Numerical viscosity from the Crank-Nicolson method

Question

Consider the following nonlinear diffusion PDE \begin{equation}\frac{\partial U}{\partial t} = \frac{\partial^2}{\partial x^2}[\Phi(U)], \end{equation}

where $\Phi$ is a smooth nonmonotone function of $U$ (footnote 1). In the paper "Shocks in Nonlinear Diffusion" (1995, Appl. Math. Lett.), Witelski makes the suggestive remark that "for numerical solutions of the above using the conservative Crank-Nicolson scheme, there is a nonlinear fourth-order numerical viscosity introduced."

I don't have much experience with this scheme and he doesn't write down any formulas or citations to clarify this, so I assume it's common knowledge somewhere or else straightforward to see from the definition of the method itself. Can someone help me write this down or else direct me to a reference where this is written down? I am particularly interested to see how it arises that this extra term is really a scalar multiple of a fourth-order finite difference approximation to a fourth-order spatial/mixed derivative of $U$.

Footnote 1: Actually I'm interested in the case where $\Phi$ is a cubic with outer branches of positive slope (i.e. the diffusion goes negative within an intermediate regime in $U$). The right-hand side of the PDE can also be written as $[D(U)U_x]_x$ if that helps.

Answer 1

The CN method is the midpoint method for $$ ∂_tU=\frac{2}{h^2}(\cosh(h∂_x)-1)\Phi(U)=∂_x^2Φ(U)+\frac{h^2}{12}∂_x^4Φ(U)+... $$ where $e^{h∂_x}$ is the Taylor shift operator in space direction.

Instead of considering the second term in the last expression as error term you can consider it as additional PDE term. The exact solution of the resulting PDE $$ ∂_tU=∂_x^2Φ(U)+\frac{h^2}{12}∂_x^4Φ(U) $$ is closer to the numerical solution than the exact solution of the original PDE. This includes qualitative properties that are connected to the viscosity property.