144
6 . Methods for Unsteady Problems
The parameter d is the ratio of time step At to the characteristic diffusion
time p ( A x ) 2 / r , which is roughly the time required for a disturbance to be
transmitted by diffusion over a distance Ax. The second quantity (c) is the
ratio of time step At to the characteristic convection time, u/Ax, the time
required for a disturbance to be convected a distance Ax. This ratio is called
Courant number and is one of the key parameters in computational fluid
dynamics.
If 4 were the temperature (a possibility), Eq. (6.25) would need to satisfy
several conditions. By virtue of diffusion, an increase in temperature at any
of the three points xi-l, xi and xi+l at the old time level should increase
the temperature at point xi at the new time level. The same can be said
for the points xipl and xi with respect to convection, assuming u > 0. If 4
represents the concentration of a substance, it should not be negative.
The possibility that some of the coefficients of 4;-' and 4
1
;
;
in Eq. (6.25)
can become negative should alert us to possible trouble and demands a more
detailed analysis. To the extent possible, we want this analysis to mimic
the one used for ordinary differential equations. A simple way to do this was
invented by von Neumannfor whom the method is named. He argued that the
boundary conditions are rarely the cause of problems (there are exceptions
that are not relevant here) so why not ignore them altogether? If that is
done, the analysis is simplified. Since this method of analysis can be applied
to essentially all of the methods discussed in this chapter, we shall describe
it in a little detail; for further details, see the book by Strikwerda (1983).
In essence the idea can be arrived at as follows. The set of Eqs. (6.25) can
be written in matrix form:
@n+l = -44" ,
(6.27)
where the elements of the tridiagonal matrix A can be derived by inspection
of Eq. (6.25). This equation gives the solution at the new step in terms of
the solution at the previous step. The solution at t,,+l can thus be obtained
by repetitive multiplication of the initial solution 4' by the matrix A. The
question is: do the differences between solutions at successive time steps (for
non-varying boundary conditions), measured in any convenient way, increase,
decrease, or stay the same as n is increased? For example, one measure is the
norm:
The differential equation requires this quantity to decrease with time through
the action of dissipation. Eventually, a steady-state solution will be obtained
if the boundary conditions do not vary. Naturally, we would like the numerical
method to preserve this property of the exact equations.
This issue is closely connected with the eigenvalues of the matrix A. If
some of them are greater than 1, it is not difficult to show that e will grow
6 . Methods for Unsteady Problems
The parameter d is the ratio of time step At to the characteristic diffusion
time p ( A x ) 2 / r , which is roughly the time required for a disturbance to be
transmitted by diffusion over a distance Ax. The second quantity (c) is the
ratio of time step At to the characteristic convection time, u/Ax, the time
required for a disturbance to be convected a distance Ax. This ratio is called
Courant number and is one of the key parameters in computational fluid
dynamics.
If 4 were the temperature (a possibility), Eq. (6.25) would need to satisfy
several conditions. By virtue of diffusion, an increase in temperature at any
of the three points xi-l, xi and xi+l at the old time level should increase
the temperature at point xi at the new time level. The same can be said
for the points xipl and xi with respect to convection, assuming u > 0. If 4
represents the concentration of a substance, it should not be negative.
The possibility that some of the coefficients of 4;-' and 4
1
;
;
in Eq. (6.25)
can become negative should alert us to possible trouble and demands a more
detailed analysis. To the extent possible, we want this analysis to mimic
the one used for ordinary differential equations. A simple way to do this was
invented by von Neumannfor whom the method is named. He argued that the
boundary conditions are rarely the cause of problems (there are exceptions
that are not relevant here) so why not ignore them altogether? If that is
done, the analysis is simplified. Since this method of analysis can be applied
to essentially all of the methods discussed in this chapter, we shall describe
it in a little detail; for further details, see the book by Strikwerda (1983).
In essence the idea can be arrived at as follows. The set of Eqs. (6.25) can
be written in matrix form:
@n+l = -44" ,
(6.27)
where the elements of the tridiagonal matrix A can be derived by inspection
of Eq. (6.25). This equation gives the solution at the new step in terms of
the solution at the previous step. The solution at t,,+l can thus be obtained
by repetitive multiplication of the initial solution 4' by the matrix A. The
question is: do the differences between solutions at successive time steps (for
non-varying boundary conditions), measured in any convenient way, increase,
decrease, or stay the same as n is increased? For example, one measure is the
norm:
The differential equation requires this quantity to decrease with time through
the action of dissipation. Eventually, a steady-state solution will be obtained
if the boundary conditions do not vary. Naturally, we would like the numerical
method to preserve this property of the exact equations.
This issue is closely connected with the eigenvalues of the matrix A. If
some of them are greater than 1, it is not difficult to show that e will grow