The unified gas kinetic scheme was initially designed to address multiscale challenges in rarefied gas dynamics and then extended to radiative transfert theory, as described by Bhatnagar-- Gross-Krook-like relaxation models. In this work, we extend its application to linear kinetic models with nonisotropic scattering collision operators, as well as Fokker--Planck models. These problems typically exhibit a fully diffusive nature in the optically thick limit (corresponding to a small Knudsen number). The scheme proposed in this work still leads to an asymptotic preserving property not only in this diffusive regime but also in the free transport limit. A series of numerical experiments confirm the effectiveness of the approach.
We present a framework based on computer-assisted proofs that turns geometry optimization simulations for atomistic structures into mathematical proofs. Starting from a numerically computed approximation of a local minimizer or saddle point, we use validated numerical computations to prove the existence of a critical point of the potential energy close to this approximation. We demonstrate this framework in two settings. In the first, we study capped carbon nanotubes modeled as minimizers of carbon interatomic potentials (harmonic, Tersoff, and a Huber potential) and obtain proven bounds on tube diameter, bond lengths, and bond angles. In particular, we show that caps induce diameter oscillations along the tube. As a second application, we consider a finite Lennard--Jones crystal in a face-centered cubic lattice and provide computer-assisted proofs of a local minimizer representing the perfect crystal, a local minimizer with a single vacancy defect, and a saddle point that connects two single-vacancy configurations on the energy landscape.
Abstract. We consider the long-time behavior of the Vlasov–Poisson–Fokker–Planck equations modeling the dynamics of two-species charged particles (e.g., electrons and ions) in a plasma under the strong effect of magnetic fields while also taking into account their small mass ratio. We derive a fluid model for both electrons and ions. In three space dimensions, a constraint occurs along the parallel direction of the magnetic field lines for each species. When neglecting the curvature of the magnetic lines and focusing on the plane motion of a plasma subjected to a magnetic field orthogonal to the plane, we provide a rigorous mathematical justification of the formally derived system in the two-dimensional setting. Our proofs rely on the modulated energy method.
When a thin sieve is submerged in a container filled with viscous fluid, an effective interface condition is needed for simulations. Supposing that this rigid porous membrane (a sieve) has periodic structure, using homogenization techniques, we rigorously derive a macroscopic interface condition in the form of Darcy's law across the sieve. More precisely, the velocity is proportional to the (large) stress jump across the sieve. We check that the derived boundary-value problem is well posed and justify the obtained model via error estimate.
In this paper, we propose a spectrally accurate solver for computing the Bogoliubovde Gennes (BdG) excitations of spin-1 Bose-Einstein condensates (BECs), which are governed by the BdG equations, around the mean-field ground state. The BdG equations are essentially a constrained eigenvalue/eigenfunction system. First, we investigate their analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Second, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free, and the operator-function evaluation is accelerated by a discrete fast Fourier transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis of the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and we investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1--3 spatial dimensions.
