Consider the scalar conservation law $u_t+f(u)_x=0.$ I understand that Lax-Wendroff scheme is second order accurate, because of the way it is derived using Taylor series. However since this scheme is not TVD there are various methods obtained from Lax-Wendroff scheme by introducing a correction term/Limiters. For example minmod, superbee, van Leer etc.
Now I have the following doubts:
- Are these schemes second order accurate always? If so how to prove?
- How to prove that these schemes converge to Kruzkov entropy solutions? (They are not monotone anymore!! So the standard monotone argument may not work)
For the first part, these slope- or flux-limiter schemes are not second-order accurate anymore. In fact, their order of accuracy measured numerically is rather around 1.6-1.8. This becomes obvious after realising that they are a kind of compromise between first-order and second-order schemes, see e.g. the 2002 book by R.J. LeVeque.
For the second part, I may find some special cases where this property is proven in literature. To be updated.