322 lines
7.7 KiB
Python
Executable File
322 lines
7.7 KiB
Python
Executable File
#!/usr/bin/env python3
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"""Ordinary Differential Equations"""
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import numpy as np
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# Integrations Schemes #
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########################
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# Single Step
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#------------
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def mk_phi_euler(f):
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""" Return a Phi computed for the Euler Method. """
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def phi_euler(x, y, h):
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return f(x[-1], y[-1])
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return phi_euler
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def mk_phi_euler_mod(f):
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""" Return a Phi computed for the Modified Euler (Collatz) Method. """
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def phi_euler_mod(x, y, h):
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return f(x[-1] + 0.5*h, y[-1] + 0.5*h*f(x[-1], y[-1]))
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return phi_euler_mod
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def mk_phi_heun(f):
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""" Return a Phi computed for the Heun Method. """
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def phi_heun(x, y, h):
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return (f(x[-1], y[-1]) + f(x[-1] + h, y[-1] + h * f(x[-1], y[-1])))/2
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return phi_heun
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def mk_phi_rk4(f):
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""" Return a Phi computed for the 4th order Runge-Kutta Method. """
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def phi_rk4(x, y, h):
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k1 = f(x[-1], y[-1])
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k2 = f(x[-1] + 0.5*h, y[-1] + 0.5*k1)
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k3 = f(x[-1] + 0.5*h, y[-1] + 0.5*h*k2)
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k4 = f(x[-1] + h, y[-1] + h*k3)
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return (k1 + 2*k2 + 2*k3 + k4)/6
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return phi_rk4
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# Multi Step
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#-----------
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def mk_phi_AB3(f, phi_short = None):
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steps = 3
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def phi_AB3(x, y, h):
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if len(x) < steps:
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if phi_short is None:
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raise ValueError("This function needs at least {} steps, {} given.".format(steps, len(x)))
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else:
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return phi_short(x, y, h)
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else:
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return ( 23*f(x[-1], y[-1]) - 16*f(x[-2], y[-2]) + 5*f(x[-3], y[-3]) )/12
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return phi_AB3
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def mk_phi_AB4(f, phi_short = None):
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steps = 4
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def phi_AB4(x, y, h):
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if len(x) < steps:
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if phi_short is None:
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raise ValueError("This function needs at least {} steps, {} given.".format(steps, len(x)))
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else:
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return phi_short(x, y, h)
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else:
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return ( 55*f(x[-1], y[-1]) - 59*f(x[-2], y[-2]) + 37*f(x[-3], y[-3]) -9*f(x[-4],y[-4]))/24
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return phi_AB4
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# Integrator #
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##############
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def integrator(x, y_0, phi, y_i = None ):
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x = np.asarray(x)
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if isinstance(y_0, (list,np.ndarray)):
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y_0 = np.asarray(y_0)
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else:
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y_0 = np.array([ y_0 ])
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N = len(x)
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M = len(y_0)
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y = np.zeros((N,M), dtype=np.float64)
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y[0] = y_0
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j = 0
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if y_i is not None:
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for i, _ in enumerate(y_i):
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y[i+1] = y_i[i]
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j = i
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h = x[1]-x[0]
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for i in range(j+1, N):
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y[i] = y[i-1] + h*phi(x[:i], y[:i], h)
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return y
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# Math Test Functions #
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#######################
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def ODEF(x, y):
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return y - x**2 + 1
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def ODEF_sol(x):
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return (x+1)**2 - 0.5*np.exp(x)
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# Testing #
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###########
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def test_integrator_singlestep():
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test_func = ODEF
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exact_sol = ODEF_sol
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N = 2e1
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# Domain on which to do the integration
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x = np.linspace(0, 1, N)
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# Generate Initial Value
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y_0 = 0.5
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schemes = [
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["Euler", mk_phi_euler],
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["Collatz", mk_phi_euler_mod],
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["Heun", mk_phi_heun],
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["RK4", mk_phi_rk4],
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]
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# Show Plot
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from matplotlib import pyplot
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pyplot.subplots()
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for name, func in schemes:
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pyplot.plot(x, integrator(x, y_0, func(test_func)), '--o', label=name)
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pyplot.plot(x, exact_sol(x), '-', label="Exact Solution")
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pyplot.xlabel("x")
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pyplot.ylabel("y")
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pyplot.legend()
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pyplot.show()
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def test_integrator_multistep():
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test_func = ODEF
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exact_sol = ODEF_sol
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N = 2e1
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# Domain on which to do the integration
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x = np.linspace(0, 1, N)
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# Calculate Initial Values using RK4
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y_0 = 0.5
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y_init = integrator(x[:4], y_0, mk_phi_rk4(test_func))
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#Name, func, steps
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multi_schemes = [
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["AB3", mk_phi_AB3, 3],
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["AB4", mk_phi_AB4, 4],
