Convergence Reports

A convergence report makes a numerical claim assessable: it identifies the observable, states the approximations, shows the refinements, and explains why the remaining uncertainty is acceptable for the conclusion. It should let another reader distinguish an accurately solved finite problem from an accurate approximation to the intended physical problem.

Required background. Convergence studies develops refinement and extrapolation; error budgets separates bias estimates, bounds, and statistical uncertainty.

The claim and the calculation

Begin with a precise target, such as the lowest even-parity eigenvalue of a declared Hamiltonian, a transition probability at a specified time, or a thermodynamic-limit estimate at fixed temperature. State units, boundary conditions, parameter values, and the observable’s normalization. “The simulation converged” has no single scientific meaning without this information.

Then describe the finite representation and algorithm: box and grid, basis truncation, sector, integrator and step, eigensolver stopping criterion, trajectory count, or bond dimension as applicable. Include the actual settings used. A requested tolerance is not the same as the achieved residual or estimator uncertainty.

An executable report should also identify the source revision, environment, run command, random-stream policy, and actual execution date. Shared running instructions and downloadable investigations belong in Labs; this page explains how the results establish numerical accuracy.

A compact worked report

Consider the exact discrete benchmark for $H=-\partial_x^2$ on $(0,1)$ with Dirichlet endpoints. Evaluating the known three-point-stencil eigenvalue gives

Target: continuum ground energy, E = pi^2
h          discrete energy       absolute continuum error
0.100      9.788696741           0.080907660
0.050      9.849327524           0.020276877
0.025      9.864532054           0.005072347

These are analytic discrete-benchmark values, not a claim about an executed eigensolver. Their successive differences give observed order $p\approx1.996$. The expected second-order extrapolation from the last two rows is $9.869600231$, with actual error about $4.17\times10^{-6}$ against $\pi^2$.

An appropriate conclusion is that this sequence exhibits second-order continuum convergence and successful leading-error cancellation in the tested regime. It does not show that the finest raw grid has $10^{-6}$ accuracy: its error is still about $5\times10^{-3}$. Nor does it test an eigensolver, roundoff at much finer grids, or a different boundary condition.

A report of an actual solver run would add residuals against the assembled matrix and differences from the discrete closed form. That separates solver verification from continuum convergence.

Interpreting a plateau

Show at least enough refinements to distinguish a trend from an isolated agreement. Report differences with enough digits to resolve the claimed order, while rounding the final physical result to its supported accuracy. Explain how other numerical parameters were held fixed or made negligible.

For state comparisons, specify phase alignment, interpolation, and norm. For a degenerate cluster, report a subspace metric instead of unstable eigenvector labels. For a stochastic estimate, give standard errors or intervals with the autocorrelation and sample-count conventions. A fit line without its uncertainty or fitting range is difficult to interpret.

A log–log plot is helpful for power-law error only when the error is nonzero and the reference is credible. Plotting the error against the finest available numerical result forces that last point to zero and can distort the apparent rate. Label such a reference as a surrogate and test its adequacy.

Negative results and unresolved errors

If refinement stagnates, report the plateau and the diagnostic experiments: tighter solver tolerances, increased arithmetic precision, a larger box, or an independent representation. A failed benchmark should remain visible in the report until understood. Removing inconvenient parameter points without explaining why destroys the value of the study.

Avoid presenting an extrapolated value with many digits merely because a fit prints them. Different plausible fit windows or error models can shift the intercept. Quantify that sensitivity when it affects the conclusion, and state when the data do not support extrapolation.

Keep modeling limitations separate from numerical ones. A highly converged approximate Hamiltonian can still omit a relevant physical effect. The report should say which question its calculations answer and where additional physics or data would be needed.

Exercises

A paper reports energies differing by $10^{-8}$ between two grids, but it uses the same finite box and gives no box study. What conclusion is supported?

Solution

Only that the tested grid change altered the energy by $10^{-8}$ at that box and other fixed settings. It does not bound the continuum-grid error without a convergence model, and says nothing about finite-box bias. A separate domain study is needed if the intended problem is unbounded.

References

• Higham, N. J., Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, 2002 — numerical error interpretation and reporting.
• LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007, author resources — refinement experiments and convergence analysis.