# Is there a method to examine numerical diffusion for non-linear PDE?

I have a nasty non-linear partial differential equation. I wonder if there exists a method that would allow me to examine what numerical errors (like numerical diffusion or dispersion) are introduced by different approximation schemes. Name of the method or a link to a paper would be great.

EDIT: Equation $$\frac{\partial f}{\partial t}+\frac{\partial}{\partial x} \left(\sqrt{c-\frac{\partial f}{\partial x}} \right)=0$$ $c$ is a constant.

• It would be helpful if you wrote out the equations here or linked to them. – Bill Barth Jan 22 '14 at 14:29
• @BillBarth I have added it. – Misery Jan 22 '14 at 18:31
• Thanks, @Misery, but that's a formula, not an equation. Do you set that equal to zero? Also, do you have a constraint that $\partial f/\partial x < c$, or do you accept that this might blow up? – Bill Barth Jan 22 '14 at 18:34
• @BillBarth ooops! Sorry. – Misery Jan 22 '14 at 19:29
• @BillBarth constant $c>0$ and I do assume that derivative is less than $c$ – Misery Jan 23 '14 at 9:05

The usual way of evaluating the accuracy of a solution scheme is to compare a numerical solution with an analytically known one. If you can't find a solution to the exact problem you have here (as is likely, given that it's nonlinear), you can always construct one using the Method of Manufactured Solutions. There are many descriptions of this method, one of which is here.

To examine the numerical diffusion you need to do the standard consistency analysis in the space variable, that is, you insert the exact solution into your numerical scheme, expand terms of the form $f(x_i+h)$ into a Taylor series, and estimate the terms that you cut off in your approximation. The terms that are of the form $c_d \frac{\partial ^2 f}{\partial x^2}$ and that do not cancel out, are then responsible for numerical diffusion.

This is straight forward for discretizations of convection terms, but I am not sure what this will give for your square-root example.

Numerical dispersion is harder to analyse. It occurs when your discretization 'supports' so-called checkerboard modes, i.e. unwanted decoupling of the variables that become visible as oscillations in the numerical solution.

Both issues are well understood for upwind discretization (leading to numerical diffusion) and discretization via central differences (leading to numerical dispersion) in the Finite Differences or Finite Volumes approximation of convection-diffusion equations. Maybe, you can transfer some ideas from there to your particular case.