  First-order hydrostatic reconstruction (HR) schemes for shallow water
equations are highly diffusive whereas high-order schemes can produce
entropy-violating solutions. Our goal is to develop a flux correction with
maximum antidiffusive fluxes to obtain entropy solutions of shallow water
equations with variable bottom topography. For this purpose, we consider a
hybrid explicit HR scheme whose flux is a convex combination of first-order
Rusanov flux and high-order flux. The conditions under which the explicit
first-order HR scheme for shallow water equations satisfies the fully discrete
entropy inequality have been studied. The flux limiters for the hybrid scheme
are calculated from a corresponding optimization problem. Constraints for the
optimization problem consist of inequalities that are valid for the first-order
HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy
inequality with the proper numerical entropy flux to single out a physically
relevant solution to the shallow water equations. A nontrivial approximate
solution of the optimization problem yields expressions to compute the required
flux limiters. Numerical results of testing various HR schemes on different
benchmarks are presented.
