Semi-Lagrangian scheme
Encyclopedia
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.
is given by
where
can be a scalar or vector field and 
is the velocity field. The first term on the right-hand side of the above equation is the local or Eulerian rate of change of 
and the second term is often called the advection term. Note that the Lagrangian rate of change is also known as the material derivative.
It can be shown that the equations governing atmospheric motion can be written in the Lagrangian form
where the components of the vector
are the (dependent) variables describing a parcel of air (such as velocity, pressure, temperature etc) and the function 
represents source and/or sink terms.
In a Lagrangian scheme, individual air parcels are traced but there are clearly certain drawbacks: the number of parcels can be very large indeed and it may often happen for a large number of parcels to cluster together, leaving relatively large regions of space completely empty. 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.
Numerical weather prediction
Numerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic...
models for the integration of the equations governing atmospheric motion. A Lagrangian description of a system (such as the atmosphere
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
) 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.
Some background
The Lagrangian rate of change of a quantity is given bywhere
can be a scalar or vector field and is the velocity field. The first term on the right-hand side of the above equation is the local or Eulerian rate of change of and the second term is often called the advection term. Note that the Lagrangian rate of change is also known as the material derivative.It can be shown that the equations governing atmospheric motion can be written in the Lagrangian form
where the components of the vector
are the (dependent) variables describing a parcel of air (such as velocity, pressure, temperature etc) and the function represents source and/or sink terms.In a Lagrangian scheme, individual air parcels are traced but there are clearly certain drawbacks: the number of parcels can be very large indeed and it may often happen for a large number of parcels to cluster together, leaving relatively large regions of space completely empty. 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
Smoothed particle hydrodynamics
Smoothed-particle hydrodynamics is a computational method used for simulating fluid flows. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography...
, 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.