142
6 . Methods for Unsteady Problems
6.2.4 Other Methods
One can approximate the integrals on both sides of Eq. (6.2) by mean values
over the integration interval.
An implicit three-level second order scheme can be constructed by integrating over a time interval At centered around tn+l (i.e. from tn+l - At12
to tn+1 + At/2) and applying the midpoint rule to both the left and right
hand sides of the equation. The time derivative at tn+l can be approximated
by differentiating a parabola forced through solutions at three time levels,
tn-I, tn, and L + i :
This leads to the following method:
The scheme is implicit, since f is evaluated at the new time level. It is of
second order and very easy to implement, but as it is implicit, it requires
iteration at each time step.
6.3 Application to the Generic Transport Equation
We next consider application of some of the methods given above to the
generic transport equation (1.28). In Chaps. 3 and 4, discretization of the
convective and diffusive fluxes and source terms for steady problems was
discussed. These terms can be treated in the same way for unsteady flows;
however, the question of the time at which the fluxes and sources are to be
evaluated must be answered.
If the conservation equation is rewritten in a form which resembles the
ordinary differential equation (6.1), eg.:
a(p4) = -div (pqh) + div (r grad 4) + qb = f (t, 4(t)) ,
a t
any method of time integration can be used. The function f (t, 4) represents
the sum of convective, diffusive and source terms, all of which now appear on
the right hand side of the equation. Since these terms are not known, we must
use one of the quadrature approximations introduced above. The convective,
diffusive and source terms are discretized using one of the methods presented
in Chaps. 3 and 4 at one or more time levels. If an explicit method is used for
time integration, these terms have to be evaluated only at times for which the
solution is already known, so they can be calculated. For an implicit method,
6 . Methods for Unsteady Problems
6.2.4 Other Methods
One can approximate the integrals on both sides of Eq. (6.2) by mean values
over the integration interval.
An implicit three-level second order scheme can be constructed by integrating over a time interval At centered around tn+l (i.e. from tn+l - At12
to tn+1 + At/2) and applying the midpoint rule to both the left and right
hand sides of the equation. The time derivative at tn+l can be approximated
by differentiating a parabola forced through solutions at three time levels,
tn-I, tn, and L + i :
This leads to the following method:
The scheme is implicit, since f is evaluated at the new time level. It is of
second order and very easy to implement, but as it is implicit, it requires
iteration at each time step.
6.3 Application to the Generic Transport Equation
We next consider application of some of the methods given above to the
generic transport equation (1.28). In Chaps. 3 and 4, discretization of the
convective and diffusive fluxes and source terms for steady problems was
discussed. These terms can be treated in the same way for unsteady flows;
however, the question of the time at which the fluxes and sources are to be
evaluated must be answered.
If the conservation equation is rewritten in a form which resembles the
ordinary differential equation (6.1), eg.:
a(p4) = -div (pqh) + div (r grad 4) + qb = f (t, 4(t)) ,
a t
any method of time integration can be used. The function f (t, 4) represents
the sum of convective, diffusive and source terms, all of which now appear on
the right hand side of the equation. Since these terms are not known, we must
use one of the quadrature approximations introduced above. The convective,
diffusive and source terms are discretized using one of the methods presented
in Chaps. 3 and 4 at one or more time levels. If an explicit method is used for
time integration, these terms have to be evaluated only at times for which the
solution is already known, so they can be calculated. For an implicit method,