2.2 Stability and Convergence
45
Since I -
> 0 for all wave numbers except the trivial case k = 0, the
preceding inequality reduces to
which is identical to the condition (2.22) obtained using the energy method. As
discussed previously in connection with (2.22), this stability condition may be
expressed as
ctxt
0<-<1.
-
-
Inspect ion of (2.26) shows that the
wave grows most rapidly in any integration performed with an unstable value of u, Thus, as it "blows up," an unstable
solution becomes dominated by large-amplitude
waves. Most other finitedifference approximations to the advection equation exhibit the same tendency:
When solutions become unstable, they usually become contaminated by largeamplitude short waves. The upstream scheme is, nevertheless, unusual in that all
waves become unstable for the same critical value of u, In many other schemes,
such as the leapfrog-tirne centered-space formulation (2.91), there exist values
of u. for which only a few of the shorter wavelengths are unstable. One might
suppose that such nominally unstable values of u. could still be used in numerical integrations if the initial data were filtered to remove all amplitude from the
unstable finite Fourier components; however, even if the initial data have zero amplitude in the unstable modes, round-off error in the numerical computations will
excite the unstable modes and trigger the instability.
2.2.3 The Courant-Fredrichs-Lewy Condition
The basic idea of the Courant-Fredrichs-Lewy (CFL) condition is that the solution of a finite-difference equation must not be independent of the data that
determines the solution to the associated partial differential equation. The CFL
condition can be made more precise by defining the domain 0/influence of a point
(xo, to) as that region ofthe x-t plane where the solution to some particular partial
differential equation is influenced by the solution at (xo, to). A related concept, the
domain 0/dependence of a point (xo, to), is defined as the set of points containing
(xo, to) within their domains of influence. The domain of dependence of (xo, to)
will therefore consist of all points (x, t) at which the solution has some influence
on the solution at (xo, to). A similar concept applicable to the discretized problem is the numerical domain 0/dependence of a grid point (noßt ,
consists of the set of all nodes on the space-time grid (n St ,
value ofthe numerical solution influences the numerical solution at
which
at which the
The CFL condition requires that the numerical domain 0/dependence 0/a finite -
difference scheme include the domain 0/ dependence 0/ the associated partial
differential equation. Satisfaction of the CFL condition is a necessary condition
for stability, but is not sufficient to guarantee stability.
45
Since I -
> 0 for all wave numbers except the trivial case k = 0, the
preceding inequality reduces to
which is identical to the condition (2.22) obtained using the energy method. As
discussed previously in connection with (2.22), this stability condition may be
expressed as
ctxt
0<-<1.
-
-
Inspect ion of (2.26) shows that the
wave grows most rapidly in any integration performed with an unstable value of u, Thus, as it "blows up," an unstable
solution becomes dominated by large-amplitude
waves. Most other finitedifference approximations to the advection equation exhibit the same tendency:
When solutions become unstable, they usually become contaminated by largeamplitude short waves. The upstream scheme is, nevertheless, unusual in that all
waves become unstable for the same critical value of u, In many other schemes,
such as the leapfrog-tirne centered-space formulation (2.91), there exist values
of u. for which only a few of the shorter wavelengths are unstable. One might
suppose that such nominally unstable values of u. could still be used in numerical integrations if the initial data were filtered to remove all amplitude from the
unstable finite Fourier components; however, even if the initial data have zero amplitude in the unstable modes, round-off error in the numerical computations will
excite the unstable modes and trigger the instability.
2.2.3 The Courant-Fredrichs-Lewy Condition
The basic idea of the Courant-Fredrichs-Lewy (CFL) condition is that the solution of a finite-difference equation must not be independent of the data that
determines the solution to the associated partial differential equation. The CFL
condition can be made more precise by defining the domain 0/influence of a point
(xo, to) as that region ofthe x-t plane where the solution to some particular partial
differential equation is influenced by the solution at (xo, to). A related concept, the
domain 0/dependence of a point (xo, to), is defined as the set of points containing
(xo, to) within their domains of influence. The domain of dependence of (xo, to)
will therefore consist of all points (x, t) at which the solution has some influence
on the solution at (xo, to). A similar concept applicable to the discretized problem is the numerical domain 0/dependence of a grid point (noßt ,
consists of the set of all nodes on the space-time grid (n St ,
value ofthe numerical solution influences the numerical solution at
which
at which the
The CFL condition requires that the numerical domain 0/dependence 0/a finite -
difference scheme include the domain 0/ dependence 0/ the associated partial
differential equation. Satisfaction of the CFL condition is a necessary condition
for stability, but is not sufficient to guarantee stability.