Iterative Splitting Methods: Almost Asymptotic Symplectic Integrator for Stochastic Nonlinear Schr\"odinger Equation

In this paper we present splitting methods which are based on iterative schemes and applied to stochastic nonlinear Schroedinger equation. We will design stochastic integrators which almost conserve the symplectic structure. The idea is based on rewriting an iterative splitting approach as a successive approximation method based on a contraction mapping principle and that we have an almost symplectic scheme. We apply a stochastic differential equation, that we can decouple into a deterministic and stochatic part, while each part can be solved analytically. Such decompositions allow accelerating the methods and preserving, under suitable conditions, the symplecticity of the schemes. A numerical analysis and application to the stochastic Schroedinger equation are presented.Comment: 17 page

Numerical Methods of the Maxwell-Stefan Diffusion Equations and Applications in Plasma and Particle Transport

In this paper, we present a model based on a local thermodynamic equilibrium, weakly ionized plasma-mixture model used for medical and technical applications in etching processes. We consider a simplified model based on the Maxwell-Stefan model, which describe multicomponent diffusive fluxes in the gas mixture. Based on additional conditions to the fluxes, we obtain an irreducible and quasi-positive diffusion matrix. Such problems results into nonlinear diffusion equations, which are more delicate to solve as standard diffusion equations with Fickian's approach. We propose explicit time-discretisation methods embedded to iterative solvers for the nonlinearities. Such a combination allows to solve the delicate nonlinear differential equations more effective. We present some first ternary component gaseous mixtures and discuss the numerical methods.Comment: 1

Iterative Splitting Methods for Coulomb Collisions in Plasma Simulations

In this paper, we present splitting methods that are based on iterative schemes and applied to plasma simulations. The motivation arose of solving the Coulomb collisions, which are modeled by nonlinear stochastic differential equations. We apply Langevin equations to model the characteristics of the collisions and we obtain coupled nonlinear stochastic differential equations, which are delicate to solve. We propose well-known deterministic splitting schemes that can be extended to stochastic splitting schemes, by taking into account the stochastic behavior. The benefit decomposing the different equation parts and solve such parts individual is taken into account in the analysis of the new iterative splitting schemes. Numerical analysis and application to various Coulomb collisions in plasma applications are presented.Comment: 27 page

Simulation of a Heat Transfer in Porous Media

We are motivated to model a heat transfer to a multiple layer regime and their optimization for heat energy resources. Such a problem can be modeled by a porous media with different phases (liquid and solid). The idea arose of a geothermal energy reservoir which can be used by cities, e.g. Berlin. While hot ground areas are covered to most high populated cites, the energy resources are important and a shift to use such resources are enormous. We design a model of the heat transport via the flow of water through the heterogeneous layer of the underlying earth sediments. We discuss a multiple layer model, based on mobile and immobile zones. Such numerical simulations help to economize on expensive physical experiments and obtain control mechanisms for the delicate heating process.Comment: 2

Recent Spectroscopy Results from ZEUS

Recent results on light hadron spectroscopy are reported, with special emphasis on the evidence for a narrow baryonic state decaying to Ks p and Ks pbar, compatible with the pentaquark state theta^+ observed by fixed target experiments. The data were collected with the ZEUS detector at HERA using an integrated luminosity of 121 pb^-1. The analyses were performed in the central rapidity region of inclusive deep inelastic scattering at an ep centre-of-mass energy of 300-318 GeV. Evidence for a narrow resonance in the Ks p and Ks pbar invariant mass spectrum is obtained, with mass 1521.5 \pm 1.5(stat)^{+2.8}_{-1.7}(syst) and width consistent with the experimental resolution. If the Ks p part of the signal is identified with the strange pentaquark theta^+, the Ks pbar part is the first evidence for its antiparticle, thetabar^-. Supporting results on other light hadron resonances are also discussed.Comment: To appear in the proceedings of 39th Rencontres de Moriond on QCD and High-Energy Hadronic Interactions, La Thuile, Italy, 28 Mar - 4 Apr 200

Embedded Zassenhaus Expansion to Operator Splitting Schemes: Theory and Application in Fluid Dynamics

