  In this study, the hyperbolic method is adopted to explore the flow field
states of incompressible flow in a four-sided lid-driven square cavity. In
particular, we focus on the flow bifurcation obtained at the critical Reynolds
number $R_e \simeq 130$. In the hyperbolic method, the diffusive term is
transformed into an hyperbolic one by introducing a diffusion flux term, which
is the solution of an additional equation. A classical Riemann-like solver with
a finite-volume discretization is thus employed for the full flux (splitted
into advective and diffusive parts), in order to solve the steady-state
incompressible Navier-Stokes equation. The incompressibility of the flow is
treated via the artificial pseudo-compressibility method. It is shown that our
numerical code is able to detect the bifurcation, by the analysis of the
residual term relaxation during the pseudo-time iteration procedure. Moreover,
depending on the combination choice of slope limiters for the two spatial
directions, our method is able to select the first or the second stable
solution among the double flow field state obtained when the Reynolds number is
higher than the critical value that is estimated to be $129.4$ in our study.
