Tested code with example 5.3 from 2. To my surprise, it works although in the book they state that it should be unstable (perhaps I did not use the same fluxes as they did). If EV is turned on, then solution changes, because of artificial diffusion. Also the CFL has to be lowered when EV is used.
I went back to 2. While reading it the following questions came up which I think should be answered before continuing any further:
Answer: I guess it is.
Answer: I think the splitting was made for convinience, because in the proof of stability that follows in the discussion, one could choose the \(\phi, \pi\) functions to account for an asymmetric splitting.