6. Methods for Unsteady Problems
6.1 Introduction
In computing unsteady flows, we have a fourth coordinate direction to consider: time. Just as with the space coordinates, time must be discretized. We
can consider the time "grid" in either the finite difference spirit, as discrete
points in time, or in a finite volume view as "time volumes". The major difference between the space and time coordinates lies in the direction of influence:
whereas a force at any space location may (in elliptic problems) influence the
flow anywhere else, forcing a t a given instant will affect the flow only in
the future - there is no backward influence. Unsteady flows are, therefore,
parabolic-like in time. This means that no conditions can be imposed on the
solution (except a t the boundaries) at any time after the initiation of the calculation, which has a strong influence on the choice of solution strategy. To
be faithful to the nature of time, essentially all solution methods advance in
time in a step-by-step or LLmarching" manner. These methods are very similar
to ones applied to initial value problems for ordinary differential equations
(ODEs) so we shall give a brief review of such methods in the next section.
6.2 Methods for Initial Value Problems in ODEs
6.2.1 Two-Level Methods
For initial value problems, it is sufficient to consider the first order ordinary
differential equation with an initial condition:
The basic problem is to find the solution q5 a short time At after the initial
point. The solution a t tl = to + At, il, can be regarded as a new initial
condition and the solution can be advanced to t2 = tl + A t , ts = t2 + A t , . . .
etc.
The simplest methods can be constructed by integrating Eq. (6.1) from
t, to t,+l = t , + At:
6.1 Introduction
In computing unsteady flows, we have a fourth coordinate direction to consider: time. Just as with the space coordinates, time must be discretized. We
can consider the time "grid" in either the finite difference spirit, as discrete
points in time, or in a finite volume view as "time volumes". The major difference between the space and time coordinates lies in the direction of influence:
whereas a force at any space location may (in elliptic problems) influence the
flow anywhere else, forcing a t a given instant will affect the flow only in
the future - there is no backward influence. Unsteady flows are, therefore,
parabolic-like in time. This means that no conditions can be imposed on the
solution (except a t the boundaries) at any time after the initiation of the calculation, which has a strong influence on the choice of solution strategy. To
be faithful to the nature of time, essentially all solution methods advance in
time in a step-by-step or LLmarching" manner. These methods are very similar
to ones applied to initial value problems for ordinary differential equations
(ODEs) so we shall give a brief review of such methods in the next section.
6.2 Methods for Initial Value Problems in ODEs
6.2.1 Two-Level Methods
For initial value problems, it is sufficient to consider the first order ordinary
differential equation with an initial condition:
The basic problem is to find the solution q5 a short time At after the initial
point. The solution a t tl = to + At, il, can be regarded as a new initial
condition and the solution can be advanced to t2 = tl + A t , ts = t2 + A t , . . .
etc.
The simplest methods can be constructed by integrating Eq. (6.1) from
t, to t,+l = t , + At: