Finite differences and spectral methods

Take a periodic function sampled at $N$ equally spaced points, $x_j=jh$ with $h=2\pi /N$, and ask for its derivative at those points. The central difference $(f_{j+1}-f_{j-1})/2h$ looks at two neighbors. Its error is $\frac{h^2}{6}{f}^{\prime \prime \prime }$, so doubling $N$ divides it by 4. The spectral derivative fits the trigonometric polynomial through all $N$ samples and differentiates that exactly: transform, multiply mode $k$ by $ik$, transform back. Every sample contributes to every derivative value.

A smooth function

For $e^{\sin x}$, which is smooth and periodic, the Fourier coefficients fall off faster than any power of $k$, and so does the spectral derivative's error. The two lines for $e^{\sin x}$ in the figure show it: the spectral error reaches rounding by 32 points, while the central difference is still losing a factor of 4 per doubling.

A jump

Now add a unit step to $\sin x$, so that the function jumps at $x=0$ and $x=\pi$. Next to a jump neither method is right, and the largest error over all points grows with $N$ for both. Away from the jumps they behave differently. The central difference at a point $\pi /4$ or farther from a jump only sees $\sin x$, and its error falls like $h^2$ as before. The spectral derivative at that point still includes the jump: in the spectral differentiation matrix the weight of a sample at distance $d$ is $\frac{1}{2}\cot (d/2)$ in size, and that does not shrink as $N$ grows. The two lines marked far points show it: the central difference's error keeps falling, the spectral one stays near $0.7$.

What the two claims cover

The two claims, claim spectral-derivative ✓ checked and claim central-difference-near-a-jump ✓ checked, cover one smooth function and one function with jumps, both periodic, on equally spaced points. Functions on an interval that are not periodic, which spectral methods handle with Chebyshev points, and the cost of each method per time step, are not tested here.