Convergence Studies
Refine the mesh or time step systematically and watch the solution approach a limit; the trend, not any single run, is the evidence.
The Idea
A discrete method approximates a continuous problem, and the approximation should improve as the mesh spacing or time step shrinks toward zero. A convergence study runs the same problem on a sequence of systematically refined discretizations and examines whether the solution settles toward a limit. If it does not, either the method is not consistent or the implementation is broken.
Systematic Refinement
The refinements must be controlled. A common practice halves the spacing each level (a refinement ratio of two), keeping the mesh family self-similar so that comparisons are clean. Reporting must include the refinement ratio, the quantity of interest, and the norm used to measure change, because an error that shrinks in one norm can stagnate in another.
What You Look For
- Monotone approach of a scalar quantity of interest toward a limiting value.
- A shrinking difference between successive levels, ideally at the theoretical rate.
- No sudden reversals, which usually signal an under-resolved feature or a bug.
Grid Convergence and Time Convergence
Spatial and temporal convergence are separate axes. Refining only the mesh while holding a coarse time step fixed will stall on the temporal error floor, and vice versa. A clean study refines one at a time, or refines both together with a fixed ratio, so the observed behavior can be attributed correctly.
Convergence studies are the empirical backbone of solution verification. They convert the abstract promise that a scheme is order-p accurate into a measured claim about the actual code on the actual problem, which is the only claim that matters for a design decision. Where a limit is reached before machine precision, the remaining gap is the discretization error to be entered in the error budget.