arXiv subjects

François Dubois

Publications and source records attributed to François Dubois.

At least 19 recordsLinked to original sources

arXiv2026

Isotropy and Galilean invariance of Lattice Boltzmann Method: Theoretical and numerical analysis using oblique dipole benchmark *

This work focuses on the two-dimensional, nine-velocity (D2Q9) lattice Boltzmann model. First, we show that the D2Q9 scheme cannot achieve secondorder accuracy unless the cubic velocity terms are neglected, and we explain how some of these parasitic terms can be eliminated. Second, we demonstrate that the standard choice of the equilibrium distribution has no effect on the equivalent PDE at second order. Finally, we numerically investigate the effect of these cubic terms and study different choices of equilibrium distributions using a new benchmark called the Oblique Dipole Benchmark, which describes obliquely propagating 2D vortex dipoles with periodic boundary conditions.

math.AP↗

arXiv2026

Analytic solutions and numerical method for a coupled thermo-neutronic problem

We consider in this contribution a simplified idealized one-dimensional model in a nuclear core reactor coupling the diffusion equation on the neutron flux with the enthalpy equation for the water which collects the heat produced by this idealized nuclear core. These equations are coupled through the dependency of the coefficients of the diffusion equation in terms of the enthalpy. We propose a numerical method treating globally the coupled problem for finding its unique solution. Simultaneously, we use incomplete elliptic integrals to represent analytically the density of neutrons and the enthalpy in the fluid. Both methods lead to the same solution with high accuracy. However, another quantity, generally used as a benchmark for comparing results, depends considerably on the approximation used for the coefficients of the diffusion equation.

math.NA↗

arXiv2025

Multiresolution relaxation times lattice Boltzmann schemes with projection

We propose to extend the multiresolution relaxation times lattice Boltzmann schemes with an additional projection step. For the explicit example of the D2Q9 scheme, we define this extended method. We prove that in general the projection step does not change the asymptotic partial differential equations at second order. We present four numerical test cases. One concerns linear stability with a Fourier analysis with a single-vertex scheme. Three bidimensional fluid flows with a coarse mesh have been tested: the Minion and Brown sheared flow, the Ghia, Ghia and Shin lid-driven cavity and an unsteady acoustic wave. Our results indicate that the bulk viscosity can be dramatically reduced with a better stability than the initial scheme.

nlin.CG↗

arXiv2025

Eigenlogic and Probabilistic Inference; when Bayes meets Born

This paper shows how inference is treated within the context of Eigenlogic projection operators in linear algebra. In Eigenlogic operators represent logical connectives, their eigenvalues the truth-values and the associated eigenvectors the logical models. By extension, a probabilistic interpretation is proposed using vectors outside the eigensystem of the Eigenlogic operators. The probability is calculated by the quantum mean value (Born rule) of the logical projection operators. We look here for possible connections between the Born rule in quantum mechanics and Bayes' theorem from probability theory and show that Eigenlogic offers an innovative approach to address the probabilistic version of logical inference (material implication) in a quantum context.

quant-ph↗

arXiv2025

Simpson variational integrator for nonlinear systems: a tutorial on the Lagrange top

This contribution presents an integration method based on the Simpson quadrature. The integrator is designed for finite-dimensional nonlinear mechanical systems that derive from variational principles. The action is discretized using quadratic finite elements interpolation of the state and Simpson's quadrature, leading to discrete motion equations. The scheme is implicit, symplectic, and fourth-order accurate. The proposed integrator is compared with the implicit midpoint variational integrator on two examples of systems with inseparable Hamiltonians. First, the example of the nonlinear double pendulum illustrates how the method can be applied to multibody systems. The analytical solution of the Lagrange top is then used as a reference to analyze accuracy, convergence, and precision of the numerical method. A reduced Lagrange top system is also proposed and solved with a classical fourth-order method. Its solution is compared with the Simpson solution of the complete system, and the convergence order of the difference between both is consistent with the order of the classical method.

math.NA↗

arXiv2025

A variational symplectic scheme based on Lobatto's quadrature

We present a variational integrator based on the Lobatto quadrature for the time integration of dynamical systems issued from the least action principle. This numerical method uses a cubic interpolation of the states and the action is approximated at each time step by Lobatto's formula. Numerical analysis is performed on both a harmonic oscillator and a nonlinear pendulum. The geometric scheme is conditionally stable, sixth-order accurate, and symplectic. It preserves an approximate energy quantity. Simulation results illustrate the performance and the superconvergence of the proposed method.

math.NA↗

arXiv2025

General fourth order Chapman-Enskog expansion of lattice Boltzmann schemes

In order to derive the equivalent partial differential equations of a lattice Boltzmann scheme,the Chapman Enskog expansion is very popular in the lattive Boltzmann community. A maindrawback of this approach is the fact that multiscale expansions are used without any clearmathematical signification of the various variables and operators. Independently of thisframework, the Taylor expansion method allows to obtain formally the equivalent partialdifferential equations. In this contribution, we prove that both approaches give identicalresults with acoustic scaling for a very general family of lattice Boltzmann schemes and upto fourth order accuracy. Examples with a single scalar conservation illustrate our purpose.

math.NA↗

arXiv2024

Discrete harmonics for stream function-vorticity Stokes problem

We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity. For this problem, the classical finite elements method of degree one converges only to order one-half for the L2 norm of the vorticity. We propose to use harmonic functions to approach the vorticity along the boundary. Discrete harmonics are functions that are used in practice to derive a new numerical method. We prove that we obtain with this numerical scheme an error of order one for the L2 norm of the vorticity.

