
The coupled set of governing equations (Equation 18.56) 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 steadystate solution is reached. Temporal discretization of the coupled equations is accomplished by either an implicit or an explicit timemarching algorithm. These two algorithms are described below. To learn how to apply the two formulations, see this section in the separate User's Guide.
Explicit Formulation
In the explicit scheme a multistage, timestepping algorithm [ 146] is used to discretize the time derivative in Equation 18.56. The solution is advanced from iteration to iteration with an stage RungeKutta scheme, given by
where and is the stage counter for the stage scheme. is the multistage coefficient for the stage. The residual is computed from the intermediate solution and, for Equation 18.56, is given by
The time step is computed from the CFL (CourantFriedrichsLewy) condition
where is the cell volume, is the face area, and is the maximum of the local eigenvalues defined by Equation 18.59.
For steadystate solutions, convergence acceleration of the explicit formulation can be achieved with the use of local time stepping, residual smoothing, and fullapproximation 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 timestepping scheme.
Residual smoothing, on the other hand, increases the bound of stability limits of the timestepping scheme and hence allows for the use of a larger CFL value to achieve fast convergence (Section 18.5.4).
The convergence rate of the explicit scheme can be accelerated through use of the fullapproximation storage (FAS) multigrid method described in Section 18.6.4.
By default, ANSYS FLUENT uses a 3stage RungeKutta scheme based on the work by Lynn [ 211] for steadystate flows that use the densitybased explicit solver.
Implicit Residual Smoothing
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:
(18.515) 
This equation can be solved with the following Jacobi iteration:
(18.516) 
Two Jacobi iterations are usually sufficient to allow doubling the time step with a value of .
Implicit Formulation
In the implicit scheme, an Euler implicit discretization in time of the governing equations (Equation 18.56) is combined with a Newtontype linearization of the fluxes to produce the following linearized system in delta form [ 370]:
The center and offdiagonal coefficient matrices, and are given by
and the residual vector and time step are defined as in Equation 18.513 and Equation 18.514, respectively.
Equation 18.517 is solved using either Incomplete Lower Upper factorization (ILU) by default or symmetric point GaussSeidel algorithm, in conjunction with an algebraic multigrid (AMG) method (see Section 18.6.3) 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:
By specifying a value less than the default value of 1 for , the variables in the solution vector will be underrelaxed and the convergence history can be improved. For information on how to set this value, see this section in the separate User's Guide.

Note that explicit relaxation is available for the densitybased implicit solver in steady state mode only.
