A CONSISTENT QUASI - SECOND ORDER STAGGERED SCHEME FOR THE TWO-DIMENSIONAL SHALLOW WATER EQUATIONS

R Herbin (1) , J.-C Latché (2) , Y Nasseri (1) , N Therme (3)

1 I2M - Institut de Mathématiques de Marseille
2 IRSN - Institut de Radioprotection et de Sûreté Nucléaire
3 CESTA - Centre d'études scientifiques et techniques d'Aquitaine

A quasi-second order scheme is developed to obtain approximate solutions of the shallow water equationswith bathymetry. The scheme is based on a staggered finite volume scheme for the space discretization:the scalar unknowns are located in the discretisation cells while the vector unknowns are located on theedges (in 2D) or faces (in 3D) of the mesh. A MUSCL-like interpolation for the discrete convectionoperators in the water height and momentum equations is performed in order to improve the precisionof the scheme. The time discretization is performed either by a first order segregated forward Eulerscheme in time or by the second order Heun scheme. Both schemes are shown to preserve the waterheight positivity under a CFL condition and an important state equilibrium known as the lake at rest.Using some recent Lax-Wendroff type results for staggered grids, these schemes are shown to be Lax-consistent with the weak formulation of the continuous equations; besides, the forward Euler schemeis shown to be consistent with a weak entropy inequality. Numerical results confirm the efficiency andaccuracy of the schemes.

Mots clés: AMS Subject Classification Primary 65M08; 75M12; Secondary 65M12; Finite volume; shallow water; staggered grid; MAC scheme

Domaines: Analyse numérique [math.NA]

R Herbin, J.-C Latché, Y Nasseri, N Therme. A CONSISTENT QUASI - SECOND ORDER STAGGERED SCHEME FOR THE TWO-DIMENSIONAL SHALLOW WATER EQUATIONS. 2021. ⟨hal-02940981v2⟩