Sponsor: CEA
Supervisor: P. Tremblin
1. Context
This project builds on recent advances in quadrature-based moment methods and quantum-inspired numerical techniques to develop efficient solvers for kinetic and fluid equations. In particular, two-node moment methods approximate velocity distributions using a minimal set of weighted Dirac deltas, enabling accurate modeling of non-equilibrium effects such as particle trajectory crossing and wall bouncing in dilute flows. Recent work has shown that such models, when extended to include 13 moments, provide a robust and efficient alternative to full kinetic simulations. In parallel, quantum-inspired approaches such as tensor-train and matrix product state decompositions have been successfully applied to compress high-dimensional PDE solutions, exploiting the low interscale correlations typical of turbulent or multiscale flows. These methods have demonstrated significant reductions in computational cost while preserving accuracy, suggesting a promising direction for combining low-rank tensor structures with quadrature-based closures.
2. Description and objectives
The objective of this project is to explore the potential of quantum-inspired numerical methods to enhance the efficiency and expressiveness of quadrature-based moment models for partial differential equations. Starting from a two-node, 13-moment method as a prototype for representing velocity distributions in kinetic equations, the project aims to
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implement and analyze this method in one spatial dimension,
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investigate its ability to capture non-equilibrium phenomena such as crossing trajectories and discontinuities,
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reformulate the inversion and evolution of moments using low-rank tensor techniques inspired by quantum many-body physics. Ultimately, the goal is to assess whether these tensor representations can reduce computational cost and improve scalability without compromising physical fidelity.