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- # FROM: "Comparing Numerical Methods for Ordinary Differential Equations"
- # T.E. Hull, W.H. Enright, B.M. Fellen and A.E. Sedgwidh
- # SIAM J. Numer. Anal. vol 9, no 4, December 1972, pp: 603-637
- def deriv_B1(y, x):
- return [2.*(y[0]-y[0]*y[1]), -(y[1]-y[0]*y[1])] # "growth of two conflicting populations"
- def deriv_B4(y, x):
- A = 1./np.sqrt(y[0]**2 + y[1]**2)
- return [-y[1] - A*y[0]*y[2], y[0] - A*y[1]*y[2], A*y[0]] # "integral surface of a torus"
- def deriv_C1(y, x):
- return [-y[0]] + [y[i]-y[i+1] for i in range(8)] + [y[8]] # a radioactive decay chain
- def deriv_D1toD5(y, x):
- A = -(y[0]**2 + y[1]**2)**-1.5
- return [y[2], y[3], A*y[0], A*y[1]] # dimensionless orbit equation
- deriv_D1, deriv_D5 = deriv_D1toD5, deriv_D1toD5
- def deriv_E1(y, x):
- return [y[1], -(y[1]/(x+1.0) + (1.0 - 0.25/(x+1.0)**2)*y[0])] # derived from Bessel's equation of order 1/2
- def deriv_E3(y, x):
- return [y[1], y[0]**3/6.0 - y[0] + 2.0*np.sin(2.78535*x)] # derived from Duffing's equation
- import numpy as np
- from scipy.integrate import odeint as ODEint
- import matplotlib.pyplot as plt
- import timeit
- y0_B1 = [1.0, 3.0]
- y0_B4 = [3.0, 0.0, 0.0]
- y0_C1 = [1.0] + [0.0 for i in range(9)]
- ep1, ep5 = 0.1, 0.9
- y0_D1 = [1.0-ep1, 0.0, 0.0, np.sqrt((1.0+ep1)/(1.0-ep1))]
- y0_D5 = [1.0-ep5, 0.0, 0.0, np.sqrt((1.0+ep5)/(1.0-ep5))]
- y0_E1 = [0.6713968071418030, 0.09540051444747446] # J(1/2, 1), Jprime(1/2, 1)
- y0_E3 = [0.0, 0.0]
- x = np.linspace(0, 20, 51)
- xa = np.linspace(0, 20, 2001)
- derivs = [deriv_B1, deriv_B4, deriv_C1, deriv_D1, deriv_D5, deriv_E3]
- names = ["deriv_B1", "deriv_B4", "deriv_C1", "deriv_D1", "deriv_D5", "deriv_E3"]
- y0s = [y0_B1, y0_B4, y0_C1, y0_D1, y0_D5, y0_E3]
- timeit_dict, answer_dict, info_dict = dict(), dict(), dict()
- ntimes = 10
- tols = [10.**-i for i in range(6, 14)]
- def F(): # low density of time points, no output for speed test
- ODEint(deriv, y0, x, rtol=tol, atol=tol)
- def Fa(): # hight density of time points, full output for plotting
- return ODEint(deriv, y0, xa, rtol=tol, atol=tol, full_output=True)
- for deriv, y0, name in zip(derivs, y0s, names):
- timez = [timeit.timeit(F, number=ntimes)/float(ntimes) for tol in tols]
- timeit_dict[name] = timez
- alist, dlist = zip(*[Fa() for tol in tols])
- answer_dict[name] = np.array([a.T for a in alist])
- info_dict[name] = dlist
- plt.figure(figsize=[10,6])
- for i, name in enumerate(names):
- plt.subplot(2, 3, i+1)
- for thing in answer_dict[name][-1]:
- plt.plot(xa, thing)
- plt.title(name[-2:], fontsize=16)
- plt.show()
- plt.figure(figsize=[10, 8])
- for i, name in enumerate(names):
- plt.subplot(2,3,i+1)
- a = answer_dict[name]
- a13, a10, a8 = a[-1], a[-4], a[-6]
- d10 = np.abs(a10-a13).max(axis=0)
- d8 = np.abs(a8 -a13).max(axis=0)
- plt.plot(xa, d10, label="tol(1E-10)-tol(1E-13)")
- plt.plot(xa, d8, label="tol(1E-08)-tol(1E-13)")
- plt.yscale('log')
- plt.ylim(1E-11, 1E-03)
- plt.title(name[-2:], fontsize=16)
- if i==3:
- plt.text(3, 1E-10, "1E-10 - 1E-13", fontsize=14)
- plt.text(2, 2E-05, "1E-08 - 1E-13", fontsize=14)
- plt.show()
- fs = 16
- plt.figure(figsize=[12,6])
- plt.subplot(1,3,1)
- for name in names:
- plt.plot(tols, timeit_dict[name])
- plt.title("timing results", fontsize=16)
- plt.xscale('log')
- plt.yscale('log')
- plt.text(1E-09, 5E-02, "D5", fontsize=fs)
- plt.text(1E-09, 4.5E-03, "C1", fontsize=fs)
- plt.subplot(1,3,2)
- for name in names:
- a = answer_dict[name]
- e = a[:-1] - a[-1]
- em = [np.abs(thing).max() for thing in e]
- plt.plot(tols[:-1], em)
- plt.title("max difference from smallest tol", fontsize=16)
- plt.xscale('log')
- plt.yscale('log')
- plt.xlim(min(tols), max(tols))
- plt.text(1E-09, 3E-03, "D5", fontsize=fs)
- plt.text(1E-09, 8E-11, "C1", fontsize=fs)
- plt.subplot(1,3,3)
- for name in names:
- nsteps = [d['nst'][-1] for d in info_dict[name]]
- plt.plot(tols, nsteps, label=name[-2:])
- plt.title("number of steps", fontsize=16)
- plt.xscale('log')
- plt.yscale('log')
- plt.ylim(3E+01, 3E+03)
- plt.legend(loc="upper right", shadow=False, fontsize="large")
- plt.text(2E-12, 2.3E+03, "D5", fontsize=fs)
- plt.text(2E-12, 1.5E+02, "C1", fontsize=fs)
- plt.show()
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