Semismooth Newton Method for Piecewise Linear Systems in Fluid Flow Models

• Grant number: 25/13449-3
• Support Opportunities: Regular Research Grants
• Start date: October 01, 2025
• End date: September 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Gabriel Haeser
• Grantee: Gabriel Haeser
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/08706-1 - Numerical optimization, AP.TEM

Abstract

This project proposes the study, implementation, and analysis of the semismooth Newton method for the efficient solution of piecewise linear systems arising from the mathematical modeling of flows in aquifers with two regions - a saturated and an unsaturated zone - according to the formulation by (L. Li, D.A. Lockington, D.A. Barry, J.-Y. Parlange, P. Perrochet (2003), Confined-unconfined flow in a horizontal confined aquifer, Journal of Hydrology 271, 150-155). This type of flow is described by differential equations with two moving boundaries, representing the saturated and unsaturated zones, with physical constraints that ensure non-negative water tables. Discretization of the model results in a piecewise linear system, which is efficiently solved by the semismooth Newton method (N. F. Armijo, Y. Bello-Cruz, G. Haeser (2023), On the convergence of iterative schemes for solving a piecewise linear system of equations, Linear Algebra and Its Applications, 665, pp. 291-314). This project aims to implement the method for such problems and to analyze its efficiency, robustness, and accuracy, compared to analytical and benchmark numerical solutions. (AU)