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.


1 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.

  • $\begingroup$ Thanks Lutz, this is great. A couple questions: 1) Do you have some reference where I can see how this is derived in some more detail? You mention elasticity, so I imagine I might be able to find something in the continuum mechanics/ fluids literature. 2) Does this argument extend if I were to add a reaction term $R(U)$ to my PDE? $\endgroup$
    – ithmath
    May 19, 2022 at 13:13
  • $\begingroup$ It was meant to refer to the stated property, for some reason this came out as elasticity where the original is viscosity. $\endgroup$ May 20, 2022 at 13:10
  • $\begingroup$ Sure. Let me ask another question: when you say that "the exact solution of the resulting PDE... is closer to the numerical solution than that of the original PDE," what does this mean exactly? In an attempt to clarify this, let $U_1(x,t)$ be a solution of the higher-order pde and $U_0(x,t)$ be a solution of the original pde. Is "closer" here measured by saying that $U_1(x_j,t_n)$ gives strictly higher-order error estimates than $U_0(x_j,t_n)$, when we use these solutions as data for the CN iteration step at a given grid point $(x_j,t_n)$? $\endgroup$
    – ithmath
    May 21, 2022 at 7:35

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