Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun mathematics, preceded by well or ill The condition of being
posed in the stated manner
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Examples

Using a mathematical perturbation of smooth, largescale atmospheric flow in the presence of vertical shear (jet stream), Browning and Kreiss (1984) showed that the hydrostatic equations can lead to unbounded exponential growth (ill posedness) and the nonhydrostatic equations can lead to rapid exponential growth on the time scale of a few hours when the wave length resolved by a numerical model approaches 10 km.

Will you provide additional information relative to the basis of the illposedness of hydrostatic models?

If there is overturning caused by numerics, physics, or ill posedness in the hydrostatic system, the model must be stabilized either by large dissipation or an ad hoc adjustment to reinstate the hydrostatic balance and usually both.

And at finer resolutions the hydrostatic model blows up even sooner as expected from the illposedness.

The ill posedness of the IBVP for the hydrostatic system was shown mathematically in 1976.

You seem to have numerical accuracy, numerical stability, ill posedness and fast exponential growth confused.

Once we have these facts, we can discuss analytically the posedness and accuracy of the process.

This modification was made in order to overcome the ill posedness of the hydrostatic system for open boundary problems IBVP in the midlatitudes so that the approximate system could be used to supply accurate, well posed boundary conditions for fine mesh limited area models.

Auxiliary differential equation models can also affect the well or illposedness of the overall equation system.

The latter, although in some agreement with the nonhydrostatic model at the lower resolution up to 12 hr all runs start with the same balanced largescale initial conditions, in the higher resolution run blows up because of the illposedness of the hydrostatic system as expected from the mathematical theory in the reference cited above.
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