Analysis and development of parallel-in-time numerical methods for fluid dynamics problems

• Grant number: 25/16494-0
• Support Opportunities: Regular Research Grants
• Start date: August 01, 2026
• End date: July 31, 2029
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: João Guilherme Caldas Steinstraesser
• Grantee: João Guilherme Caldas Steinstraesser
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• City of the host institution: São Paulo
• Associated researchers: Fabricio Rodrigues Lapolli ; Lucas Franceschini ; Pedro da Silva Peixoto
• Associated scholarship(s): 26/25784-4 - Introduction to model order reduction for the numerical solution of partial differential equations, BP.IC

Abstract

Parallel-in-time methods aim to replace the serial numerical integration of time-dependent differential equations with an iterative process relying on the simultaneous use of a coarse and a fine discretization of the problem, seeking to reduce the computing time of numerical simulations. Nevertheless, parallel-in-time methods present instabilities and slow convergence when applied to simple hyperbolic problems, discouraging their application to complex advection-dominated problems, such as those arising in fluid dynamics.This research project aims to develop and analyze parallel-in-time methods, focusing on complex applications in fluid dynamics, such as weather prediction and flow in wind farms. The central hypothesis is that the choice of coarse discretization is crucial, such that it needs to be sufficiently accurate and computationally low expensive. The proposed study will combine efforts in numerical analysis, scientific computing, and knowledge of physical processes, being conducted on two fronts: (i) we aim to formulate theoretical convergence and stability results of the parallel-in-time methods, with numerical verification, as a function of the choice of coarse discretization; and (ii) we seek to formulate, develop and analyze coarse models based on learning approaches (reduced order modeling and machine learning), aiming to incorporate the fine dynamics of the problem into the coarse discretization. The proposed research will allow to develop the study around temporal parallelization within the national scientific community, which is still incipient despite the growing international interest in the topic. (AU)