6
Semi-Lagrangian Methods
Most of the fundamental equations in fluid dynamics can be derived from first
principles in either a Lagrangian form or an Eulerian form. Lagrangian equations
describe the evolution ofthe flow that would be observed following the motion of
an individual parcel of fluid. Eulerian equations describe the evolution that would
be observed at a fixed point in space (or at least at a fixed point in a coordinate
system such as the rotating Earth whose motion is independent of the fluid). If
S(x , t) represents the sources and sinks of a chemical tracer 1{!(x, r), the evolution
of the tracer in a one-dimensional flow field may be altematively expressed in
Lagrangian form as
d1{!
.
-=S,
dt
(6.1)
or in Eulerian form as
CJ1{! + u CJ1{! = S.
CJt
CJx
The mathematical equivalence of these two equations follows from the definition
of the total derivative,
d
CJ
dx CJ
- = - + - - ,
dt
and the definition of the velocity,
(Jt
dt CJx
dx
-=u.
dt
(6.2)
One strategy for the solution of (6.1) as an initial value problem would be to
choose a regularly spaced distribution of fluid parcels at the initial time, assign a
D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics