B-Spline Methods for Partial Differential Equations toward Applications in Hydrogeological Modeling

This thesis investigates the use of B-spline basis functions for the numerical solution of elliptic and time-dependent partial differential equations. The work originates from an internship at the Model Validation Office of Risk Management at Unipol Assicurazioni SPA, motivated by the application of spline-based methods to the modeling and forecasting of hydrogeological phenomena. In the one-dimensional elliptic setting, two spatial discretization strategies are compared: the Collocation and the Galerkin methods. The Galerkin is more accurate than the Collocation, and is further benchmarked against classical Lagrangian finite elements, demonstrating the advantages of B-splines in achieving high-order approximations with a negligible increase in the number of degrees of freedom. The analysis is then extended to the two-dimensional Poisson problem via tensor product construction, where a convergence study confirms the higher accuracy of the Galerkin method, at the cost of increased computational effort compared to Collocation. The second part of the thesis addresses time-dependent problems relevant to hydrogeological risk. In particular, the nonlinear Burgers' equation is investigated in both one and two space dimensions, using the Method of Lines. In this setting, the one-dimensional results show consistency with the existing literature and, in some cases, demonstrate improved performance. Moreover, the two-dimensional results are validated against the Lagrangian finite element method, and they show better accuracy. The findings confirm the approximation quality of B-spline basis functions and suggest promising directions for future research.