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_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()
def test_multi_step():
"""
Проверяем методы Адамса
Q: сравните правые графики для обоих случаев и объясните разницу
"""
y0 = np.array([0., 1.])
t0 = 0
t1 = np.pi
dt = 0.1
f = Harmonic(y0, 1, 1)
ts = np.arange(t0, t1 + dt, dt)
exact = f[ts].T
for one_step_method in [
RungeKuttaMethod(collection.rk4_coeffs),
ExplicitEulerMethod(),
]:
fig, (ax1, ax2) = plt.subplots(1, 2)
ax1.plot(ts, [e[0] for e in exact], 'k', label='Exact')
for p, c in adams_coeffs.items():
f.clear_call_counter()
t_adams, y_adams = adams(f,
y0,
ts,
c,
one_step_method=one_step_method)
n_calls = f.get_call_counter()
print(
f'{p}-order multi-step with one-step {one_step_method.name}: {n_calls} function calls'
)
err = get_accuracy(exact, y_adams)
label = f"Adams's order {p}"
ax1.plot(t_adams, [y[0] for y in y_adams], '.--', label=label)
ax2.plot(t_adams, err, '.--', label=label)
ax1.legend(), ax1.set_title('y(t)')
ax2.legend(), ax2.set_title('accuracy')
fig.suptitle(
f'test_multi_step\none step method: {one_step_method.name}')
fig.tight_layout()
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_multi_step():
"""
test Adams method
Q: compare the right plot for both cases and explain the difference
"""
y0 = np.array([0., 1.])
t0 = 0
t1 = 1.
dt = 0.1
f = Harmonic(y0, 1, 1)
ts = np.arange(t0, t1 + dt, dt)
exact = f[ts].T
for one_step_method in [
RungeKuttaMethod(collection.rk4_coeffs),
ExplicitEulerMethod(),
]:
_, (ax1, ax2) = plt.subplots(1, 2)
ax1.plot(ts, [e[0] for e in exact], 'k', label='Exact')
for p, c in adams_coeffs.items():
f.clear_call_counter()
t_adams, y_adams = adams(f,
y0,
ts,
c,
one_step_method=one_step_method)
n_calls = f.get_call_counter()
print(
f'{p}-order multi-step with one-step {one_step_method.name}: {n_calls} function calls'
)
err = get_accuracy(exact, y_adams)
label = f"Adams's order {p}"
ax1.plot(t_adams, [y[0] for y in y_adams], '.--', label=label)
ax2.plot(t_adams, err, '.--', label=label)
ax1.set_xlabel('t'), ax1.set_ylabel('y'), ax1.legend()
ax2.set_xlabel('t'), ax2.set_ylabel('accuracy'), ax2.legend()
plt.suptitle(
f'test_multi_step\none step method: {one_step_method.name}')
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()