The scheme is -contracting

the scheme is -contracting,

Lemma 5.3.8 [96, page 166]. Consider the numerical scheme

If the numerical solution is total variation diminishing, and if the numerical solution is constant for large spatial indices, meaning that
there exists so that for all there exists
so that for all
and for all
then the scheme is monotonicity-preserving.
Proof It suffices to prove the conclusion for and non-decreasing . Note that for ,

Similarly, for . Next, we note that since is nondecreasing and ,

It follows that

which implies that for all .
Now that we have examined the inter-relationships of some of these nonlinear stability concepts, let us review their implications. First, a theorem due to Godunov shows that any linear monotonicity-preserving scheme is at best firstorder accurate. Lemma 5.3.8 shows that total variation diminishing schemes are monotonicity-preserving. Total variation diminishing schemes are convergent [ 96 , page 166], however, convergence to the entropy-satisfying solution is not guaranteed. Lemma 5.3.7 shows that an -contracting scheme is total variation diminishing. Lemma 5.2.10. shows that monotone schemes are -contracting, and the Harten-Hyman-Lax theorem 5.2.6 showed that monotone schemes converge to the entropy-satisfying solution, and are at best first-order accurate.
In summary, if we want to guarantee convergence to the entropy-satisfying solution then we can use a monotone scheme, provided that we are satisfied with first-order accuracy. If we want to guarantee convergence then we can use a total variation diminishing scheme, but such a scheme will be monotonicitypreserving and therefore first-order if it is linear. Thankfully, these are not our only options. Our results so far indicate that if we want to achieve better than first-order accuracy and simultaneously preserve monotonicity, then we cannot use a linear scheme.