  The efficiency of several first-order numerical schemes for two-layer shallow
water equations are evaluated in this paper by considering different eigenvalue
solutions. Specifically, the accuracy and computational cost of numerical,
analytical, and approximated eigenvalue solvers are analysed when implemented
in Roe, Intermediate Field CaPturing (IFCP) and Polynomial Viscosity Matrix
(PVM) schemes. Several numerical tests are performed to examine the overall
efficiency of numerical schemes with different eigenvalue solvers when
computing two-layer shallow-water flows. The results show that analytical
solutions are much faster than numerical solvers, with a computational cost
closer to approximate expressions. Consequently, Roe schemes with analytical
solutions to the eigenstructure are much faster, with overall efficiency equal
to IFCP scheme. On the other hand, IFCP and PVM schemes with analytical
solutions to eigenvalues are found to be equally efficient as those with
approximated expressions. Analytical eigenvalues show slightly better results
when dealing with a larger density difference between the layers.
