Well-balanced Positivity Preserving Unstaggered Central And Hydrostatic Reconstruction Schemes

In this paper,the new unstaggered central scheme(NUCS),the modified hydrostatic reconstruction(MHR)scheme,the surface reconstruction(SR)scheme,and the interface hydrostatic reconstruction(IHR)scheme are mainly introduced to solve the initial boundary value problems of shallow water and related hyperbolic partial differential equations.The unstaggered central scheme and the SR,MHR,IHR schemes belong to the finite volume method.The SR,MHR and IHR schemes belong to a class of upwind schemes.The core difference between the unstaggered central scheme and the upwind scheme is that the upwind scheme requires approximate or exact Riemann solvers to define the numerical flux function across the cell interface and the unstaggered central scheme avoids using the Riemann solvers due to the use of staggered grids.To solve the shallow water equations,the numerical scheme needs to satisfy the well-balanced property,that is,to maintain the steady-state solution and to guarantee the water depth to be non-negative.The source term of shallow water equations makes it difficult to design a numerical scheme that satisfies the well-balanced property.Thanks to the physical and numerical constraints,the numerical scheme needs to guarantee the positivity of the water depth,otherwise the calculation process maybe break down and lose the conservation.Losing the well-balanced property,which leads to the numerical scheme that produces a large error and loses physical meaning.The shallow water equations satisfy a class of entropy conditions which are used to select the entropy solutions.One of the difficulties of the unstaggered central scheme in solving shallow water equations is the discretization of the source terms on the staggered cells.Based on the constant water level to discretize the source term,although the numerical scheme satisfies the hydrostatic equilibrium of deep water,the numerical scheme produces spurious oscillations when the computational domain contains wet/dry fronts.The unstaggered central scheme needs to deal with the discrete problem of source terms on staggered cells and avoid using staggered cells.Based on the hydrostatic reconstruction method,we discretized the source terms on the staggered cell and constructed the mapping between the water level and it's cell average such that the numerical scheme could preserve the hydrostatic balance at the dry-wet fronts.The NUCS scheme can be used to solve the Ripa system using the analogous method in solving the shallow water equations.The numerical scheme can also satisfy the well-balanced and positivity properties.The two-layer shallow water equations are not strictly hyperbolic,and the non-conserved product term appears in the source term,which makes it difficult to design the numerical scheme which is robust and efficient.We use the interface hydrostatic reconstruction method to define the water depth value of the interface of unstaggered cells such that the scheme can deal with non-conservative product terms.When solving shallow water equations with varying river width using the unstaggered central scheme,the difficulty of designing well-balanced property lies in the discretization of source terms.We rewrite the source terms and use hydrostatic reconstruction method to discrete the integral form of the source term,so that the numerical scheme can satisfy the well-balanced property.We adopt the method of ”draining time step” to satisfy the positivity of the numerical scheme.The unstaggered central scheme can be used to solve the shallow water equations with two layers of variable river width.We use the splitting method to solve the Saint-Venant-Exner equations.We use the unstaggered central scheme to discrete the hydrodynamic equations and the upwind scheme to discrete the morphodynamic equations.The numerical scheme satisfies the well-balanced property and can guarantee the water depth to be positivity.When the MHR scheme solves the shallow water equations,the core is to reconstruct the bottom topography for preserving the monotone property.The MHR scheme can be seen as the modified hydrostatic reconstruction.The MHR scheme is wellbalanced,positivity preserving and efficient for the large bottom discontinuity.When the SR scheme is used to solve the shallow water and related equations,the core is to reconstruct the value of free surface at the cell interface which is used to redefine the Riemann states.The SR scheme can accurately maintain the still water steady-state solution,even the computational domain contains wet-dry fronts.The SR scheme can also keep the water depth to be non-negative.The IHR scheme is used to solve the two-layer shallow water equations,the core is to reconstruct the lower layer height on the cell interface for dealing with the Kelvin-Helmholtz instabilities and well-balanced properties.In order to verify the well-balanced,positivity preserving and robust properties of numerical schemes,we give some examples of solving shallow water and related equations using the unstaggered central scheme and the upwind schemes.