arXiv subjects

Denise Aregba-Driollet

Publications and source records attributed to Denise Aregba-Driollet.

6 recordsLinked to original sources

arXiv2026

A vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes equations

We introduce a second-order accurate vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes system, inspired by a discrete-velocity kinetic approximation proposed by Carfora and Natalini [ESAIM: M2AN, 42(1), 93-112, 2008]. Advantages and drawbacks compared to relaxation schemes are investigated by providing spectral analyses in the linearized case, and numerical validations on the genuinely non-linear problem.

math.NA↗

arXiv2026

Convergence of a two-relaxation-times kinetic approximation towards the solution of a scalar conservation law

We introduce a two-relaxation-times (TRT) kinetic approximation for scalar non-linear conservation laws in one space dimension, addressing the convergence of relaxation approximations to entropy solutions. The proposed TRT system, derived from a lattice Boltzmann scheme, generalizes the classical BGK (single-relaxationtime) framework. Requesting quasi-monotonicity of the relaxation operator, we establish global existence of solutions for the TRT system and prove their convergence, using a lattice Boltzmann scheme, to the entropy solution of the original conservation law as the relaxation times vanish. Moreover, we propose a qualitative analysis, clarifying the role of relaxation parameters and equilibrium coefficients in shaping solutions, via Chapman-Enskog expansion and looking at travelling wave solutions.

math.AP↗

arXiv2025

Equilibrium boundary conditions for vectorial multi-dimensional lattice Boltzmann schemes

The concept of equilibrium is a general tool to fill the gap between macroscopic and mesoscopic information, both within kinetic systems and kinetic schemes. This work explores the use of equilibria to devise numerical boundary conditions for multi-dimensional vectorial lattice Boltzmann schemes tackling systems of hyperbolic conservation laws. In the scalar case, we prove convergence for schemes with monotone relaxation to the weak entropy solution by Bardos, Leroux, and N{é}delec [Commun. Partial Differ. Equ., 4 (9), 1979], following the path by Crandall and Majda [Math. Comput., 34, 149 (1980)]. Numerical experiments are conducted both for scalar and vectorial problems, and demonstrate the effectiveness of equilibrium boundary conditions in capturing significant physical phenomena.

math.NA↗

arXiv2025

Monotonicity and convergence of two-relaxation-times lattice Boltzmann schemes for a non-linear conservation law

We address the convergence analysis of lattice Boltzmann methods for scalar non-linear conservation laws, focusing on two-relaxation-times (TRT) schemes. Unlike Finite Difference/Finite Volume methods, lattice Boltzmann schemes offer exceptional computational efficiency and parallelization capabilities. However, their monotonicity and $L^{\infty}$-stability remain underexplored. Extending existing results on simpler BGK schemes, we derive conditions ensuring that TRT schemes are monotone and stable by leveraging their unique relaxation structure. Our analysis culminates in proving convergence of the numerical solution to the weak entropy solution of the conservation law. Compared to BGK schemes, TRT schemes achieve reduced numerical diffusion while retaining provable convergence. Numerical experiments validate and illustrate the theoretical findings.

math.NA↗

arXiv2014

Godunov scheme for Maxwell's equations with Kerr nonlinearity

We study the Godunov scheme for a nonlinear Maxwell model arising in nonlinear optics, the Kerr model. This is a hyperbolic system of conservation laws with some eigenvalues of variable multiplicity, neither genuinely nonlinear nor linearly degenerate. The solution of the Riemann problem for the full-vector 6x6 system is constructed and proved to exist for all data. This solution is compared to the one of the reduced Transverse Magnetic model. The scheme is implemented in one and two space dimensions. The results are very close to the ones obtained with a Kerr-Debye relaxation approximation.

math.NA↗

arXiv2012

Time Asymptotic High Order Schemes for Dissipative BGK Hyperbolic Systems

We introduce a new class of finite differences schemes to approximate one dimensional dissipative semilinear hyperbolic systems with a BGK structure. Using precise analytical time-decay estimates of the local truncation error, it is possible to design schemes, based on the standard upwind approximation, which are increasingly accurate for large times when approximating small perturbations of constant asymptotic states. Numerical tests show their better performances with respect to those of other schemes.

math.NA↗