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]
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# Show Plot
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from matplotlib import pyplot
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pyplot.subplots()
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# Exact Solution
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pyplot.plot(x, exact_sol(x), '-', label="Exact Solution")
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# Plot the schemes
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for name, func, steps in multi_schemes:
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try:
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pyplot.plot(x, integrator(x, y_0, func(test_func), y_init[:steps]), '--o', label=name)
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except Exception as e:
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print(e)
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pass
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pyplot.xlabel("x")
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pyplot.ylabel("y")
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pyplot.legend()
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pyplot.show()
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def plot_integration_errors():
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test_func = ODEF
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exact_sol = ODEF_sol
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# Domain on which to do the integration
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x = np.arange(0, 2.001, step=0.02)
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# Calculate Initial Values using RK4
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y_0 = 0.5
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y_init = integrator(x[:4], y_0, mk_phi_rk4(test_func))
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single_schemes = [
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["Euler", mk_phi_euler],
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["Collatz", mk_phi_euler_mod],
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["Heun", mk_phi_heun],
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["RK4", mk_phi_rk4],
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]
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multi_schemes = [
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["AB3", mk_phi_AB3, 3],
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["AB4", mk_phi_AB4, 4],
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]
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# Show Plot
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from matplotlib import pyplot
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pyplot.subplots()
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# Pre calculate the exact solution
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exact = exact_sol(x)
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exact = np.reshape(exact, (-1,1))
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# Plot Single Step Schemes
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for name, func in single_schemes:
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try:
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pyplot.plot(x, np.abs(exact - integrator(x, y_0, func(test_func))), '--o', label=name)
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except Exception as e:
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print(e)
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pass
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# Plot Multi Step Schemes
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for name, func, steps in multi_schemes:
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try:
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pyplot.plot(x, np.abs(exact - integrator(x, y_0, func(test_func), y_init[:steps])), '--o', label=name)
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except Exception as e:
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print(e)
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pass
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pyplot.xlabel("x")
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pyplot.ylabel("absolute error $|\\bar{y} - y|$")
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pyplot.title("absolute error between an approach and the exact solution")
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pyplot.legend()
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pyplot.show()
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def accuracy_per_stepsize( debug=True ):
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stepsizes = np.asarray([1e-1, 1e-2, 1e-3, 1e-5])
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# define the problem to be solved
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test_func = lambda x,y: x
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exact_sol = lambda x: np.exp(x)
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# set domain and intial value
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x_min = 0
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x_max = 2
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y_0 = 1
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exact_end = exact_sol(x_max)
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# define the schemes to use
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schemes = [
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["Euler", mk_phi_euler, 1],
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["Collatz", mk_phi_euler_mod, 1],
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["Heun", mk_phi_heun, 1],
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["RK4", mk_phi_rk4, 1],
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["AB3", mk_phi_AB3, 3],
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["AB4", mk_phi_AB4, 4],
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]
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# get max steps needed for the multistep schemes
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max_steps = 0
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for _, _, steps in schemes:
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if steps > max_steps:
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max_steps = steps
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# pre calculate for multistep integrations
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y_init = np.zeros((len(stepsizes),1))# Note the dimensionality of y_0
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max_steps += 1
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for i, h in enumerate(stepsizes):
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x = x_min + np.linspace(0, h * max_steps, max_steps, True)
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# Note the dimensionality of y_0
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# TODO: fix up this ugliness
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z = integrator(x[:4], y_0, mk_phi_rk4(test_func))
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y_init[i] = z[1,:]
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# plot schemes
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from matplotlib import pyplot
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pyplot.subplots()
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for name, scheme, steps in schemes:
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diffs = np.zeros(len(stepsizes))
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if debug:
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print("Calculating {}".format(name))
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for i, h in enumerate(stepsizes):
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N = int(abs(x_max-x_min)/h)
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x = np.linspace(x_min, x_max, N, True)
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y = integrator(x, y_0, scheme(test_func), y_init[:steps])
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diffs[i] = np.abs(exact_end - [-1])
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pyplot.plot(stepsizes, diffs, '--o', label=name)
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pyplot.xlabel('Stepsize $h$')
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pyplot.ylabel('Absolute Error')
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pyplot.title('Absolute error at the end of integration')
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pyplot.legend()
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pyplot.show()
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def pendulum_problem():
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pass
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if __name__ == "__main__":
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np.random.seed(0)
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test_integrator_singlestep()
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#test_integrator_multistep()
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#plot_integration_errors()
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#accuracy_per_stepsize()
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