Convergence and Grid Independence

A simulated number is only real if it stops changing as the grid and timestep are refined; otherwise we are reporting the mesh, not the physics.

Refine until the answer stops moving

Any discretized simulation approximates a continuous problem on a finite grid. If the answer keeps changing as the grid refines, the reported number reflects the grid as much as the physics. Convergence testing refines resolution until the result stabilizes to within tolerance, and only then treats the number as meaningful.

What we measure

• Grid independence: the result changes negligibly under further spatial refinement.
• Timestep independence: the result is stable under further temporal refinement.
• Observed order: error falls at the rate the scheme promises.
• A converged residual: iterative solvers reach a tight, pre-set stopping criterion.

Convergence and the error budget

The residual discretization error at the chosen resolution is a line in the error budget, not something to ignore because it is small. We report it, so a reader knows how much of the uncertainty on a number like Q_sci 3.424 is numerical versus physical. A result quoted without evidence of convergence is incomplete.

Convergence testing also guards against a seductive failure: a coarse grid that happens to land on the desired answer. Refining past it would reveal the truth, so skipping the refinement is a way to fool oneself. We refine anyway, and report where the number settles, not where it first looked right. Shown alongside the result, a convergence study lets a reader confirm for themselves that a headline figure describes the physics and not the mesh -- a check that costs a plot and buys a great deal of trust.