Patterns for the application of modern informatics to the integration of PDEs: the case of the Boussinesq Equation

Today Francesco Serafin graduated finishing his master in civil and environmental  engineering. In brief, the scope of his thesis was to implement a series of classes, eventually ported to OMS, to solve the groundwater Boussinesq equation (under the link, please find also specific reference to previous work), but with a more large scope to envision an object oriented structure which could work for any PDE. Let Francesco's introduction talk:

"Mathematical models play a fundamental role in many scientific and engineering fields in today’s world. They are used for example in geotechnics to evalute the hillslope stability, in weather science to predict weather trends and produce weather reports, in structural design to study the resistance to stress, and in fluid dynamics to compute fluid flows and air flows.

Consequently mathematical models are evolving all the time: more and more new numerical methods are being invented to solve the Partial Differential Equations (PDE)s that describe physical problems with increasing precision, and more and more complex and efficient processor units are being created to reduce the computational time.
Therefore, the code into which the mathematical models are translated has to be “dynamic” in order to be easily updated on the basis of the continuous developments (Formetta et al. (2014) [16]).
On the other hand, completely different physical problems are often de- scribed using similar PDEs. For this reason, the numerical methods which provide solutions to different problems can be the same. This suggest the implementation of an IT infrastructure that hosts a standard structure for solving PDEs and that can serve various disciplines with the minimum of hassles.

This work is focused on the application of what is envisioned above, with the main purpose of the creation of an abstract code for implementing every type of mathematical model described by PDEs.

We work on hydrological topics but we hope to design a structure of general interest. Obviously the final goal of any work of this type is to find a proper numerical solver, and therefore, part of the thesis is devoted to the analysis of the problem under scrutiny, and the description of the solution found."

The thesis can be found here. The presentation given at the thesis discussion is here.  The code can be found here.