math.NA↗

arXiv2024

On single distribution lattice Boltzmann schemes for the approximation of Navier Stokes equations

In this contribution we study the formal ability of a multi-resolution-times lattice Boltzmann scheme to approximate isothermal and thermal compressible Navier Stokes equations with a single particle distribution. More precisely, we consider a total of 12 classical square lattice Boltzmann schemes with prescribed sets of conserved and nonconserved moments. The question is to determine the algebraic expressions of the equilibrium functions for the nonconserved moments and the relaxation parameters associated to each scheme. We compare the fluid equations and the result of the Taylor expansion method at second order accuracy for bidimensional examples with a maximum of 17 velocities and three-dimensional schemes with at most 33 velocities. In some cases, it is not possible to fit exactly the physical model. For several examples, we adjust the Navier Stokes equations and propose nontrivial expressions for the equilibria.

math.NA↗

arXiv2024

Numerical approximations of a lattice Boltzmann scheme with a family of partial differential equations

In this contribution, we address the numerical solutions of high-order asymptotic equivalent partial differential equations with the results of a lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial dimension. We first derive a family of equivalent partial differential equations at various orders, and we compare the lattice Boltzmann experimental results with a spectral approximation of the differential equations. For an unsteady situation, we show that the initialization scheme at a sufficiently high order of the microscopic moments plays a crucial role to observe an asymptotic error consistent with the order of approximation. For a stationary long-time limit, we observe that the measured asymptotic error converges with a reduced order of precision compared to the one suggested by asymptotic analysis.

math.NA↗

arXiv2024

On Macroscopic Intricate States

The present contribution is in the field of quantum modelling of macroscopic phenomena. The focus is on one enigmatic aspect of quantum physics, namely the Einstein-Podolsky-Rosen paradox and entanglement. After a review of the state of the art concerning macroscopic quantum effects and quantum interaction, this contribution proposes a link between embryology and acupuncture in the framework of macroscopic intricate states induced by quantum mechanics.

physics.bio-ph↗

arXiv2024

On quantum models for opinion and voting intention polls

In this contribution, we construct a connection between two quantum voting models presented previously. We propose to try to determine the result of a vote from associated given opinion polls. We introduce a density operator relative to the family of all candidates to a particular election. From an hypothesis of proportionality between a family of coefficients which characterize the density matrix and the probabilities of vote for all the candidates, we propose a numerical method for the entire determination of the density operator. This approach is a direct consequence of the Perron-Frobenius theorem for irreductible positive matrices. We applyour algorithm to synthetic data and to operational results issued from the French presidential election of April 2012.

physics.soc-ph↗

arXiv2024

A quantum approach for determining a state of the opinion

We propose to define a notion of state of the opinion in order to link politician popularity estimations and voting intentions. We present two ways of modelling: a classical approach and quantum modelling. We test these ideas on data obtained during the French presidential election of April 2012.

physics.soc-ph↗

arXiv2024

Theory of the lattice Boltzmann method: discrete effects due to advection

Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space derivatives. Several previous works have been devoted to analyzing the accuracy of these models with special emphasis on deviations from pure Newtonian viscous behaviour, related to higher order space derivatives of even order. The presentcontribution concentrates on possible inaccuracies of the advection behaviour linked to space derivatives of odd order. Detailed properties of advection-diffusion and athermal fluids are presented for two-dimensional situations allowing to propose situations that are accurate to third order in space derivatives. Simulations of the advection of a gaussian dot or vortex are presented. Similar results are discussed in appendices for three-dimensional advection-diffusion.

math.NA↗

arXiv2024

On Voting Process and Quantum Mechanics

In this communication, we propose a tentative to set the fundamental problem of measuring process done by a large structure on a microsopic one. We consider the example of voting when an entire society tries to measure globally opinions of all social actors in order to elect a delegate. We present a quantum model to interpret an operational voting system and propose a quantum approach for grading step of Range Voting, developed by M. Balinski and R. Laraki in 2007.

physics.soc-ph↗

arXiv2024

Conditions aux limites fortement non lin{é}aires pour les {é}quations d'Euler de la dynamique des gaz

We study various formulations of the boundary conditions for the Euler equations of gas dynamics from a mathematical and numerical point of view. In the case of one space dimension, we recall the classical results, based on an analysis of the linearized problem. Then we present a more recent formulation of the problem, which allows for nonlinear effects at the boundary of the study domain. This formulation fits naturally into a finite volume discretization, and we present a significant one-dimensional test case.

math.NA↗

arXiv2024

Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes

We propose a formal expansion of multiple relaxation times lattice Boltzmann schemes in terms of a single infinitesimal numerical variable. The result is a system of partial differential equations for the conserved moments of the lattice Boltzmann scheme. The expansion is presented in the nonlinear case up to fourth order accuracy. The asymptotic corrections of the nonconserved moments are developed in terms of equilibrium values and partial differentials of the conserved moments. Both expansions are coupled and conduct to explicit compact formulas. The new algebraic expressions are validated with previous results obtained with this approach. The example of isothermal D2Q9 lattice Boltzmann scheme illustrates the theoretical framework.

math.NA↗

arXiv2024

Simpson's quadrature for a nonlinear variational symplectic scheme

We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state between two time steps and we approximate the action in one time step by the Simpson's quadrature formula. The resulting scheme is nonlinear and symplectic. First numerical experiments concern a nonlinear pendulum and we have observed experimentally very good convergence properties.

math.NA↗