Geography Reference
In-Depth Information
In an actual prediction model the velocity field is predicted rather than known
as in this simple example. Thus, for a two-dimensional field
q (x, y, t
+
δt)
=
q (x
−
uδt, y
−
vδt, t)
(13.25)
where the velocity components at time t can be used to estimate the fields at t
δt;
once these are obtained they can be used to provide more accurate approximations
to the velocities on the right in (13.25). The right side in (13.25) is again estimated
by interpolation, which now must be carried out in two dimensions.
As shown in Fig. 13.1, the semi-Lagrangian scheme guarantees that the domain
of influence in the numerical solution corresponds to that of the physical problem.
Thus, the scheme is computational stability for time steps much longer than possi-
ble with an explicit Eulerian scheme. The semi-Lagrangian scheme also preserves
the values of conservative properties quite accurately and is particularly useful for
accurately advecting trace constituents such as water vapor.
+
13.3.6
Truncation Error
To be useful it is necessary not only that a numerical solution be stable, but that
it also provide an accurate approximation to the true solution. The difference
between the numerical solution to a finite difference equation and the solution to
the corresponding differential equation is called the
discretization error
. If this
error approaches zero with δt and δx, the solution is called
convergent
. The differ-
ence between a differential equation and the finite difference analog to it is referred
to as a
truncation error
because it arises from truncating the Taylor series approx-
imation to the derivatives. If this error approaches zero as δt and δx go to zero,
the scheme is called
consistent.
According to the Lax equivalence theorem,
1
if the
finite difference formulation satisfies the consistency condition, then stability is
the necessary and sufficient condition for convergence. Thus, if a finite difference
approximation is consistent and stable, one can be certain that the discretization
error will decrease as the difference intervals are decreased, even if it is not possible
to determine the error exactly.
Because numerical solutions are as a rule sought only when analytic solutions
are unavailable, it is usually not possible to determine the accuracy of a solution
directly. For the linear advection equation with constant advection speed consid-
ered in Section (13.3.3), it is possible, however, to compare the solutions of the
finite difference equation (13.8) and the original differential equation (13.6). We
can then use this example to investigate the accuracy of the difference method
introduced above.
From the above discussion we already can conclude that the magnitude of the
truncation error in the present case will be of order δx
2
and δt
2
. It is possible
1
See Richtmyer and Morton (1967).
