This article demonstrates how numerical methods for atmospheric models can be validated by showing that they give the theoretically predicted rate of convergence to relevant asymptotic limit solutions. This procedure is necessary because the exact solution of the Navier–Stokes equations cannot be resolved by production models. The limit solutions chosen are those most important for weather and climate prediction. While the best numerical algorithms for this purpose largely reflect current practice, some important limit solutions cannot be captured by existing methods. The use of Lagrangian rather than Eulerian averaging may be required in these cases.