What is the origin of the spurious oscillations in the Crank-Nicolson scheme?

Question

I was reading about the Crank-Nicolson method, and it is often said that it can produce "spurious oscillations" or that this method is prone to "ringing", especially for large time step and stiff initial value.

I'm curious about that phenomena but I can't find any satisfying answer. I read on this website that this could be because the Crank-Nicolson scheme was not L-stable.

What is the definition of L-stable ? How this is related to the production of oscillations ?

I found in this paper a beginning of an explanation thanks to a Fourier transform (see eq. 9, p. 255), but I really can't understand why the frequency $\varphi$ is only between $-\pi$ and $\pi$, and why $\varphi \approx \pm \pi$ should correspond to "high frequencies". Can you help me with that, too ?

Finally, I would like to know if this spurious oscillations problem always appear with a Crank-Nicolson scheme, or does it depend on the equation you discretize ?

To sum up, my questions are :

1. What is the origin of the spurious oscillations when using a Crank-Nicolson scheme ?
2. Does it depend on the equation we consider ?
3. What is the L-stability and in what it is related with the oscillations ? (if it is)
4. Can you help me to understand eq. 9 of this paper, why $-\pi < \varphi < \pi$ and not $\varphi \in \mathbb{R}$ ?

Answer 1

There is something very basic that you should know about hyperbolic problems. Consider the most basic example $\partial_tu+a\partial_xu=0$ with a numerical marching scheme of the form $$u_j^{n+1}=\sum_kc_ku_{j+k}^n$$. This covers all explicit schemes, and all implicit schemes like Crank Nicolson also, if you begin by solving the tridiagonal system. It actually covers ALL numerical methods, of any kind, that are linear under addition and multiplication. Godunov's theorem, known since 1959, asserts that any method of that form that is better than first-order accurate will produce spurious oscillations. LeVeque's book will fill you in.

The important thing to stress is that this has little to do with stability, but is absolutely pervasive in CFD because the F stands for Fluid and fluids can distort a lot, so advection terms are very important, and there things that cannot be done, even in the most trivial cases.

The rather amazing conclusion is that even linear hyperbolic problem must be solved by nonlinear numerical methods to achieve reasonable accuracy. All successful methods involve an element of feedback between the developing solution and the numerical scheme.