Polytopal Discontinuous Galerkin Methods for Geophysical Fluid Dynamics

• Grant number: 25/02678-1
• Support Opportunities: Scholarships abroad - Research
• Start date: January 01, 2026
• End date: December 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Pedro da Silva Peixoto
• Grantee: Pedro da Silva Peixoto
• Host Investigator: Paola Francesca Antonietti
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: Politecnico di Milano, Italy
• Associated research grant: 21/06176-0 - Numerical methods for a new generation of weather and climate models, AP.PFPMCG.JP2

Abstract

Numerical methods for geophysical fluid dynamics are central to weather and climate modelling. Important modern atmospheric models, including a recent development of the next-generation global Brazilian atmospheric model (MONAN), employ spherical unstructured polygonal (Voronoi) grids. The numerical solution of the fluid dynamics relies on low-order finite volume schemes, which are highly sensitive to grid regularity and the shape of the spherical polygonal cell.Recent developments of Polytopal Discontinuous Galerkin (PolyDG) methods show accurate results on highly complex polygonal grids in various applications, including wave propagation (elastodynamics) and multiphysics brain modelling. PolyDG methods can provide accuracy, scalability, and geometric flexibility that typical low-order finite-volume methods cannot afford. Therefore, they have great potential for geophysical fluid dynamics, yet to be fully explored.This project aims to investigate PolyDG methods in geophysical fluid dynamics, focusing on both theoretical and numerical aspects. The theoretical components involve extending existing PolyDG theory for curved (spherical) polygons and developing a numerical analysis framework for these methods, taking into account prototype spherical problems. Additionally, ensuring the desired conservation and structural mimetic properties necessitates careful development of the schemes within the context of Geophysical Fluid Dynamics. On the numerical front, we aim to implement the examined PolyDG schemes on bi-periodic and spherical polygonal (Voronoi) grids for the rotating shallow water equations. This introduces challenges, particularly concerning the development of efficient numerical integration for curved polygonal cells. These objectives serve as the foundation for the potential implementation of such schemes in weather and climate models that currently utilise spherical polygonal grids, such as the MONAN model. The project brings together a FAPESP-funded research group in Numerical Geophysical Fluid Dynamics from the University of São Paulo and researchers from the Laboratory for Modelling and Scientific Computing (MOX) in the Department of Mathematics at Politecnico Milano, Italy, where cutting-edge research in DG methods on polytopes is conducted.