A von Neumman eigenvalue analysis is performed on three classic problems associated with large-scale dynamics: the Rossby wave problem and the Eady and Charney baroclinic instability problems. In the Rossby problem, the results show surprisingly large phase speed errors (often greater than 25 percent) relative to the theoretical solution over much of the relevant parameter space. The errors appear despite the use of time steps for which the mode is absolutely stable and well-resolved in time. Both the Eady and Charney problems show similar errors for both the phase speed and the growth rate. The use of implicit-biasing in the small-step solver (as introduced by Durran and Klemp (1983)) decreases the errors somewhat, although large values of the biasing coefficient are needed to reduce the errors to acceptable values. The errors are found to be relatively insensitive to both divergence damping and Asselin time filtering.
Further results show that the integration errors described above can be virtually eliminated by modifying the KW scheme so that buoyancy is integrated on the small-time-step cycle (as introduced for stability purposes by Skamarock and Klemp (1993)). This increased accuracy comes with only a minor increase in computational cost and requires no added damping, filtering, or biasing.