# Euler Method Harmonic Oscillator

I have written very simple Python code to solve the simple harmonic oscillator using Euler method, but I am not sure if the program is correct or not. I would be very grateful if anyone can look at my code and suggest further improvements since I am very new to this programming thing.

import matplotlib.pyplot as plt
v=0.0   #initial velocity
h=0.01  #time step
x=5.0   #initial position

t=0.0
ta,xa=[],[]

while t<1.0:
ta.append(t)
xa.append(x)

v=v-(10.0/1.0)*x*h    #k=10.0, m=1.0
x=x+v*h
t=t+h
plt.figure()
plt.plot(ta,xa,'--')
plt.xlabel('$t(s)$')
plt.ylabel('$x(m)$')
plt.show()

• In what sense is this the "Euler" method? It just looks like a discrete-timestep simulation to me. – 200_success May 29 '17 at 2:09
• For dy/dx=f(x,y), we can find the value of y at x1=x0+h as y(x1)=y(x0)+hf(x0,y0)\$. Isn't this what the Euler's method precisely about ? – Roshan Shrestha May 29 '17 at 2:22
• I don't see what role does g play. Besides, h is really dt. That said, it is far from harmonic oscillator. – vnp May 29 '17 at 4:37
• Yes, it seems I did a mistake by calling variable g which is not needed for this one since I was doing free fall motion earlier. So, haven't I used the Euler method correctly in here. – Roshan Shrestha May 29 '17 at 5:50

The code is quite monolithic: calculation and presentation are separate concerns, and while I appreciate the context given by showing the presentation, it would be better to make clear that they're separate concerns by separating them with whitespace and commenting briefly.

Also on structure: separate constants (h) from variables (v, x, t).

    v=v-(10.0/1.0)*x*h    #k=10.0, m=1.0
x=x+v*h
t=t+h


Two things: firstly, what are k and m? It seems that they should be constants with comments explaining their physical significance.

Secondly, this isn't Euler's method. Euler's method is $\vec{x}_{n+1} = \vec{x}_n + hf(t_n, \vec{x}_n)$ where $f = \frac{d\vec{x}}{dt}$. Here $\vec{x} = (v, x)$ and $f(t, v, x) = (\frac{dv}{dt}, \frac{dx}{dt}) = (-\frac{k}{m}x, v)$. In other words, to be Euler's method you should update x according to the previous value of v, not the updated value of v. Since you're using Python, you can take advantage of simultaneous assignment:

    v,x=v-(k/m)*x*h,x+v*h
t=t+h


(As it happens your buggy implementation works better than Euler's method, but if it was intended to implement Euler's method then it's still technically buggy).

• Thanks @Taylor. I would definitely take that into consideration, and I should have updated x first. – Roshan Shrestha May 29 '17 at 10:34
• @RoshanShrestha, no, that would be equally wrong. You should update them simultaneously, which depending on the language may mean using temporary variables. – Peter Taylor May 29 '17 at 10:52
• But, I have used temporary variables for x,v and t outside the while loop. So, does it mean that if I order my code for Euler method x=x+vh v=v-(10.0/1.0)*xh will still be wrong ? – Roshan Shrestha May 29 '17 at 10:56
• @RoshanShrestha, I wouldn't call those temporary variables. Also, I've realised that the code is Python (I had initially misread and thought it was Matlab, which I don't know), so that means it supports simultaneous assignment. I've edited sample code into the answer. – Peter Taylor May 29 '17 at 11:22