About Under-Relaxation Factors

The app can estimate the optimal under-relaxation factors automatically for all equations required by your simulation. In general, automatically calculated under-relaxation factors exhibit faster convergence rates than the fixed default values. In addition, using this method allows the app to automatically adjust the under-relaxation factors when the mesh is modified. These adjustments remove the degradation in convergence exhibited when the under-relaxation factors are kept constant while the mesh is refined.

In general, permit the solvers to estimate automatically the under-relaxation factors for all equations, particularly when the discretized equations required for the simulation include the advection term from the Navier-Stokes advection-diffusion equation. However, if there is an explicit body force (or heat source) $f_i$ on the model, specify the under-relaxation factor manually. An explicit body force is not a function of the flow property quantity $\varphi$ , so it cannot be linearized by the solvers and accounted for in the automatic under-relaxation factor calculation. For more information about which terms each discretized equation contains, see Flow Property Equations.

If the convergence rate obtained with the automatic option is not satisfactory or the solver fails to converge, specify the under-relaxation factor manually and adjust each coefficient to improve the convergence.

When a discretized equation does not contain the advection term from the Navier-Stokes advection-diffusion equation, specify its under-relaxation factor manually. Specifying this parameter is not trivial—a bad choice could potentially increase (in orders of magnitude) the number of iterations required to achieve convergence. For more information about which terms each discretized equation contains, see Flow Property Equations.

For any given flow and mesh discretization, it is possible to increase the convergence rate of the numerical algorithm by increasing the under-relaxation factor up to a point, after which the convergence rate degrades. Alternatively, keeping the under-relaxation factor constant and further refining the mesh discretization also degrades the convergence rate. By increasing or decreasing the under-relaxation factor, you can balance the speed of convergence with code robustness; that is, the likelihood that the solver will converge on a solution.

Despite the fact that it is virtually impossible to know before the analysis the optimal under-relaxation factor for any flow and mesh discretization, a few general rules can help you choose suitable values:

1. Specify a value in the range $0.5\le \omega <1$ for the under-relaxation factor of each transport equation, excluding the pressure equation.
2. Compute the under-relaxation factor for the pressure equation using the momentum equation under-relaxation factor, as follows: ${\omega }_p=1-{\omega }_m$ .
3. 0.8 is a suitable starting value for the under-relaxation factor for all transport equations, excluding the pressure equation. Following the relationship specified in rule 2, 0.2 is a suitable starting value for the under-relaxation factor for the pressure equation.
4. If your model exhibits large regions of mesh refinement (for example, boundary layers or mesh clusters), increase the under-relaxation factor for non-pressure transport equations to values in the range $0.9\le \omega <1$ . Following the relationship specified in rule 2, you can adjust the under-relaxation factor for the pressure equation to a value in the range $0<\omega \le 0.1$ .
5. If you continue to refine your model's mesh, always increase the under-relaxation factors while staying within the limits described in rule 4.

The table below provides the range of values to use when specifying an under-relaxation factor for each equation. It includes specifications even for those equation types where enabling automatic calculation of the under-relaxation factor for all equations is preferred. An em dash (—) indicates that the under-relaxation factors do not apply to a particular equation type.

Equation Type	Under-Relaxation Factor Specification
	Automatic	Manual
Coupled	—	—
Pressure		< 0.5
Momentum		> 0.5
Turbulence		> 0.5
Energy		> 0.5
Species		> 0.5
VOF	—	—
Electromagnetics	—	—

The convergence rate of the steady-state integration algorithm is controlled by the under-relaxation factor that is specified for each equation.

The effect of the under-relaxation factor is to increase the diagonal dominance of the matrix, which represents the discretization of the transport equations, to enhance the convergence of the linear solver. By under-relaxing the matrix, a pseudo-transient term is introduced on each element of the mesh. This artificial transient term uses a local time-step that depends on a Courant-Friedrichs-Lewy (CFL) number and a characteristic time scale that is a function of the physics involved in the flow; for example, its advection, diffusion, and body forces.

The CFL number used in the pseudo-transient term is an input parameter that is obtained from the specified under-relaxation factor, $\omega$ :

For example, for $\omega =0.8$ , $CFL=4$ ; while for $\omega =0.99$ , $CFL=99$ . The CFL number increases in a highly nonlinear manner as the under-relaxation approaches 1.

In general, for steady-state calculations, the information propagates faster when large CFL numbers are used because each cell is able to use large time steps to move the flow throughout the domain. Unfortunately, the impact of the CFL number on the convergence of the numerical algorithm depends on the mesh resolution and the physics of the flow. Therefore, there is no single CFL number or under-relaxation factor that, in general, produces an optimum convergence rate on every flow and mesh discretization. Consequently, you should adjust the under-relaxation factor every time the flow physics or mesh discretization changes.