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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!

jcneek
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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 $$ \frac{a \Delta x}2 \frac{1-\Gamma^2}{\Gamma} u_{xx} + \dots = O(\Delta x). $$

EditPiAf
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  • And since you've enforced proportionality between $\Delta x$ and $\Delta t$, this is the same as $\mathcal{O}(\Delta x + \Delta t)$. Thank you for your answer! – jcneek Jan 13 '23 at 15:15