Approximating Sin wave with euler method in python

Question

Hi I'm trying to use euler method in python to plot a sin wave function sin(A).

Using the following algorithm:

1. define f(t,y)

2. input t0 and y0 .

3. input step size, h and the number of steps, n .

4. for j from 1 to n do:

   4a. m=f(t0,y0)

   4b. y1=y0+h∗m

   4c. t1=t0+h

   4d. Print t1 and y1

   4e. t0=t1

   4f. y0=y1

   4g. end

in my case I am approximating the function sin(A) so my function is the deriviative of sin(A) which is cos(A).

I've implemented it in the code as below

def dSindt(A): 
    dSindt = cos(A) ;
    return dSindt; 



%matplotlib inline
import matplotlib.pyplot as plt

A0 = 0 
t0 = 0; 
tf = 3600
del_t = .1; 
num_steps = int((tf - t0)/del_t); 
A_mesh = [0]*(num_steps + 1);
time_mesh = [0]*(num_steps + 1); 

A_mesh[0] = A0;
time_mesh[0] = t0;

for i in range(num_steps):
    A_mesh[i+1] = A_mesh[i] + dTindt(A_mesh[i])*del_t 
    time_mesh[i+1] = time_mesh[i] + del_t; 

plt.plot(time_mesh,A_mesh,color='b');
plt.title('Approx. Sin Wave');
plt.xlabel('Time (min)');
plt.ylabel('A')

It seems like no matter what i do to the step size the derivative cos(A) heads towards zero but never gets to be negative. It has to be negative to make the sine wave function go down. so it can oscillate. My erroneuos results are pictures here:

I must be doing something really dumb, but i can't figure it out.

Any help is appreciated.

Answer 1

You are solving the ODE

y'(t) = cos(y(t))

which has an attracting stable point at y=pi/2 which you reach for t=10 for all graphical purposes, after that the solution is constant. As you got in your graph.

You need either a system

x' = -y
y' =  x

or on a limited interval

y' = sqrt(1-y^2)

Or in a simple integration

y'(t) = cos(t).