The coupled set of governing equations (Equation 23–81) in Ansys Fluent is discretized in time for both steady and unsteady calculations. In the steady case, it is assumed that time marching proceeds until a steady-state solution is reached. Temporal discretization of the coupled equations is accomplished by either an implicit or an explicit time-marching algorithm. These two algorithms are described below. To learn how to apply the two formulations, see Density-Based Solver Settings in the User’s Guide.
In the explicit scheme a multi-stage, time-stepping algorithm [277] is used to discretize the time
derivative in Equation 23–81. The solution is
advanced from iteration 
to iteration 
with an 
-stage Runge-Kutta scheme, given
by
where 
and 
is the stage counter for the 
-stage scheme. 
is
the multi-stage coefficient for the 
stage.
The residual 
is
computed from the intermediate solution 
and, for Equation 23–81, is given by
 | (23–88) |
The time step 
is computed from the CFL (Courant-Friedrichs-Lewy)
condition
 | (23–89) |
where 
is the cell volume, 
is the face
area, and 
is
the maximum of the local eigenvalues defined by Equation 23–84.
For steady-state solutions, convergence acceleration of the explicit formulation can be achieved with the use of local time stepping, residual smoothing, and full-approximation storage multigrid.
Local time stepping is a method by which the solution at each control volume is advanced in time with respect to the cell time step, defined by the local stability limit of the time-stepping scheme.
Residual smoothing, on the other hand, increases the bound of stability limits of the time-stepping scheme and hence allows for the use of a larger CFL value to achieve fast convergence (Implicit Residual Smoothing).
The convergence rate of the explicit scheme can be accelerated through use of the full-approximation storage (FAS) multigrid method described in Full-Approximation Storage (FAS) Multigrid.
By default, Ansys Fluent uses a 3-stage Runge-Kutta scheme based on the work by Lynn [402] for steady-state flows that use the density-based explicit solver.
The maximum time step can be further increased by increasing the support of the scheme through implicit averaging of the residuals with their neighbors. The residuals are filtered through a Laplacian smoothing operator:
 | (23–90) |
This equation can be solved with the following Jacobi iteration:
 | (23–91) |
where 
is the number of neighbors. Two Jacobi iterations
are usually sufficient to allow doubling the time step with a value
of 
.
In the implicit scheme, an Euler implicit discretization in time of the governing equations (Equation 23–81) is combined with a Newton-type linearization of the fluxes to produce the following linearized system in delta form [696]:
 | (23–92) |
The center and off-diagonal coefficient matrices, 
and 
are given by
 | (23–93) |
 | (23–94) |
and the residual vector 
and
time step 
are defined as in Equation 23–88 and Equation 23–89, respectively.
Equation 23–92 is solved using either Incomplete Lower Upper factorization (ILU) by default or symmetric point Gauss-Seidel algorithm, in conjunction with an algebraic multigrid (AMG) method (see Algebraic Multigrid (AMG)) adapted for coupled sets of equations.
Explicit relaxation can improve the convergence to steady state
of the implicit formulation. By default, explicit relaxation is enabled
for the implicit solver and uses a factor of 0.75. You can specify a factor 
to control the amount that the solution
vector 
changes between iterations after the end of the
algebraic multigrid (AMG) cycle:
 | (23–95) |
By specifying a value less than the default value of 1 for 
, the variables in the solution vector will
be under-relaxed and the convergence history can be improved. For
information on how to set this value, see Specifying the Explicit Relaxation in the User’s Guide.
Important: Note that explicit relaxation is available for the density-based implicit solver in steady state mode only.
When the flow is solved on meshes containing highly stretched cells (cells with large aspect ratio typically found near wall boundaries), the traditional implicit solution method will suffer from convergence slow down due to the small local time step values defined by equation Equation 23–89. The larger the cell aspect ratio, the smaller the local time step and the slower the local rate change of the solution. The time step definition in equation Equation 23–89 is based effectively on the minimum cell characteristic distance. A more optimum time step definition for the implicit solution method can be based on the maximum characteristic length of the cell [364] to give a new time step definition:
 | (23–96) |
where AR is the cell aspect ratio defined
as the ratio of maximum length to minimum length in the cell 
Effectively, the new definition will vary the local CFL value from one cell to another proportional to the value of the cell aspect ratio. When the cell aspect ratio is near unity, the local CFL value will be similar to the one entered by you. On the other hand, on stretched high aspect ratio cells, the local CFL value is a multiple of the cell aspect ratio. The new time step definition will accelerate the solution convergence particularly on meshes with Y+ near 1.