Hans Peter Stimming

LaBonte's method revisited: An effective steepest descent method for micromagnetic energy minimization

We present a steepest descent energy minimization scheme for micromagnetics. The method searches on a curve that lies on the sphere which keeps the magnitude of the magnetization vector constant. The step size is selected according to a modified Barzilai-Borwein method. Standard linear tetrahedral finite elements are used for space discretization. For the computation of quasistatic hysteresis loops, the steepest descent minimizer is faster than a Landau-Lifshitz micromagnetic solver by more than a factor of two. The speed up on a graphic processor is 4.8 as compared to the fastest single-core central processing unit (CPU) implementation.

Fluctuations and Stochastic Processes in One-Dimensional Many-Body Quantum Systems

We study the fluctuation properties of a one-dimensional many-body quantum system composed of interacting bosons and investigate the regimes where quantum noise or, respectively, thermal excitations are dominant. For the latter, we develop a semiclassical description of the fluctuation properties based on the Ornstein-Uhlenbeck stochastic process. As an illustration, we analyze the phase correlation functions and the full statistical distributions of the interference between two one-dimensional systems, either independent or tunnel-coupled, and compare with the Luttinger-liquid theory.

On the time evolution of Wigner measures for Schrodinger equations

In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.

Highly nonlocal optical nonlinearities in atoms trapped near a waveguide

Nonlinear optical phenomena are typically local. Here, we predict the possibility of highly nonlocal optical nonlinearities for light propagating in atomic media trapped near a nano-waveguide, where long-range interactions between the atoms can be tailored. When the atoms are in an electromagnetically induced transparency configuration, the atomic interactions are translated to long-range interactions between photons and thus to highly nonlocal optical nonlinearities. We derive and analyze the governing nonlinear propagation equation, finding a roton-like excitation spectrum for light and the emergence of order in its output intensity. These predictions open the door to studies of unexplored wave dynamics and many-body physics with highly nonlocal interactions of optical fields in one dimension.

FFT-based Kronecker product approximation to micromagnetic long-range interactions

We derive a Kronecker product approximation for the micromagnetic long-range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi-linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results.

On the non-equivalence of perfectly matched layers and exterior complex scaling

Abstract The perfectly matched layers (PML) and exterior complex scaling (ECS) methods for absorbing boundary conditions are analyzed using spectral decomposition. Both methods are derived through analytical continuations from unitary to contractive transformations. We find that the methods are mathematically and numerically distinct: ECS is complex stretching that rotates the operators spectrum into the complex plane, whereas PML is a complex gauge transform which shifts the spectrum. Consequently, the schemes differ in their time-stability. Numerical examples are given.

Numerical studies for nonlinear Schrödinger equations: the Schrödinger–Poisson-Xα model and Davey–Stewartson systems

We consider the dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapour phase. As the basic mathematical model we use the Euler equations for a sharp interface approach and local and global versions of the NavierStokes-Korteweg equations for the diffuse interface approach. The mathematical models are discussed and we introduce discretization methods for both approaches. Finally numerical simulations in one and two space dimensions are presented.This work is motivated by the numerical simulation of the generation and break-up of droplets after the impact of a rigid body on a tank filled with a compressible fluid. This paper splits into two very different parts. The first part deals with the modeling and the numerical resolution of a spray of liquid droplets in a compressible medium like air. Phenomena taken into account are the breakup effects due to the velocity and pressure waves in the compressible ambient fluid. The second part is concerned with the transport of a rigid body in a compressible liquid, involving reciprocal effects between the two components. A new one-dimensional algorithm working on a fixed Eulerian mesh is proposed. The GENJET (GENeration and breakup of liquid JETs) project has been proposed by the Centre d’Études de Gramat (CEG) of the Délégation Générale de l’Armement (DGA). It concerns the general study of the consequences of a violent impact of a rigid body against a reservoir of fluid. Experiments show that once the solid has pierced the shell of the reservoir, it provokes a dramatic increase of the pressure inside the reservoir, whose effect is the ejection of some fluid through the pierced hole. The generated liquid jet then expands into the ambient air, where it can interact with some air pressure waves, leading to a fragmentation of the jet into small droplets. These experiments show that after having pierced the shell, the projectile behaves as a rigid body. They also show that the liquid inside the reservoir behaves as a compressible fluid (indeed, the projectile velocity, around 1000 m.s, is in the same order of magnitude than the sound speed in the liquid). The modeling of such a complex flow requires to take into account very different regimes, from the pure compressible and/or incompressible flow condition to a droplet regime (such a regime sharing some similarities with kinetic modeling of Liquid jet generation and breakup 3 particles). Moreover many scales are needed to correctly describe the complete experiments, from the large hydrodynamic scale to the small droplet scale. The study done during CEMRACS 2004 focused on the fluid regime and on the droplet regime, since some important difficulties are still there for both regimes separately. • Concerning the breakup of droplets in the air, we have focused on physical and numerical modeling issues. • Concerning the fluid regime, an important difficulty at the numerical level is that we want to get an accurate numerical description of the transport of a rigid body inside a compressible fluid. Even if the rigid body is of course not a fluid, the situation shares at the numerical level a lot of similarities with the coupling an incompressible fluid with a compressible one. Thus this part of the study concerns more numerical algorithms than the modeling. The present paper follows this cutout of the study. Section 1 presents the modeling of the breakup of droplets, whereas section 2 treats the coupling of the rigid body and the fluid. In both sections, numerical results are reported. In view of the main goal of the GENJET project, a natural perspective of the work described below would be the coupling of the models, algorithms and numerical methods. 1. A kinetic modeling of a breaking up spray with high Weber numbers In this section, we aim to model a spray of droplets which evolve in an ambient fluid (typically the air). That kind of problem was first studied by Williams for combustion issues [32]. The works of O’Rourke [20] helped to set the modeling of such situations and their numerical simulation through an industrial code, KIVA [1]. The main phenomenon that occuring in the spray is the breakup of the droplets. Any other phenomena, such as collisions or coalescence, will be neglected in this work, but they are reviewed in [3] for example. Instead of using the TAB model (see [2]), which is more accurate for droplets with low Weber numbers, we choose the so-called Reitz wave model [27], [21], [4]. Then this breakup model is taken into account in a kinetic model [14], [2]. The question of the spray behavior with respect to the breaking up has arised in the context of the French military industry. One aims to model with an accurate precision the evolution of a spray of liquid droplets inside the air. In that situation, the droplets of the spray are assumed to remain incompressible (the mass density ρd is a constant of the problem) and spherical. We also assume that the forces on the spray are negligible with respect to the drag force, at least at the beginning of the computations. After a few seconds, the gravitation may become preponderant. Note that the aspects of energy transfer will not be tackled in this report.In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADER-DG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N +2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction.In this paper, we introduce a new PIC method based on an adaptive multi-resolution scheme for solving the one dimensional Vlasov–Poisson equation. Our approach is based on a description of the solution by particles of unit weight and on a reconstruction of the density at each time step of the numerical scheme by an adaptive wavelet technique: the density is firstly estimated in a proper wavelet basis as a distribution function from the current empirical data and then “de-noised” by a thresholding procedure. The so-called Landau damping problem is considered for validating our method. The numerical results agree with those obtained by the classical PIC scheme, suggesting that this multi-resolution procedure could be extended with success to plasma dynamics in higher dimensions.