146
6. Methods for Unsteady Problems
d < 0.5 means that, each time the spatial mesh is halved, the time step
has to be reduced by a factor of four. This makes the scheme unsuitable
for problems which do not require high temporal resolution (slowly varying
solutions, solutions approaching steady state); these are normally the cases
in which one would like to use first order methods in time. An example which
illustrates the stability problem will be shown below.
The problem of instability in connection with the condition of Eq. (6.33)
was recognized in the 1920's by Courant and Friedrichs and they suggested
a cure which is still used today. They noted that, for problems dominated by
convection, it is possible for the coefficient of 4:;; in Eq. (6.25) to be negative
and suggested that the problem could be cured by using upwind differences.
Instead of evaluating the convective term by a CDS approximation as we
have done above, we use UDS (see Chap. 3). We then get, in place of Eq.
(6.24):
which yields, in place of Eq. (6.25):
Since the coefficients of the neighbor values are always positive, they cannot
contribute to unphysical behavior of instability. However, the coefficient $1
can be negative, thus creating a potential problem. For this coefficient to be
positive, the time step should satisfy the following condition:
When convection is negligible, the restriction on the time step required for
stability is the same as Eq. (6.32). For negligible diffusion, the criterion to be
satisfied is:
i.e. the Courant number should be smaller than unity.
One can also apply the von Neumann stability analysis to this problem;
the analysis is in accord with the conclusions just reached. Thus, unlike the
central difference method, the upwind approximation provides some stability
which, combined with its ease of use, made this type of method very popular for many years. It continues to be used today. When both convection
and diffusion are present, the stability criterion is more complicated. Rather
than dealing with this complexity, most people require that each individual
criterion be satisfied, a condition that may be a little more restrictive than
necessary but is safe.
6. Methods for Unsteady Problems
d < 0.5 means that, each time the spatial mesh is halved, the time step
has to be reduced by a factor of four. This makes the scheme unsuitable
for problems which do not require high temporal resolution (slowly varying
solutions, solutions approaching steady state); these are normally the cases
in which one would like to use first order methods in time. An example which
illustrates the stability problem will be shown below.
The problem of instability in connection with the condition of Eq. (6.33)
was recognized in the 1920's by Courant and Friedrichs and they suggested
a cure which is still used today. They noted that, for problems dominated by
convection, it is possible for the coefficient of 4:;; in Eq. (6.25) to be negative
and suggested that the problem could be cured by using upwind differences.
Instead of evaluating the convective term by a CDS approximation as we
have done above, we use UDS (see Chap. 3). We then get, in place of Eq.
(6.24):
which yields, in place of Eq. (6.25):
Since the coefficients of the neighbor values are always positive, they cannot
contribute to unphysical behavior of instability. However, the coefficient $1
can be negative, thus creating a potential problem. For this coefficient to be
positive, the time step should satisfy the following condition:
When convection is negligible, the restriction on the time step required for
stability is the same as Eq. (6.32). For negligible diffusion, the criterion to be
satisfied is:
i.e. the Courant number should be smaller than unity.
One can also apply the von Neumann stability analysis to this problem;
the analysis is in accord with the conclusions just reached. Thus, unlike the
central difference method, the upwind approximation provides some stability
which, combined with its ease of use, made this type of method very popular for many years. It continues to be used today. When both convection
and diffusion are present, the stability criterion is more complicated. Rather
than dealing with this complexity, most people require that each individual
criterion be satisfied, a condition that may be a little more restrictive than
necessary but is safe.