Semi-Lagrangian scheme

The Semi-Lagrangian scheme (SLS) is a numerical method that is widely used in numerical weather prediction models for the integration of the equations governing atmospheric motion. 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 range 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 Lagrangian rate of change of a quantity ${\displaystyle F}$ is given by

${\displaystyle {\frac {DF}{Dt}}={\frac {\partial F}{\partial t}}+(\mathbf {v} \cdot {\vec {\nabla }})F,}$

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