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_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_one_step():
"""
Проверяем методы Эйлера и Рунге-Кутты
"""
y0 = np.array([0., 1.])
t0 = 0
t1 = np.pi
ode = Harmonic(y0, 1, 1)
for dt in [0.1, 0.01]:
ts = np.arange(t0, t1 + dt, dt)
exact = ode[ts].T
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8, 4))
ax1.plot(ts, [e[0] for e in exact], 'k', label='Exact')
colors = 'rgbcmyk'
for i, method in enumerate([
ExplicitEulerMethod(),
ImplicitEulerMethod(),
RungeKuttaMethod(collection.rk4_coeffs),
RungeKuttaMethod(collection.dopri_coeffs),
]):
ode.clear_call_counter()
_, y = fix_step_integration(method, ode, y0, ts)
n_calls = ode.get_call_counter()
print(
f'One-step {method.name}: {len(y)-1} steps, {n_calls} function calls'
)
ax1.plot(ts, [_y[0] for _y in y],
f'{colors[i]}.--',
label=method.name)
ax2.plot(ts,
get_accuracy(exact, y),
f'{colors[i]}.--',
label=method.name)
ax1.legend(), ax1.set_title('y(t)')
ax2.legend(), ax2.set_title('accuracy')
fig.suptitle(f'test_one_step, dt={dt}')
fig.tight_layout()
plt.show()
def test_one_step():
"""
test Euler and RK methods
"""
y0 = np.array([0., 1.])
t0 = 0
t1 = np.pi / 2
dt = 0.1
f = Harmonic(y0, 1, 1)
ts = np.arange(t0, t1 + dt, dt)
exact = f[ts].T
_, (ax1, ax2) = plt.subplots(1, 2)
ax1.plot(ts, [e[0] for e in exact], 'k', label='Exact')
colors = 'rgbcmyk'
for i, method in enumerate([
ExplicitEulerMethod(),
ImplicitEulerMethod(),
RungeKuttaMethod(collection.rk4_coeffs),
RungeKuttaMethod(collection.dopri_coeffs),
]):
f.clear_call_counter()
_, y = fix_step_integration(method, f, y0, ts)
n_calls = f.get_call_counter()
print(
f'One-step {method.name}: {len(y)-1} steps, {n_calls} function calls'
)
ax1.plot(ts, [_y[0] for _y in y], f'{colors[i]}.--', label=method.name)
ax2.plot(ts,
get_accuracy(exact, y),
f'{colors[i]}.--',
label=method.name)
ax1.set_xlabel('t'), ax1.set_ylabel('y'), ax1.legend()
ax2.set_xlabel('t'), ax2.set_ylabel('accuracy'), ax2.legend()
plt.suptitle('test_one_step')
plt.show()
def test_one_step():
"""
test Euler and RK methods
"""
y0 = np.array([0., 1.])
t0 = 0
t1 = np.pi / 2
dt = 0.1
f = Harmonic(y0, 1, 1)
ts = np.arange(t0, t1 + dt, dt)
exact = f[ts].T
plt.figure()
plt.subplot(1, 2, 1)
plt.plot(ts, [e[0] for e in exact], 'k', label='Exact')
colors = 'rgbcmyk'
for i, method in enumerate([
ExplicitEulerMethod(),
ImplicitEulerMethod(),
RungeKuttaMethod(collection.rk4_coeffs),
RungeKuttaMethod(collection.dopri_coeffs),
]):
_, y = fix_step_integration(method, f, y0, ts)
print(f'len(Y): {len(y)}')
print(f'Function calls: {f.get_call_counter()}')
plt.subplot(1, 2, 1), plt.plot(ts, [_y[0] for _y in y],
f'{colors[i]}.--',
label=method.name)
plt.subplot(1, 2, 2), plt.plot(ts,
get_log_error(exact, y),
f'{colors[i]}.--',
label=method.name)
plt.subplot(1, 2, 1), plt.xlabel('t'), plt.ylabel('y'), plt.legend()
plt.subplot(1, 2, 2), plt.xlabel('t'), plt.ylabel('accuracy'), plt.legend()
plt.suptitle('test_one_step')
plt.show()