Joint work with K. Ait Ameur (X-CMAP).
Finite volume schemes on staggered grids are popular among thermalhydraulics engineers for their correct low Mach number asymptotic expansion and the absence of checkerboard type spurious oscillations. However, they are generally non conservative, and their stability analysis has historically been performed with a heuristic approach and the tuning of numerical parameters. In this talk, we first investigate the linear \(L^2\)-stability of staggered schemes by analysing their numerical diffusion operator. For most classical staggered schemes, our analysis shows that the numerical diffusion is highly nonlinear and we are able to prove the linear stability only in the case of constant sign velocities. We then propose a new class of conservative linearly \(L^2\)-stable staggered schemes for averaged multiphase flow models on staggered grids. The schemes are based on a carefully chosen numerical diffusion operator and the proof of stability follows from the symmetrisation of the system. An important remark is that unlike Godunov type schemes on colocated grids, the numerical diffusion operator of a symmetric system is not symmetric. This property is fundamental to ensure precision and avoid spurious checkerboard modes oscillations for low Mach number flows. We give examples of such new conservative staggered schemes and present some numerical results showing the good behaviour of the method for 1D and 2D multiphase flows at both low and high Mach number.