6.1 Tbc Scalar Advection Equation
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In the case of the unforced scalar advection equation, it is helpful to perforrn the
Taylor series expansions about the point (xj , tn) because this isolates the errors in
the trajectory calculations from those generated by the interpolation of the tracer
field. Let 1/fd = 1/f(xj, r") ; then the error produced by linear interpolation is
Since the wind speed is constant, the backward trajectory is exact and 1/fj+1 = 1/Id.
This can be verified by expanding 1/fj+I in a Taylor series. Defining s = x j - xj ,
= 1/Id + Ilt 81/1 I + s 81/f I
J
8t d
8x d
+ (1lt)2 8
2 1/1 I + s Ilt 8
2 1/f I + s2 8
2 1/f I +.. .
(6.8)
2 8t 2 d
8t8x d 2 8x 2 d
'
and since s = U Si, the preceding reduces to
1/f n+1 .
J
= 1/fd + Ilt ( 8
-
8t
+ U - 8) 1/f I + -
Bx
(llt)2 (
- - + U -
8
8)2 I
1/f + ...
d
2
8t
8x
d
Substituting (6.7) and (6.9) into (6.6) yields
1/f'r
1
- [(I - cx)1/fj_p + cx1/f'j_P _I] <>< _cx(l - cx) (llx)2 8 2 1/f I
.(6.10)
If the Courant number is held constant as Ilt
0 and Ilx
0 and Ilt
0, the truncation
0, the crror is
crror is clearly O(llx). If Sx] Ilt
0 as Ilx
no larger than O(llx). It may appear that the semi-Lagrangian scheme could be
inconsistent in the limit I'::1t I Ilx
unity, cx = U I'::1t I Ilx, and using
0, but once the Courant number drops below
2
1/f
2 82 1/f
8
- - U -
8t 2 -
8x 2 '
the leading-ordcr truncation error reduces to
This is identical to the leading-ordcr truncation error in Eulerian upstream differencing (2.12). The global truncation error of the semi-Lagrangian scheme (6.5) is
therefore of first order in space and time .
Consistent with (6.10) , the preceding semi-Lagrangian scheme is exact whenever the Courant number is an integer, i.e., whenever the departure point exactly