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6. Semi-Lagrangian Methods
6.5 Eulerian or Semi-Lagrangian?
The relative efficiency ofEulerian and semi-Lagrangian methods can vary considerably between different physieal applications. Semi-Lagrangian methods require
more work per time step than their Eulerian counterparts because additional effort is required to compute the backward trajectories. Thus to be more efficient,
semi-Lagrangian methods must produce stable and accurate solutions using larger
time steps than comparable Eulerian schemes. The feasibility of taking a large
semi-Lagrangian time step is primarily deterrnined by two factors: the ease with
whieh an accurate trajectory can be computed that extends several grid intervals
upstream and the extent to whieh the frequency of the forcing in the Lagrangian
reference frame is reduced relative to that in the Eulerian frame .
One application where the semi-Lagrangian approach can have a distinct advantage is in the simulation of tracer transport in a smooth, slowly varying flow
field. If the tracer is conservative and the flow is inviscid, there is no forcing in the
Lagrangian reference frame, and the only factor deterrnining the time step is the
need to compute accurate backward trajectories. On the other hand, the highestfrequency forcing in the Eulerian reference frame, WE, is deterrnined by the product of the velocity and the largest wave number resolved on the spatial mesh.
Stability constraints (for explicit methods) and accuracy considerations require
the time step of the Eulerian scheme to be small enough that IWE.MI < 0(1).
It follows that if the spatial mesh required to adequately resolve the tracer field
is much finer than that required to define the flow field, the maximum time step
suitable for use with a semi-Lagrangian scheme can be much greater than that
suitable for an Eulerian method.
Semi-Lagrangian methods also have an advantage in solving problems in spherieal geometry. The most natural coordinate system for such problems is a latitudelongitude grid, but the convergence of the meridians near the poles greatly
decreases the east-west distance between grid points in the polar regions. In applications such as global atmospheric modeling, the spatial scale of the disturbances
near the poles is similar to that in middle latitudes, and the extra resolution in the
polar regions is not required to accurately capture the meteorologieally significant
phenomena. The maximum stable time step of an Eulerian method rnust, nevertheless, be small enough to ensure that the CFL condition defined with respect
to the wind speed is less than order unity in the polar regions . Semi -Lagrangian
methods are free from this time-step restriction, although some care is required in
order to accurately compute backward trajectories near the poles (Ritchie 1987;
Williamson and Rasch 1989; McDonald and Bates 1989).
In those problems where the frequency of the forcing in the Lagrangian reference frame is similar to that in the Eulerian frame, semi-Lagrangian schemes tend
to be at a disadvantage because accuracy considerations often require that both
methods use similar time steps. In some cases, such as flow over a topographie
barrier, forcing that is stationary in the Eulerian coordinate system is Doppler
shifted to a higher frequency in the Lagrangian coordinate frame (Pinty et al.
1995; Hereil and Laprise 1996). One situation in whieh the frequency of the forc-
6. Semi-Lagrangian Methods
6.5 Eulerian or Semi-Lagrangian?
The relative efficiency ofEulerian and semi-Lagrangian methods can vary considerably between different physieal applications. Semi-Lagrangian methods require
more work per time step than their Eulerian counterparts because additional effort is required to compute the backward trajectories. Thus to be more efficient,
semi-Lagrangian methods must produce stable and accurate solutions using larger
time steps than comparable Eulerian schemes. The feasibility of taking a large
semi-Lagrangian time step is primarily deterrnined by two factors: the ease with
whieh an accurate trajectory can be computed that extends several grid intervals
upstream and the extent to whieh the frequency of the forcing in the Lagrangian
reference frame is reduced relative to that in the Eulerian frame .
One application where the semi-Lagrangian approach can have a distinct advantage is in the simulation of tracer transport in a smooth, slowly varying flow
field. If the tracer is conservative and the flow is inviscid, there is no forcing in the
Lagrangian reference frame, and the only factor deterrnining the time step is the
need to compute accurate backward trajectories. On the other hand, the highestfrequency forcing in the Eulerian reference frame, WE, is deterrnined by the product of the velocity and the largest wave number resolved on the spatial mesh.
Stability constraints (for explicit methods) and accuracy considerations require
the time step of the Eulerian scheme to be small enough that IWE.MI < 0(1).
It follows that if the spatial mesh required to adequately resolve the tracer field
is much finer than that required to define the flow field, the maximum time step
suitable for use with a semi-Lagrangian scheme can be much greater than that
suitable for an Eulerian method.
Semi-Lagrangian methods also have an advantage in solving problems in spherieal geometry. The most natural coordinate system for such problems is a latitudelongitude grid, but the convergence of the meridians near the poles greatly
decreases the east-west distance between grid points in the polar regions. In applications such as global atmospheric modeling, the spatial scale of the disturbances
near the poles is similar to that in middle latitudes, and the extra resolution in the
polar regions is not required to accurately capture the meteorologieally significant
phenomena. The maximum stable time step of an Eulerian method rnust, nevertheless, be small enough to ensure that the CFL condition defined with respect
to the wind speed is less than order unity in the polar regions . Semi -Lagrangian
methods are free from this time-step restriction, although some care is required in
order to accurately compute backward trajectories near the poles (Ritchie 1987;
Williamson and Rasch 1989; McDonald and Bates 1989).
In those problems where the frequency of the forcing in the Lagrangian reference frame is similar to that in the Eulerian frame, semi-Lagrangian schemes tend
to be at a disadvantage because accuracy considerations often require that both
methods use similar time steps. In some cases, such as flow over a topographie
barrier, forcing that is stationary in the Eulerian coordinate system is Doppler
shifted to a higher frequency in the Lagrangian coordinate frame (Pinty et al.
1995; Hereil and Laprise 1996). One situation in whieh the frequency of the forc-