This is part of a larger problem regarding the Lax-Friedrichs scheme applied to the PDE $u_t+au_x=0$. I have that the consistency error of the scheme to leading order is
$$
C\Big (\Delta t + \frac{(\Delta x)^2}{\Delta t}\Big)
$$
and we have the CFL number $a\Delta t /\Delta x\leqslant 1$ for stability.
My question is whether we can write the consistency error as $\mathcal{O}((\Delta t)^p+(\Delta x)^q)$ for some $p$, $q$ as large as possible. The error that I have is of course conditional on how the spacings relate, as is the CFL condition, and using the latter, I know the consistency error is
$$
\geqslant \mathcal{O}(\Delta t + \Delta x)
$$
but I'm not sure how we could arrive at a consistency error of the given form.
Any insight would be great!
Best Answer
According to this post, one should find that the consistency error of the scheme to leading order is $$ \frac12 \left(\frac{\Delta x^2}{\Delta t} - a^2\Delta t\right) u_{xx} + \dots $$ One more assumption is needed to conclude, namely that enforcing a constant Courant number $\Gamma = a \Delta t/\Delta x$ such that $\Gamma \leqslant 1$. This way, we rewrite the local truncation error as $$ \frac{a \Delta x}2 \frac{1-\Gamma^2}{\Gamma} u_{xx} + \dots = O(\Delta x). $$