We construct the DFLUS finite difference scheme for a hyperbolic 2x2 system of conservation laws which models a polymer flooding process in enhanced oil recovery. This scheme is of Godunov type and its construction is achieved without implementing wave interaction. It is based on previous work on scalar conservation laws with a flux function discontinuous in space. Indeed, in the finite difference scheme, each equation in the system is seen as a scalar equation in one unknown with a discontinuous flux in the other unknown. L∞​ stability and convergence of the scheme is studied. Numerical performance of this new scheme is presented and compared with other schemes.