ENACT - Efficient numerical schemes and optimal control methods for time-dependent partial differential equations
Abstract:
MO1: High order conservative semi-Lagrangian (SL) schemes
MO2: Optimal control problems and computational social dynamics
MO3: Mathematical models and numerical methods for fluid-structure interaction (FSI) problems
MO4: High order numerical schemes for moving complex-shaped domains coupled with multiphysics
The usage of modern high-performance computing (HPC) with very big computations is foreseen.
Dynamic processes in continuum physics are modeled using time-dependent partial differential equations, which are based on the conservation of some physical quantities, such as mass, momentum and energy. Most of the physical systems exhibit multiple space and time scales, like low Mach number flows in classical fluid mechanics, or rarefied gas dynamics with small Knudsen number in kinetic models. Improving the efficiency of state-of-the-art numerical methods for PDE while keeping high accuracy and physical properties of the solution is the main goal of this project.
In MO1 the SL approach is exploited to enlarge the stability region of numerical schemes for the solution of advection problems. Conservation, asymptotic and positivity preservation properties of the novel methods will be assessed. The design of efficient schemes for optimal control problems of large systems of differential equations will be investigated in MO2. The research activity will focus on efficient time integration techniques which become crucial while dealing with multiscale models such as hyperbolic relaxation systems. Since kinetic models can be employed for the description of collective behaviors, we plan to develop computational methods for collective motion of intelligent systems such as swarm of drones and opinion formation, including the modeling and simulation of epidemic spreading with heterogeneous space and population structure (MO2). To overcome the coupling of different physical models in multi-material simulations, in MO3 a unified model for continuum mechanics will be solved at the aid of the Arbirtrary-Lagrangian-Eulerian (ALE) method applied to fluid-structure interaction problems on moving unstructured domains. Moving objects with curved boundaries in a multiphysics context are tackled in MO4, where high order ghost-point methods are proposed with multigrid techniques.
In this ambitious project we aim at making innovative contributions in applied mathematics and scientific computing with new algorithms that go beyond the state of the art. We will demonstrate the applicability of the proposed schemes and their utility by considering a variety of applications that range from academic benchmarks to more realistic scenarios and computations close to the industrial sector.
Risultati attesi:
This project aims to develop advanced high-order numerical methods for simulating and controlling multiscale, multiphysics systems. Applications span fluid dynamics, kinetic theory, epidemiology, and social sciences. The research is structured around four main objectives (MOs), which are listed hereafter along with the expected achievements.
MO1 focuses on high-order semi-Lagrangian (SL) schemes for hyperbolic systems. These methods will allow large time steps while preserving accuracy and physical properties like conservation and positivity. Coupled with IMEX time integrators, they will achieve stability and asymptotic preservation, especially in stiff regimes such as the low Mach number limit. Applications include compressible flows, kinetic equations (e.g., BGK), and new SL-ALE methods for moving domains.
MO2 targets optimal control and learning in systems modeled by mean-field and kinetic equations. Techniques like Model Predictive Control, reinforcement learning, and Boltzmann-type approaches will be explored for problems in crowd motion, traffic, epidemics, and opinion dynamics. These methods will address uncertainty and reduce computational cost in high-dimensional settings.
MO3 proposes a unified approach to fluid-structure interaction using a single hyperbolic PDE system for both fluids and solids. This removes the need for interface tracking and enables a monolithic numerical treatment. IMEX and SL-Arbitrary-Lagrangian-Eulerian methods will ensure efficiency and robustness, with applications in biofluid dynamics and soft materials.
MO4 develops a high-order solver for incompressible flows in complex, moving domains using ghost-point and multigrid techniques. It will integrate with the MO3 framework, enabling accurate and efficient simulations of deformable structures in fluids, with potential applications in aerospace, renewable energy, and biomedical devices.
Together, these advances will provide accurate, stable, and efficient tools for next-generation simulation challenges.
Dettagli progetto:
Referente scientifico: Boscheri Walter
Fonte di finanziamento: Bando PRIN 2022
Data di avvio: 28/09/2023
Data di fine: 28/09/2025
Contributo MUR: 7.095 €
Cofinanziamento UniFe: 6.825 €
Partner:
- Politecnico di FERRARA (capofila)
- Università degli Studi di VERONA
- Università degli Studi CATANIA
- Università degli Studi di TRENTO