Convergence and Discretization tests. In the example above (Figure 3) it was shown that insufficient iterative convergence introduces numerical errors which may offset (mask) more fundamental modelling errors. A general recommendation is that iterative convergence is demonstrated by showing at least 3 orders of magnitude degrease in the normalized residuals for each equation solved (Xxxxxxx et al). Xxx (2005) suggests that iterative error should be no more than 1% of the discretization error on the finest mesh employed. For time dependent problems, iterative convergence at each time step should be checked, and convergence trends should be shown on selected variables. An example might be the evaluation of the global mass error at each timestep in a domain9. Systematic mesh refinement The asymptotic behavior of the discretized equations would be observed in the limit by taking the discretization parameters to zero. In practice a realistic and pragmatic evaluation of discretization error is required. Generally, the discretization error is evaluated by computing solutions on a number of systematically refined grids. Comparing an appropriate error norm (L1, L2 etc) as a function of the discretization parameter, one can evaluate whether this relationship reflects the expected (formal) order of accuracy). At least three meshes are typically required, this is to demonstrate that the solutions are in the asymptotic part of the domain Figure 1. Error norms (left) and observed order of accuracy (right) as a function of systematic grid refinement (Xxxxxxxxx and Xxx, 2010). Local versus systematic global refinement In many practical CFD simulations, local refinement is undertaken to improve spatial resolution of high gradient phenomena in particular volumes in a domain. However, this is not equivalent to the systematic uniform grid refinement that is required to evaluate discretization error. Grid dependent model terms Care is required that the refinement does not inadvertently change the governing equations being solved. This can be the case when sub-models are incorporated which have thresholds/switches/filters which are dependent on grid size. An example would be LES modelling. In such circumstances the practitioner should identify these models which are grid/discretization scale dependent, and if possible fix these to be constant in all verification tests. 9 Xxxxxxx et al; “Processes and Procedures for Application of CFD to Nuclear Reactor Safety Analysis”. Idaho National Laboratory, 2006.
