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I am trying to numerically solve a differential equation but I am having trouble getting the convergence test to run properly. The problem is as follows:


Consider an ODE

$$y'(t) \enspace = \enspace f(t,y) \quad , \qquad y(0) \enspace = \enspace y_0 \quad . \tag{1}$$

Let now $y^{h}$ be a numerical solution to $(1)$ with step size $h$. My numerical solver is of order $\mathcal{O}(h^n)$. As far as I understand, I can test convergence of the solution by verifying the limit

$$ \lim_{h \longrightarrow 0} \frac{ || y^{h/2} - y^{h} || }{ || y^{h/4} - y^{h/2} || } \enspace = \enspace 2^n \quad .$$

In order to keep the code for this question nice and simple and clean, I will use a standard Euler-Forward approach of order $\mathcal{O}(h^2)$ and do not care about efficiency etc. of the implementation itself. My working example will be the differential equation

$$ y'(t) \enspace = \enspace - \sin(t) \quad , \qquad y(0) = 1 $$

with known exact solution

$$ y(t) \enspace = \enspace \cos(t) $$

Here is the broken-down and simplified code in PYTHON (Note that I am using grid point number $N$ instead of step size $h$):

import numpy as np
import matplotlib.pyplot as plt


def Solve( N ) :

   y = np.zeros(N)
   y[0] = 1
   
   t = np.linspace( 0, 2*np.pi, num=N )
   h = t[1] - t[0]
   
   def func( x ) :
       return -np.sin(x)
   
   for i in range(1,len(t)) :
       y[i] = y[i-1] - h*np.sin(t[i-1])

   return y


def main() :
       N = 10
       for i in range( 10 ):   
               a = Solve( N*2**i )
               b = Solve( N*2**(i+1) )
               c = Solve( N*2**(i+2) ) 
       
               res = np.linalg.norm( b[::2] - a ) / np.linalg.norm( c[::2] - b )        

               plt.scatter( N*2**i, res )


main()

This should by what I stated above converge to $2^2 = 4$. However, it rather converges to some different value (probably $\sqrt{2}$, see picture) and I have no clue why. Any ideas?

enter image description here

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1 Answer 1

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  • The Euler method has global error order 1, not 2

  • To get step size $h=1/N$, you need $N$ steps, which gives $N+1$ nodes in the time subdivision. Currently you compare sequences with step sizes $\frac1{N-1}$, $\frac1{2N-1}$ and $\frac1{4N-1}$. These are not exact multiples, and so in the difference of the sequences you compare values at slightly different points.

  • You have indeed a built-in $\sqrt2$, as in the denominator the vector for the norm has twice as many as the one in the numerator. So the limit is indeed $\frac{2^1}{\sqrt2}$ The easiest way to repair this is to use the same points below as above

               res = np.linalg.norm( b[::2] - a ) / np.linalg.norm( c[::4] - b[::2] )
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  • $\begingroup$ Thank you for your answer! I accepted it gladly, because point 1 and point 3 were very enlightening. However, I dont quite understand point 2. I am taking the limit h -> 0, so this discrepancy shouldnt have any influence on the outcome of the convergence test. Was this just intended as a general comment? $\endgroup$
    – Octavius
    Nov 20 at 15:59
  • $\begingroup$ That is correct, the influence of this error goes to zero. It is just a question of style to avoid such systematic errors. $\endgroup$ Nov 20 at 16:16
  • 1
    $\begingroup$ Also, the time scales do not match. This is without consequence for order 1 methods, but can degrade the computation to order 1 also for higher order methods. $\endgroup$ Nov 20 at 16:59

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