Judging Convergence

Residual monitor plots are useful for judging the convergence (or divergence) of a solution.

Residual monitor plots are useful for judging the convergence (or divergence) of a solution, and they are created automatically within every simulation. However, it is important to understand both the significance of residuals and their limitations. While it is true that the residual quantity tends toward a small number when the solution is converged, the residual monitors cannot be relied on as the only measure of convergence. The limitations of residuals are as follows:

• The amount that a residual decreases by depends on the particulars of the simulation. Therefore, a three-order-of-magnitude drop in residuals is possibly acceptable for one simulation, but not another. The initial guess also strongly influences the amount that residuals are reduced. If the initial solution satisfies the discretized equations perfectly, the residuals do not drop at all.
• There are two types of discretization errors: dissipative errors and dispersive errors. Dissipative errors are characteristic of first-order upwind schemes; they are inherently stabilizing and produce residual plots that tend to decrease monotonically. Dispersive errors are characteristic of second-order upwind schemes which tend to “smear” solutions less than first-order schemes. While dispersive errors tend to produce residual plots that are not monotonic. This outcome is generally an acceptable price to pay for the enhanced accuracy.

  In some cases, often because of poor mesh quality, dispersive errors result in oscillating solutions (that is, changing from one iteration to the next) within a few cells. The result is that the residual plots can indicate that the solution is not “converged”. You have a choice to either accept the solution, or to try to stabilize it by choosing a lower-order numerical scheme. Frequently, it is better to accept the solution.

• Residuals do not necessarily relate to quantities of engineering interest in the simulation such as integrated forces, pressure losses, or mass flow rates.

With the issues above in mind, it is advisable to monitor quantities of engineering interest, such as integrated forces, pressure changes, or mass flow rates as well as the residuals. Simcenter STAR-CCM+ features such as scenes and plots can help you examine these quantities while the solution progresses. The features are described in detail in the chapter on analyzing.

The choice of the engineering quantity, as well as the convergence criterion, is your judgment call. In the example below, taken from a large external aerodynamics solution, both lift and drag coefficients are monitored as well as the residuals. It is clear that not all quantities reach an asymptotic limit at the same time. Use your judgment and decide which coefficient is the most critical.

Monitor-based stopping criteria can be used to stop the simulation when an appropriate level of convergence has been reached, based on an appropriate combination of residual monitors or monitors of other physical quantities.