Mathematical modeling and computational methods for two-layer shallow water systems: theory, numerics and applications

• Grant number: 25/07662-6
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: July 27, 2025
• End date: August 09, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Eduardo Cardoso de Abreu
• Grantee: Eduardo Cardoso de Abreu
• Visiting researcher: Gerardo Hernandez Duenas
• Visiting researcher institution: Universidad Nacional Autónoma de México, Juriquilla (UNAM), Mexico
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

The two-layer nonlinear shallow water equations are used to model a wide range of atmospheric and geo-physical flows. They can be derived from the Euler equations through cross-sectional averaging and are particularly suited for describing flows that are predominantly horizontal. These equations form a system of nonlinear hyperbolic conservation laws with geometric source terms (i.e., such models are nonlinear hyperbolic balance laws) that account for the effects of topography and flow-constraining geometry. Our goal is to develop a novel numerical scheme that offers several advantages over existing approaches. In particular, we are interested in applying the new scheme to the nontrivial two-layer shallow water models. For São Paulo, FAPESP and UNICAMP/IMECC this project firms an opportunity to the study of a state of the art numerical-analytical mathematics innovation on the study of numerical analysis and potential applications that are related to the study extreme events in climate and other systems and thus with a possible positive impact on the climate change adaptation strategy for the State of São Paulo. In recent years, Dr. G. Hernández-Dueñas and his collaborators have focused on developing models based on partial differential equations, with a particular emphasis on hyperbolic conservation laws. This special class of models is well-suited for analyzing a wide range of physical phenomena. One of their key characteristics is that information propagates at a finite speed, and shock waves can form in finite time, even when the initial conditions are smooth. Dr. Hernández-Dueñas has designed efficient, accurate, and robust numerical schemes specifically tailored for hyperbolic conservation laws. In this project proposal, we aim to extend these techniques by integrating them with the Lagrangian-Eulerian methods developed by Dr. Abreu and his research group, recently introduced as approximation methods for hyperbolic balance laws and hyperbolic conservation laws with relaxation and analyzed via weak asymptotic methods. A delicate balance between the flux gradients and the geometric source terms gives rise to a variety of interesting flow behaviors, including non-trivial equilibrium solutions. In this proposal, we focus on both two-dimensional shallow water flows and their one-dimensional counterparts, which describe flow through channels with variable cross-sectional areas. In such cases, the interaction between the bottom topography and the channel's contraction plays a crucial role in shaping the resulting flow dynamics. Several of these schemes have been applied - or newly adapted - to simulate, analyze, or reconstruct real-world geophysical flows such as gravity currents or straight flows. These scenarios typically introduce additional complexities, including non-symmetric channel geometries derived from bathymetric data and the need to model processes such as entrainment. Hyperbolic conservation/balance law models (Shallow water systems, one-layer, two-layers or multi-layers are nontrivial particular submodels) differ from other types of evolutionary nonlinear partial differential equations due to the possible loss of regularity of solutions in finite time: formation of shock waves. Therefore, the mathematical theory and the corresponding numerical analysis have to deal with weak solutions. Any progress in the field might contribute to a deeper understanding in several other models and topics, ranging from turbulent flow (Navier-Stokes) motion to hyperbolic transport-convection in porous media (Darcy flows) models. These models can also help predict and mitigate the impacts of extreme events. We plan to extend previous approaches introduced by Dr. G. Hernández-Dueñas and Dr. Abreu with the aim to apply them to two-layer shallow water systems. (AU)