## Wednesday, March 19, 2014

### Ubiquitous Diffusion

These are the slides for the lecture I gave to Michael Dumbser' class on Environmental Modelling. I tried to show the non linear diffusion equations that can be found in analysing the water and energy budget of the soil-snow (with freezing soil) continuum. In practice I used the material from my class in hydrology (the whole stuff here) and other material from GEOtop's talks, and the presentation is, at the moment, in Italian and English.
Incidentally two of these equations present  discontinuities due to phase transitions and the three of them require special numerical methods to be integrated. Here I suggest that a good method could be the Nested Newton one, introduced recently by Casulli and Zanolli (for integrating Richards), and before by  Brugnano and Casulli (for integrating Boussinesq equation).
Below you can find also the audio (in Italian) of the lecture: Richards equation (21.9 Mb); Frozen Soil (18.4 Mb); Snow (7.1 Mb). I gave longer presentations on Richards equation, in Todini Symposium (here), and at the summer School on Landslide Modelling in Praia a Mare (here).
An update. A new treatment of part of this matter is given by Niccolo Tubini in his Master Thesis. The slides he used in the 2017 lecture are here.

Essential References

L. Brugnano and V. Casulli, Iterative solution of piecewise linear systems and applications
to ﬂows in porous media, SIAM J. Sci. Comput., 31 (2009), pp. 1858–1873.

Casulli, V., and Zanolli, P., A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form, SIAM J. Sci. Comput., 32(4), 2255–2273,  Volume 32, Issue 4, 2010.

Cordano E., and Rigon R., A mass-conservative method for the integration of the two-dimensional groundwater (Boussinesq) equation,  Water Resour. Res., 49, doi:10.1002/wrcr.20072, 2013.

Dall’Amico, M.; Endrizzi, S., Gruber, S; and Rigon, R. (2011), An energy-conserving model of freezing variably-saturated soil, The Cryosphere.

Endrizzi S., Gruber S., Dall’Amico M., Rigon R., GEOtop 2.0.: Simulating the combined energy and water balance at and below the land surface accounting for soil freezing, snow cover and terrain effects, Geosci. Model Dev., 2015