The coupled set of governing equations (Equation 25.5-6) in 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.
In the explicit scheme a multi-stage, time-stepping algorithm [ 161] is used to discretize the time derivative in Equation 25.5-6. 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 25.5-6, is given by
The time step is computed from the CFL (Courant-Friedrichs-Lewy) condition
where is the cell volume, is the face cell, and is the maximum of the local eigenvalues defined by Equation 25.5-9.
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 (Section 25.5.4).
The convergence rate of the explicit scheme can be accelerated through use of the full-approximation storage (FAS) multigrid method described in Section 25.6.4.
By default, FLUENT uses a 3-stage Runge-Kutta scheme based on the work by Lynn [ 225] for steady-state flows that use the density-based 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:
This equation can be solved with the following Jacobi iteration:
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 25.5-6) is combined with a Newton-type linearization of the fluxes to produce the following linearized system in delta form [ 394]:
The center and off-diagonal coefficient matrices, and are given by
and the residual vector and time step are defined as in Equation 25.5-13 and Equation 25.5-14, respectively.
Equation 25.5-17 is solved using either Incomplete Lower Upper factorization (ILU) by default or asymmetric point Gauss-Seidel algorithm, in conjunction with an algebraic multigrid (AMG) method (see Section 25.6.3) adapted for coupled sets of equations.