Even vs uneven spacing

Even vs uneven spacing#

Evenly spaced data#

As long as this data is evenly spaced with step size \(h\), we can use finite difference directly.

BUT: Be aware of the boundaries! The central difference is the superior choice for the interior points, but you will need forward and backward differences at the boundaries.

Example: Find the derivative of a discretized function#

Say we have data for a function \(y(x)\) that looks like a parabola. We measured it every \(h\) steps in the range [a,b], so we have data for \(x_0 = a\) to \(x_n = b\)

23.03.01-Finite-difference.png

x = np.array([0,1,2,3,4,5,6,7,8,9])
y = np.array([0.5, 1.48765432, 2.22839506, 2.72222222, 2.9691358, 2.9691358,
 2.72222222, 2.22839506, 1.48765432, 0.5 ])

plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('y vs x')
plt.show()

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[1], line 1
----> 1 x = np.array([0,1,2,3,4,5,6,7,8,9])
      2 y = np.array([0.5, 1.48765432, 2.22839506, 2.72222222, 2.9691358, 2.9691358,
      3  2.72222222, 2.22839506, 1.48765432, 0.5 ])
      5 plt.plot(x, y)

NameError: name 'np' is not defined

If we just applied central difference to this function would encounter problems at the endpoints.

The numpy function gradient identifies the endpoints and treats them with forward / backward difference:

dy = np.gradient(y,x)

plt.plot(x, dy)
plt.xlabel('x')
plt.ylabel('dy')
plt.title('dy vs x')
plt.show()

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[2], line 1
----> 1 dy = np.gradient(y,x)
      3 plt.plot(x, dy)
      4 plt.xlabel('x')

NameError: name 'np' is not defined

This looks good in the middle but the ends are funny…

What do you think?

dy2 = np.gradient(y,x, edge_order = 2)

plt.plot(x, dy)
plt.plot(x, dy2)
plt.xlabel('x')
plt.ylabel('dy')
plt.title('dy vs x')
plt.show()

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[3], line 1
----> 1 dy2 = np.gradient(y,x, edge_order = 2)
      3 plt.plot(x, dy)
      4 plt.plot(x, dy2)

NameError: name 'np' is not defined

Bingo! Using the higher order edge cases solved the problem!

Note, this is the foundation of the finite difference method for solving differential equations.

Unevenly spaced data#

When data is unevenly spaced we can always do first-order forward / backward difference (but not central difference. Why?!?). If we want higher order generally have to resort to a local polynomial fitting.

Recall the cubic Lagrange interpolation. We can take the (analytic) derivative and find:

\[ f'(x) = f(x_{i-1}) \frac{2x - x_i-x_{i+1}}{(x_{i-1}-x_i) (x_{i-1}-x_{i+1})} +f(x_{i}) \frac{2x - x_{i-1}-x_{i+1}}{(x_{i}-x_{i-1}) (x_{i}-x_{i+1})} + f(x_{i+1}) \frac{2x - x_{i-1}-x_{i}}{(x_{i+1}-x_{i-1}) (x_{i+1}-x_{i})} \]

which is of comparable accuracy to the centeral difference and recovers it if the points are equally spaced.

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