With introduction of flux-weighted concept into the compact scheme

a new flux-weighted compact scheme was presented by combining classical scheme and compact scheme.This scheme and a 2nd-order total variation diminishing(TVD) scheme were used to calculate one-dimensional square waves

combine waves and some Riemann problems

such as Sod problem and Shu problem

as well as one-dimensional steady shock.The results show that the new scheme has advantage over the 2nd-order TVDscheme in capturing various discontinuities or resolving various complex waves.The results solved by the new scheme are also highly consistent with exact solutions.