Finite difference of order 2 and order 4 on staggered grid

finite_diff_stag.py

import numpy as np
import matplotlib.pyplot as plt

x = 1.3
f = np.exp
fp = np.exp

errs2, errs4 = [], []
dxs = np.linspace(1e-4, 1e-1, 200)
for dx in dxs:
    fpx = fp(x)
    fpx_2 = (f(x+dx/2) - f(x-dx/2))/dx
    fpx_4 = (-1./24*f(x+dx*3/2) + 9/8*f(x+dx/2) - 9/8*f(x-dx/2) + 1./24*f(x-3*dx/2))/dx
    errs2.append(np.abs(fpx_2 - fpx))
    errs4.append(np.abs(fpx_4 - fpx))
plt.plot(dxs, errs2, label='order2')
plt.plot(dxs, errs4, label='order4')
plt.loglog()
plt.legend()
plt.grid();