The maximal order of semidiscrete schemes for quasilinear first order   partial differential equations
  We prove that a semidiscrete $(2r+1)$-point scheme for quasilinear first order PDE cannot attain an order higher than $2r$. Moreover, if the forward Euler fully discrete scheme obtained from the linearization about any constant state of the semidiscrete scheme is stable, then the upper bound for the order of the scheme is $2r-1$. This bound is attained for a wide range of schemes and equations. 
A. Baeza, P. Mulet, D. Zor\'io
math.NA