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Discrete error and convergence

I was asked a question today about a typical way to present the error between a numerical scheme and an exact solution, and the convergence of the method. I will demonstate one method that is typically used based on the $L_2$-norm of the error.

Consider the ODE

\begin{equation} \frac{\rm{d}x}{\rm{d}t} = x(t) \quad x(0) = 1 \end{equation}

which has the analytic solution

\begin{equation} x(t) = {\rm e}^t \end{equation}

I'm too lazy to code up anything much more sophisticated, so let's use an Euler explicit finite difference solution for $0 < t < 100$. We'll write a function that computes the $L_2$ norm of the error between the discrete solution and the exact solution. The formula for the norm is

$$ \Vert error \Vert_{L_2} = \sqrt{ \frac{1}{N} \sum_{i=1}^N(discrete(x_i) - exact(x_i))^2} $$

In [1]:
import numpy as np

def finite_diff_solution_err(N):
       Compute explicit finite difference solution and L2 error norm
       input: N - number of time steps
       output: normed error
    t = np.linspace(0., 100., num=N)
    x = np.zeros_like(t)
    n = np.arange(N)
    x[0] = 1.0
    x[1:] = (1.0 + (t[1:] - t[:-1])) ** (n[1:]) * x[0]
    exact = np.exp(t)
    err = np.sqrt(np.sum((x[:] - exact[:]) ** 2.0) / N)
    return err

Now let's solve our problem for increasing degrees-of-freedom

In [2]:
dofs = np.array([10, 100, 1000, 10000, 100000])

errs = [finite_diff_solution_err(i) for i in dofs]

Finally, we plot the results as a function of step size $h$. If we fit a straight line to the data, we get an estimate of the convergence rate of the method, so we'll do that as well.

In [3]:
%matplotlib inline
import matplotlib.pyplot as plt

#Fit a straight line
coefs = np.polyfit(np.log10(1.0 / dofs), np.log10(errs), 1)
y = 10 ** (coefs[0] * np.log10(1.0 / dofs) + coefs[1])
plt.loglog(1.0 / dofs, y, 'b-')
plt.loglog(1.0 / dofs, errs, 'b^')
plt.xlabel("$\log_{10} h$")
plt.ylabel("$\log_{10} \Vert error \Vert_{L_2}$");
No description has been provided for this image

The first term below is the slope of the least-square fit staight line which shows the convergence rate.

In [4]:
array([  0.45172826,  43.45975537])


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