs), AdaptType.EMBEDDED),
]
):
f.clear_call_counter()
ts, ys = adaptive_step_integration(method, f, y0, (t0, t1), adapt_type=adapt_type, atol=1e-6, rtol=1e-3)
print(f'{method.name}: {len(ts)-1} steps, {f.get_call_counter()} RHS calls')
plt.figure(1)
plt.plot([y[0] for y in ys],
[y[1] for y in ys],
f'{colors[i]}.--', label=method.name)
plt.figure(2)
plt.subplot(1,2,1), plt.plot(ts, [y[0] for y in ys], f'{colors[i]}.--', label=method.name)
plt.subplot(1,2,2), plt.plot(ts, [y[1] for y in ys], f'{colors[i]}.--', label=method.name)
plt.figure(3)
plt.plot(ts[:-1],
np.array(ts[1:]) - np.array(ts[:-1]),
f'{colors[i]}.--', label=method.name)
plt.figure(1)
plt.xlabel('x'), plt.ylabel('y'), plt.legend()
plt.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, y(x)')
plt.figure(2)
plt.subplot(1,2,1), plt.xlabel('t'), plt.ylabel('x'), plt.legend()
plt.subplot(1,2,2), plt.xlabel('t'), plt.ylabel('y'), plt.legend()
plt.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, x(t)')
plt.figure(3)
plt.xlabel('t'), plt.ylabel('dt'), plt.legend()
plt.suptitle(f'test_stiff: Van der Pol, mu={mu:.2f}, dt(t)')
plt.show()
def test_adaptive(f, y0):
"""
test adaptive step algorithms
"""
t0, t1 = 0, 4*np.pi
atol = 1e-6
rtol = 1e-3
tss = []
yss = []
methods = (
(EulerMethod(), AdaptType.RUNGE),
(RungeKuttaMethod(coeffs=collection.rk4_coeffs), AdaptType.RUNGE),
(EmbeddedRungeKuttaMethod(coeffs=collection.dopri_coeffs), AdaptType.EMBEDDED),
)
for method, adapt_type in methods:
f.clear_call_counter()
ts, ys = adaptive_step_integration(method=method,
func=f, y_start=y0, t_span=(t0, t1),
adapt_type=adapt_type,
atol=atol, rtol=rtol)
print(f'{method.name} took {f.get_call_counter()} function calls')
tss.append(np.array(ts))
yss.append(ys)
ts = np.array(sorted([t for ts in tss for t in ts]))
exact = f[ts].T
y0 = np.array([y[0] for y in exact])
# plots
plt.figure('y(t)'), plt.suptitle('y(t)'), plt.xlabel('t'), plt.ylabel('y')
plt.plot(ts, y0, 'ko-', label='exact')
plt.figure('dt(t)'), plt.suptitle('step sizes'), plt.xlabel('t'), plt.ylabel('dt')
plt.figure('dy(t)'), plt.suptitle('accuracies'), plt.xlabel('t'), plt.ylabel('accuracy')
for (m, _), ts, ys in zip(methods, tss, yss):
plt.figure('y(t)'), plt.plot(ts, [y[0] for y in ys], '.', label=m.name)
plt.figure('dt(t)'), plt.plot(ts[:-1], ts[1:] - ts[:-1], '.-', label=m.name)
plt.figure('dy(t)'), plt.plot(ts, get_log_error(f[ts].T, ys), '.-', label=m.name)
plt.figure('y(t)'), plt.legend()
plt.figure('dt(t)'), plt.legend()
plt.show()
def test_adaptive_order():
"""
test adaptive algorithms convergence
"""
t0, t1 = 0, 2 * np.pi
y0 = np.array([1., 1.])
f = Harmonic(y0, 1, 1)
methods = (
(ExplicitEulerMethod(), AdaptType.RUNGE),
(RungeKuttaMethod(coeffs=collection.rk4_coeffs), AdaptType.RUNGE),
(EmbeddedRungeKuttaMethod(coeffs=collection.dopri_coeffs),
AdaptType.EMBEDDED),
)
tols = 10.**-np.arange(3, 9)
plt.figure()
for i, (method, adapt_type) in enumerate(methods):
print(method.name)
fcs = []
errs = []
for tol in tols:
f.clear_call_counter()
ts, ys = adaptive_step_integration(method=method,
func=f,
y_start=y0,
t_span=(t0, t1),
adapt_type=adapt_type,
atol=tol,
rtol=tol * 1e3)
err = np.linalg.norm(ys[-1] - f[t1])
fc = f.get_call_counter()
print(f'{method.name}: {fc} RHS calls, err = {err:.5f}')
errs.append(err)
fcs.append(fc)
x = np.log10(fcs)
y = -np.log10(errs)
k, b = np.polyfit(x, y, 1)
plt.plot(x, k * x + b, 'k:')
plt.plot(x, y, 'p', label=f'{method.name} ({k:.2f})')
plt.suptitle('test_adaptive_order: check RHS evals')
plt.xlabel('log10(function_calls)')
plt.ylabel('accuracy')
plt.legend()
plt.show()
