In the MacCormick upwind scheme.

U^(n+1) = U^n - delta_t/delta_x*(f_(i+1) - f_i)

is it f_(i+1) - f_i = A_(i+0.5)*(U_(i+1) - U_i) where A = df/dU

or f = {rho*u, rho*u^2 + P...}^T

it appears that the second order upwind schemes doesn't work for the shock tube problem.

Predictor:

U^(n+1) = U^n - delta_t/delta_x*(f_(i+1)^n - f_i^n)

Corrector:

U^(n+2) = 0.5*(U^n + U^(n+1)) - delta_t/delta_x*(f_(i+1)^(n+1) - f_i^(n+1))

However the flux-splitting version does work. Is there anything i am doing wrong?

Implicit: