WEAK CONSISTENCY OF NON LINEAR CONVECTION OPERATORS ON STAGGERED MESHES. APPLICATION TO A QUASI -SECOND ORDER STAGGERED SCHEME FOR THE TWO-DIMENSIONAL SHALLOW WATER EQUATIONS
2020
In this paper a Lax-Wendroff type result of consistency is given for convection operators on staggered meshes. It is applied to a class of second order finite volume schemes developed to obtain approximate solutions of the shallow water equations with bathymetry. These schemes are based on staggered grids for the space discretization: scalar and vector unknowns are defined on different meshes. MUSCL-like interpolations for the discrete convection operators in the water height and momentum equations are performed in order to improve the precision of the scheme. The time discretization is performed either by a first order segregated forward Euler scheme in time or by the second order Heun scheme. Both schemes are shown to preserve the water height positivity under a CFL condition and an important state equilibrium known as the lake at rest. Using the above mentioned staggered Lax-Wendroff type results, these schemes are shown to be Lax-consistent with the weak formulation of the continuous equations; besides, the forward Euler scheme is shown to be consistent with a weak entropy inequality. Numerical results confirm the efficiency and accuracy of the schemes. 2010
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