A simple and efficient locally mass conserving semi-Lagrangian transport scheme

A new simple and accurate locally mass conserving semi-Lagrangian (LMCSL) scheme has been constructed. Mass conservation is obtained by introducing modified interpolation weights at the upstream departure points. Thereby the total mass given off by a given Eulerian grid point to all the surrounding semi-Lagrangian (SL) departure points is equal to the cell area represented by that grid point. The new scheme is equivalent to the cell-integrated semi-Lagrangian (CISL) transport schemes in the sense that divergence—via the weights—is determined by the trajectories and not by centred differences as in traditional SL-schemes.

The LMCSL scheme has been combined with the semi-implicit scheme in a shallow water model. Thereby a numerically stable and inherently mass conserving scheme permitting long time steps has been set up. Tests in plane horizontal geometry including topography give solutions very similar to those obtained with the traditional semi-implicit SL scheme. The well known mountain wave resonance problem appears to be reduced.

The increase in numerical cost of the new scheme relative to traditional SL models is small, particularly when there are several passive tracers, since the same weights are used for all tracers.