High-order well-balanced and positivity-preserving finite-difference AWENO scheme with hydrostatic reconstruction for shallow water equations

The shallow water equations (SWEs) admit still water steady-state solutions in which the flux gradients are exactly balanced by the source term. Furthermore, the no-water dry areas make the numerical simulation of the SWEs more difficult since the water height often loses positivity near the dry areas. In this study, we design a high-order well-balanced and positivity-preserving finite-difference AWENO scheme for solving the SWEs with or without dry areas. In this numerical framework, the hydrostatic reconstruction (HR) method is used with two main advantages: 1) The well-balanced property is achieved using the HR method with arbitrary monotone fluxes and the reformulated source terms. 2) The first-order scheme with the Lax-Friedrichs (LF) flux and the HR method maintains the positivity of the water height with an appropriate time step. Therefore, a simple positivity-preserving limiter conjugating the high-order reconstructed flux with the first-order positivity-preserving LF flux is introduced. One- and two-dimensional classical examples with or without dry areas demonstrate that the proposed scheme is high-order, well-balanced, and positivity-preserving.