For time-accurate calculations, explicit and implicit time-stepping schemes are available. (The time-implicit approach is also referred to as "dual time stepping".)
The explicit time stepping approach, is available only for the explicit scheme described in Explicit Formulation. The time step is determined by the CFL condition. To maintain time accuracy of the solution the explicit time stepping employs the same time step in each cell of the domain (this is also known as global-time step), and with preconditioning disabled. By default, Ansys Fluent uses a 4-stage Runge-Kutta scheme for unsteady flows.
The implicit-time stepping method (also known as dual-time formulation) is available in the density-based explicit and implicit formulation.
When performing unsteady simulations with implicit-time stepping (dual-time stepping), Ansys Fluent uses a low Mach number time-derivative unsteady preconditioner to provide accurate results both for pure convective processes (for example, simulating unsteady turbulence) and for acoustic processes (for example, simulating wave propagation) [661], [504].
Here we introduce a preconditioned pseudo-time-derivative term into Equation 23–74 as follows:
 | (23–97) |
where 
denotes physical-time and 
is a pseudo-time used in the time-marching procedure. Note that as
, the second term on the left side of Equation 23–97
vanishes and Equation 23–74 is recovered.
The time-dependent term in Equation 23–97 is discretized in an implicit fashion by means of either a first- or second-order accurate, backward difference in time.
The dual-time formulation is written in semi-discrete form as follows:
 | (23–98) |
where {
} gives first-order time accuracy, and {
} gives second-order time accuracy (note that this is for the fixed time step
size formulation; for the variable time step size formulation, see Equation 23–27). 
is the inner iteration counter and 
represents any given physical-time level.
The pseudo-time-derivative is driven to zero at each physical time level by a series of inner iterations using either the implicit or explicit time-marching algorithm.
Throughout the (inner) iterations in pseudo-time, 
and 
are held constant and 
is computed from 
. As 
, the solution at the next physical time level 
is given by 
.
Note that the physical time step size 
is limited only by the level of desired temporal accuracy. The pseudo time
step size 
is determined by the CFL condition of the (implicit or explicit)
time-marching scheme.
Table 23.1: Summary of the Density-Based Solver summarizes all operation modes for the density-based solver from the iterative scheme in steady-state calculations to time-marching schemes for transient calculations.
Table 23.1: Summary of the Density-Based Solver
|
Solution Method |
Density-Based Solver - Explicit Formulation |
Density-Based Solver - Implicit Formulation |
|---|---|---|
| Steady-State | – 3-stage Runge-Kutta | – local time step |
| – local time step | – time-derivative preconditioning | |
| – time-derivative preconditioning | ||
| — FAS | ||
| Unsteady - Explicit Time Stepping | – 4-stage Runge-Kutta | N/A |
| – global time step | ||
| – no time-derivative preconditioning | ||
| – No FAS | ||
| Unsteady - Implicit Time Stepping (dual-time formulation) First Order | – dual-time formulation | – dual-time formulation |
| – Physical time: first order Euler backward | – Physical time: first order Euler backward | |
| – preconditioned pseudo-time derivative | – preconditioned pseudo-time derivative | |
| – inner iteration: explicit pseudo-time marching, 3-stage Runge-Kutta | – inner iteration: implicit pseudo-time marching | |
| Unsteady - Implicit Time Stepping (dual-time formulation) Second Order | – dual-time formulation | – dual-time formulation |
| – Physical time: second-order Euler backward | – Physical time: second-order Euler backward | |
| – preconditioned pseudo-time derivative | – preconditioned pseudo-time derivative | |
| – inner iteration: explicit pseudo-time marching, 3-stage Runge-Kutta | – inner iteration: implicit pseudo-time marching |