Frequently Asked Question

A residual measures the discrete equation imbalance. Lower residuals usually indicate that the algebraic equations are being satisfied more closely, but residual reduction alone does not prove that the engineering result is converged.

Use at least three kinds of evidence: equation residuals, global conservation imbalances, and monitored quantities such as pressure drop, force, heat rate, or outlet temperature. A solution is practically converged when these quantities remain stable under additional iterations and the remaining error is smaller than the required engineering tolerance.

Example: If the pressure-drop monitor changes by less than 0.1% over several hundred iterations, inlet/outlet mass imbalance is negligible, and residuals have reached a consistent plateau below the selected threshold, the result has stronger evidence of convergence than any single metric alone.

Numerical error and convergence

A numerical solution approximates the continuous conservation laws on a finite set of cells or control volumes. Discretization error depends on characteristic size, stretching, skewness, alignment, interpolation, boundary treatment, and the output being measured. Iterative convergence is different from mesh convergence: a solver can reduce algebraic residuals while the engineering quantity continues to drift.

φ is the reported quantity and h is a characteristic mesh size. Use an analytical or benchmark reference when possible; otherwise report a systematic refinement study.

Worked example

A pressure drop is 104 Pa on a coarse grid, 101 Pa on a medium grid, and 100 Pa on a fine grid. The medium-to-fine change is approximately 1%, while the coarse-to-medium change is approximately 3%. Report mesh sizes, refinement ratio, residual state, and whether the 1% change meets the design tolerance.

Check: refine where gradients or the engineering output are sensitive, not only where the global cell count is convenient.