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Numerical methods challenge: Day 17

Nicolás Guarín-Zapata

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During October (2017) I will write a program per day for some well-known numerical methods in both Python and Julia. It is intended to be an exercise then don't expect the code to be good enough for real use. Also, I should mention that I have almost no experience with Julia, so it probably won't be idiomatic Julia but more Python-like Julia.

Euler method

Today we have the Euler method. Which is the simplest of Runge-Kutta methods, and was named after Leonhard Euler who used in the 18th century.

The method consist in making updates of the function using the slope value with the formula

\begin{equation*} y_{n + 1} = y_n + hf(t_n, y_n) \end{equation*}

Following are the codes.

Python

from __future__ import division, print_function
import numpy as np
import matplotlib.pyplot as plt


def euler(dydt, y0, t, args=()):
    ndof = len(y0)
    ntimes = len(t)
    y = np.zeros((ndof, ntimes))
    y[:, 0] = y0
    for cont in range(1, ntimes):
        h = t[cont] - t[cont - 1]
        y[:, cont] = y[:, cont - 1] + h*dydt(y[:, cont - 1], t[cont], *args)
    return y


def pend(y, t, b, c):
    theta, omega = y
    dydt = [omega, -b*omega - c*np.sin(theta)]
    return np.array(dydt)


b = 0.25
c = 5.0
y0 = [np.pi - 0.1, 0.0]
t = np.linspace(0, 10, 10001)
y = euler(pend, y0, t, args=(b, c))
plt.plot(t, y[0, :])
plt.plot(t, y[1, :])
plt.xlabel(r"$t$")
plt.legend([r"$\theta(t)$", r"$\omega(t)$"])
plt.show()

Julia

using PyPlot


function euler(dydt, y0, t; args=())
    ndof = length(y0)
    ntimes = length(t)
    y = zeros(ndof, ntimes)
    y[:, 1] = y0
    for cont = 2:ntimes
        h = t[cont] - t[cont - 1]
        y[:, cont] = y[:, cont - 1] + h*dydt(y[:, cont - 1], t[cont], args...)
    end
    return y
end


function pend(y, t, b, c)
    theta, omega = y
    dydt = [omega, -b*omega - c*sin(theta)]
    return dydt
end


b = 0.25
c = 5.0
y0 = [pi - 0.1, 0.0]
t = linspace(0, 10, 1001)
y = euler(pend, y0, t, args=(b, c))
plot(t, y[1, :])
plot(t, y[2, :])
xlabel(L"$t$")
legend([L"$\theta(t)$", L"$\omega(t)$"])
show()

In both cases the result is the following plot

Solution for a pendulum using Euler method.

Comparison Python/Julia

Regarding number of lines we have: 32 in Python and 33 in Julia. The comparison in execution time is done with %timeit magic command in IPython and @benchmark in Julia.

For Python:

%timeit euler(pend, y0, t, args=(b, c))

with result

100 loops, best of 3: 18.5 ms per loop

For Julia:

@benchmark euler(pend, y0, t, args=(b, c))

with result

BenchmarkTools.Trial:
  memory estimate:  648.33 KiB
  allocs estimate:  15473
  --------------
  minimum time:     366.236 μs (0.00% GC)
  median time:      399.615 μs (0.00% GC)
  mean time:        486.364 μs (16.96% GC)
  maximum time:     4.613 ms (80.26% GC)
  --------------
  samples:          10000
  evals/sample:     1

In this case, we can say that the Python code is roughly 40 times slower than Julia.

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