326
6. Semi-Lagrangian Methods
The velocities on the left side of the preceding carry a fluid parcel an integral
number of grid points over a time interval 2!i.t, so the left side of (6.50) can be
evaluated as a Lagrangian derivative without numerical error. The right side of
(6.50) can be integrated using a leapfrog time difference with the forcing evaluated at i t
1
12, which is the midpoint of a back trajectory computed with respect
to the velocity p!i.xl(2!i.t). Depending on whether p is even or odd, itl/2
will either coincide with a grid point or He halfway between two grid points . A
second-order finite-difference approximation to (6.50) can therefore be written in
the form
t/Jj+l -
2!i.t
= l-U'82xiflJ-P/2 + S(t/J'}_p/2) x if pis even;
-u'8Xt/J'}_p/2 + (S(t/J'}_P/2») if pis odd.
(6.51)
The stability of the constant-coefficient equivalent of the preceding scheme can
be easily investigated. Suppose that the source term is zero and the wind speed is
V ; substituting the discrete Fourier mode
t/J' ! = ei{kj 6 x - wn M )
J
into (6.51) and invoking the assumption that S = 0 yields the discrete-dispersion
relation
sin(w!i.t - k(V _ u')!i.tJ =
u'!i.t sin(k!i.x)
!i.x
2u ' At sin(k!i.xI2)
u
!i.x
if pis even;
(6.52)
if
I
°
plSO
dd •
By the choice of p,
I
so
u' !i.t I= Iv _p!i.x I
2!i.t!i.x
I !i.x
< 4
which implies that the right side of (6.52) is bounded by one-half, w is real, and
the scheme is stable independent of the value of Si ,
If the velocity is a function of space and time, the residual Courant number
Iu'!i.t1Sx I will vary as a function of x and t. Let the subscript "* " indicate that
the function is evaluated at the midpoint of the trajectory between x j and x j - p »
in which case
u'j_p/2
u* =
1
x
if pis even;
(6.53)
(U'j_P/2)
if pis odd.
According to (6.52), a necessary condition for the stability ofthe variable-velocity
algorithm is lu:!i.t1!i.x I < 4. This condition is not automatically satisfied unless