# Semi-lagrangian method

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## semi-Lagrangian method

A physically motivated numerical technique for solving the advection (transport) equation.

In the advective form, DQ/Dt = 0, where D( )/Dt is the total derivative, and "mixing ratio" Q is an invariant along a flow trajectory. By tracing (along the flow trajectory) backward in time to the "departure point", the value at the "arrival point" can be obtained by an interpolation or a remapping procedure (between the fixed Eulerian grid and a time-dependent distorted Lagrangian grid). Because of the discrete particle–like approach, total mass is generally not conserved. To ensure mass conservation, the semi-Lagrangian method can be formulated with the conservative flux form. The singular particle discretization is replaced by a finite control-volume discretization. Analogous to an Eulerian flux-form formulation, total flux from the upstream direction, computed in the Lagrangian fashion, is used for the prediction of the volume-averaged quantity, which can be the density or a density-weighted mixing ratio–like quantity. Because the size of the time step is not limited by the CFL condition, both the advective-form and the flux-form semi-Lagrangian methods are computationally efficient, particularly in spherical geometry.

Staniforth, A., and J. Cote 1991. Semi-Lagrangian integration schemes for atmospheric models – A review. Mon. Wea. Rev.. 119. 2206–2223.

Lin, S.-J., and R. B. Rood 1996. Multidimensional flux-form semi-Lagrangian transport schemes. Mon. Wea. Rev.. 124. 2046–2070.

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