Geography Reference
In-Depth Information
Fig. 13.1
Grid in x
−
t space showing the domain of dependence of the explicit finite-difference
solution at m
=
3 and s
=
2 for the one-dimensional linear advection equation. Solid circles
show grid points. The sloping line is a characteristic curve along which q(x, t)
=
q(0, 0),
and the cross shows an interpolated point for the semi-Lagrangian differencing scheme. In
this example the leapfrog scheme is unstable because the finite difference solution at point
A does not depend on q(0, 0).
The centered difference scheme for the advection equation (13.8) is an example
of an
explicit
time differencing scheme. In an explicit difference scheme, the
value of the predicted field at a given grid point for time step s
1 depends
only on the known values of the field at previous time steps. (In the case of the
leapfrog method the fields at time step s and s
+
1 are utilized.) The difference
equation can then be solved simply by marching through the grid and obtaining the
solution for each point in turn. The explicit leapfrog scheme is thus simple to solve.
However, as shown in the next section, it has disadvantages in that it introduces a
spurious “computational” mode and has stringent requirements on the maximum
value permitted for the Courant number. There are a number of alternative explicit
schemes that do not introduce a computational mode (e.g., see Problem 13.3), but
these still require that the Courant number be sufficiently small.
−
13.3.3 Computational Stability
Experience shows that solutions to finite difference approximations such as (13.8)
will not always resemble solutions to the original differential equations even when
the finite difference increments in space and time are very small. It turns out that
the character of the solutions depends critically on the
computational stability
of the difference equations. If the difference equations are not stable, numerical
solutions will exhibit exponential growth even when, as in the case of the linear
