The two-regime conjecture for finite-difference approximations of viscous Burgers equation

Let $u$ be governed by the regularised Burgers equation with viscosity parameter $\varepsilon$, and let $\delta$ be the spatial mesh size. Write $c\in\{1,0,-1\}$ according as the discretisation is right-sided, centred, or left-sided. Two-regime Burgers discretisation conjecture. For $\varepsilon\ll1$, the finite-difference approximation has the following behaviour: if $\delta\ll\varepsilon$, it converges to the viscosity solution

$$\partial_tu=-\frac12\partial_x(u^2)+\frac c4;$$

neither of the stated instabilities occurs. If $\varepsilon\ll\delta$, viscosity and noise become irrelevant and the solution behaves like the corresponding inviscid Burgers approximation; before shocks, under the stated sign condition preventing ultraviolet blow-up, it is expected to converge to the same equation with $c=0$, while after shocks stability is expected only for upwind schemes, and the specified conservative discretisations have incorrect shock speeds. The source gives this as a heuristic statement based on scaling arguments and does not prove it.