One last visit to the capillarity correction for free surface flow
Fink, James P., J.-Y. Parlange, and Aly I. El-Kadi
When a capillarity correction is incorporated into the diffusion equation to describe the sudden drawdown of a water table, the resulting differential equation preserves, instead of smoothes, irregularities in the initial data. We describe how this problem can be handled by a proper and careful interpretation of the initial conditions, explain why it works, and illustrate the resulting behavior with numerical experiments.