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()