A Lagrangian description of a system (such as the atmosphere) focuses on following individual air parcels along their trajectories as opposed to the Eulerian description, which considers the rate of change of system variables fixed at a particular point in space.
A semi-Lagrangian scheme uses Eulerian framework but the discrete equations come from the Lagrangian perspective.
The first term on the right-hand side of the above equation is the local or Eulerian rate of change of
It can be shown that the equations governing atmospheric motion can be written in the Lagrangian form
are the (dependent) variables describing a parcel of air (such as velocity, pressure, temperature etc.)
Such voids can cause computational problems, e.g. when calculating spatial derivatives of various quantities.
There are ways round this, such as the technique known as Smoothed Particle Hydrodynamics, where a dependent variable is expressed in non-local form, i.e. as an integral of itself times a kernel function.
Semi-Lagrangian schemes avoid the problem of having regions of space essentially free of parcels.
Semi-Lagrangian schemes use a regular (Eulerian) grid, just like finite difference methods.