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.

翻译：一阶静水重构（HR）格式用于浅水方程时具有高度耗散性，而高阶格式可能产生违反熵的解。本文旨在开发一种具有最大反扩散通量的通量修正方法，以获得变地形条件下浅水方程的熵解。为此，我们考虑一种混合显式HR格式，其通量为一阶Rusanov通量与高阶通量的凸组合。已研究显式一阶HR格式满足全离散熵不等式的条件。混合格式的通量限制器通过相应的优化问题计算得出。优化问题的约束条件由适用于一阶HR格式并应用于混合格式的不等式构成。我们采用具有恰当数值熵通量的离散单元熵不等式，以筛选出浅水方程的物理解。优化问题的非平凡近似解可导出计算通量限制器的表达式。给出了不同基准算例下多种HR格式的数值结果。