For transient simulations, the governing equations must be discretized in both space and
time. The spatial discretization for the time-dependent equations is identical to the
steady-state case. Temporal discretization involves the integration of every term in the
differential equations over a time step 
. The integration of the transient terms is straightforward, as shown
below.
A generic expression for the time evolution of a variable 
is given by
 | (23–16) |
where the function 
incorporates any spatial discretization. If the time derivative is discretized
using backward differences, the first-order accurate temporal discretization is given by
 | (23–17) |
and the second-order discretization is given as described in Second-Order Time Integration Using a Variable Time Step Size.
| where | |
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Once the time derivative has been discretized, a choice remains for evaluating
: in particular, which time level values of 
should be used in evaluating 
.
One method is to evaluate 
at the future time level:
 | (23–18) |
This is referred to as "implicit" integration since 
in a given cell is related to 
in neighboring cells through 
:
 | (23–19) |
This implicit equation can be solved iteratively at each time level before moving to the next time step.
The advantage of the fully implicit scheme is that it is unconditionally stable with respect to time step size.
Any independent variable could be discretized in time as
 | (23–20) |
 | (23–21) |
 | (23–22) |
Where 
, 
, 
, 
, 
are different time levels.
and 
are bounding factors for each variable at the 
and 
level.
Bounded variables include the following:
Multiphase flows: Volume fraction
Turbulent flows: Turbulence kinetic energy, dissipation rate, specific dissipation rate
Reacting flows: Species mass fraction, premixed/non-premixed variables
The following limitations exist when using the bounded second-order implicit formulation:
It is not available with the density-based solver, only the pressure-based solver.
It is not available with moving deforming meshes.
It is only applicable to the implicit volume fraction discretization schemes for multiphase, and not to the explicit volume fraction schemes.
It does not support the Singhal et al. expert cavitation model.
Ansys Fluent’s default second-order time integration schemes are based on a variable time step size formulation. This formulation introduces a generalized time derivative discretization that supports the use of variable time step sizes at any arbitrary time instance.
The generalized second-order discretization is given by
 | (23–23) |
In the previous equation, 
is a time step size ratio defined as
and
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When using fixed time steps,
and thus
Consequently, the generalized equation becomes a fixed time step size formulation defined as:
 | (23–24) |
For the bounded second-order implicit formulation, the time step size ratio must be
included as a factor acting on 
and 
, the bounding factors for each variable at 
and 
time levels, and so Equation 23–21 and Equation 23–22 become:
 | (23–25) |
 | (23–26) |
For the density-based solver, the variables 
, 
, and 
in Equation 23–98 are redefined as
the following, in order to extend the time derivative to the variable time step size
formulation.
 | (23–27) |
Important: The variable time step size formulation for second-order time integration is only available for fixed or sliding meshes. For dynamic meshes, the fixed time step size formulation (Equation 23–24) is used: for such cases, do not run a series of simulations in which you vary the time step size, as doing so creates an error that reduces with a reduction of the time step jump.
The explicit time integration is only available when using the explicit density-based
solver, in which 
is evaluated explicitly (based on the existing solution of the dependent
variable 
:
 | (23–28) |
and is referred to as "explicit" integration since 
can be expressed explicitly in terms of the existing solution values,
:
 | (23–29) |
Here, the time step 
is restricted to the stability limit of the underlying solver (that is, a
time step is limited by the Courant-Friedrichs-Lewy condition). In order to be time-accurate,
all cells in the domain must use the same time step. For stability, this time step must be the
minimum of all the local time steps in the domain. This method is also referred to as
"global time stepping".
The use of explicit time stepping is fairly restrictive. It is used primarily to capture the transient behavior of moving waves, such as shocks, because it is more accurate and less expensive than the implicit time stepping methods in such cases. You cannot use explicit time stepping in the following cases:
Calculations with the pressure-based solver or density-based implicit formulation: The explicit time stepping formulation is available only with the density-based explicit formulation. Ansys Fluent also uses multi-stage Runge-Kutta explicit time integration for the density-based solver, as detailed in Steady-State Flow Solution Methods and Unsteady Flows Solution Methods.
Incompressible flow: Explicit time stepping cannot be used to compute time-accurate incompressible flows.
FAS multigrid and residual smoothing cannot be used with explicit time stepping because they destroy the time accuracy of the underlying solver.