In this paper, we contribute operator-splitting methods improved by the Zassenhaus product for the numerical solution of linear partial differential equations. We address iterative splitting methods, that can be improved by means of the Zassenhaus product formula, which is a sequnential splitting scheme. The coupling of iterative and sequential splitting techniques are discussed and can be combined with respect to their compuational time. While the iterative splitting schemes are cheap to compute, the Zassenhaus product formula is more expensive, based on the commutators but achieves higher order accuracy. Iterative splitting schemes and also Zassenhaus products are applied in physics and physical chemistry are important and are predestinated to their combinations of each benefits. Here we consider phase models in CFD (computational fluid dynamics). We present an underlying analysis for obtaining higher order operator-splitting methods based on the Zassenhaus product. Computational benefits are given with sparse matrices, which arose of spatial discretization of the underlying partial differential equations. While Zassenhaus formula allows higher accuracy, due to the fact that we obtain higher order commutators, we combine such an improved initialization process to cheap computable to linear convergent iterative splitting schemes. Theoretical discussion about convergence and application examples are discussed with CFD problems.Comment: 17 page

Multiscale methods for Levitron Problems: Theory and Applications

In this paper, we describe a multiscale model based on magneto-static traps of neutral atoms or ion traps. The idea is to levitate a magnetic spinning top in the air repelled by a base magnet. For such a problem, we have to deal with different time and spatial scales and we propose a novel splitting method for solving the levitron problem. We focus on the multiscale problem, which we obtain by coupling the kinetic T and the potential U part of our equation. The kinetic and potential parts, can be seen as generators of flows. The main problem is based on the accurate computation of the Hamiltonian equation and we propose a novel higher order splitting scheme to obtain stable states near the relative equilibrium. To improve the splitting scheme we apply a novel method so called MPE (multiproduct expansion method), which include higher order extrapolation schemes. In numerical studies, we discuss the stability near this relative equilibrium with our improved time-integrators. Best results are obtained by iterative and extrapolated Verlet schemes in comparison to higher order explicit Runge-Kutta schemes. Experiments are applied to a magnetic top in an axisymmetric magnetic field (i.e. the Levitron) and we discuss the future applications to quantum computations.Comment: 12 pages, 6 figure

Iterative operator-splitting methods for unbounded operators: Error analysis and examples

In this paper we describe an iterative operator-splitting method for unbounded operators. We derive error bounds for iterative splitting methods in the presence of unbounded operators and semigroup operators. Here mixed applications of hyperbolic and parabolic type are allowed and discussed in the applications. Mixed experiments are applied to ordinary differential equations and evolutionary Schr\"odinger equations.Comment: 16 Page

Iterative Implicit Methods for Solving Hodgkin-Huxley Type Systems

We are motivated to approximate solutions of a Hodgkin-Huxley type model with implicit methods. As a representative we chose a psychiatric disease model containing stable as well as chaotic cycling behaviour. We analyze the bifurcation pattern and show that some implicit methods help to preserve the limit cycles of such systems. Further, we applied adaptive time stepping for the solvers to boost the accuracy, allowing us a preliminary zoom into the chaotic area of the system.Comment: 25 pages, 8 figures, 3 table

Comparison of Integrators for Electromagnetic Particle in Cell Methods: Algorithms and Applications

In this paper, we present different types of integrators for electro-magnetic particle-in-cell (PIC) methods. While the integrator is an important tool of the PIC methods, it is necessary to characterize the different conservation approaches of the integrators, e.g. symplecticity, energy- or charge-conservation. We discuss the different principles, e.g. composition, filtering, explicit and implicit ideas. While, particle in cell methods are well-studied, the combination between the different parts, i.e. pusher, solver and approximations are hardly to analyze. we concentrate on choosing the optimal pusher component, with respect to conservation and convergence behavior. We discuss oscillations of the pusher component, strong external magnetic fields and optimal conservation of energy and momentum. The algorithmic ideas are discussed and numerical experiments compare the exactness of the different schemes. An outlook to overcome the different error components is discussed in the future works.Comment: 23 page