ONE FLOW · ONE ERROR CHECK · FOUR METHODS
Compare the paper’s explicit w₂ approximation with an optimized RK4 calculation. The heat flow w₀ and first Picard iterate w₁ provide two additional reference points. Every method uses the same full-residual error check.
COMPARE LIKE WITH LIKE
All bounds below use the same normalized spatial L² velocity error and final time t = 0.25. They refer to saved, exact rational, piecewise-linear reference trajectories.
Bar lengths use approximate decimals. Displayed bounds are rounded upward; exact fractions are in the reports.
A COMMON ERROR REQUIREMENT
| Method | Time step | Error bound | Full run · ms | Min–max · ms |
|---|
These full runtimes were measured directly. They are not sums of the separate medians below. Only the selected cases are compared here.
| Method | Time step | Error bound | Generate · ms | Setup · ms | Certify · ms | Inspect |
|---|
Timings are medians for these measured runs. Generation, setup and certification are separate operations; their medians are not an end-to-end runtime.
OPEN A MEASURED CASE
Normalized spatial L² velocity error bound at t = 0.25. This is an upper limit, not the measured actual error.
Each card shows the median, with the measured minimum–maximum range. Runtime varies with hardware and execution conditions.
TRY AN ERROR REQUIREMENT
Choose a target for the certificate’s upper bound. This checks the finite set of measured cases, across every tested time step.
Membership is checked using exact rational values. This is not an adaptive solver, a search for the globally best method, or a guarantee that an untested case will meet the target.
WHAT THE RESULT MEANS
The comparison certifies rational linear reconstructions of the saved samples. It does not directly certify each formula’s continuous exponential path. The paper’s global X-norm convergence theorem is a separate claim.
A useful result on this test does not by itself establish novelty, an industrial advantage, or a general Navier–Stokes regularity result.
READ THE COSTS CORRECTLY
Sampling an explicit formula and advancing an ODE solver are different operations. Setup and certificate generation can dominate a small benchmark. The download records the individual runs so the comparison can be inspected.
REPRODUCE THE COMPARISON
The package contains the measured cases and the instructions needed to repeat the experiment.