# finding discretization error in Burger equation

I was reading the paper given in the link http://www.unige.ch/~hairer/preprints/parareal.pdf and I have a problem in understanding in page 10 for the Burger equation on implementing Parareal method for Burger equation. In section 4.4, the line "... fine time step $$\Delta t=1/100$$, which gives a numerical accuracy of $$4e-2$$."

For backward Euler truncation error is of $$O(\Delta t)$$ and thus for $$\Delta t=1/100$$ the discretization error is $$C/100$$, where C is constant. Now it seems that author calculate the constant $$C$$ and which is equal to $$4$$.

My question is how to accurately compute truncation error for $$\Delta t=1/100$$, the way authors reported?

Thanks.

• Perhaps see here for a more in-depth error analysis: arxiv.org/pdf/2203.08455.pdf At the end of the day, I believe they are using their own analysis of Sec 3 in the paper your link, rather than along the lines you are thinking (of infering $C$). Indeed such a naive truncation analysis is likely to go awry as you have two time steps interplaying (and an interation parameter). Their Sec 3 takes care of this. Of course it is possible they are doing something more crude as an initial error estimate, but the purpose of these papers is to get a more realistic error expectation. Jul 13 at 22:31

## 1 Answer

I believe what they do here is much simpler than what you are thinking. The backward Euler discretization error is not computed from a theoretical consideration but simply by comparing its output against some reference solution.

If you know the exact solution to the problem you are solving numerically, you can simply calculate the error without knowing anything about the error behaviour of your method.

What I could not find in the paper is what their reference solution is. They mention "Friedrich's N-wave" but I am not sure there is an analytic expression how this looks like.

Alternatively, they might have simply run a reference solution with a much smaller time step and taken this as the "true" solution. This is often done when an analytic solution is not available.

Either way, they should have mentioned this in the paper.

Note. In the quoted sentence, they talk about the discretization error of their fine implicit Euler integrator. Their Section III is purely about the convergence of the Parareal iteration, so not directly related to this issue. The reason they look at the fine integrator discretization error is that this gives a natural tolerance for when to stop the Parareal iteration.