On the ground states and dynamics of space fractional nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions

In this paper, we propose some efficient and robust numerical methods to compute the ground states and dynamics of Fractional Schr\"{o}dinger Equation (FSE) with a rotation term and nonlocal nonlinear interactions. In particular, a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal interaction evaluation \cite{EMZ2015}. To compute the ground states, we integrate the preconditioned Krylov subspace pseudo-spectral method \cite{AD1} and the GauSum solver. For the dynamics simulation, using the rotating Lagrangian coordinates transform \cite{BMTZ2013}, we first reformulate the FSE into a new equation without rotation. Then, a time-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the new FSE

Diffusion of energetic particles in turbulent MHD plasmas

In this paper we investigate the transport of energetic particles in turbulent plasmas. A numerical approach is used to simulate the effect of the background plasma on the motion of energetic protons. The background plasma is in a dynamically turbulent state found from numerical MHD simulations, where we use parameters typical for the heliosphere. The implications for the transport parameters (i.e. pitch-angle diffusion coefficients and mean free path) are calculated and deviations from the quasi-linear theory are discussed.Comment: Accepted for publication in Ap

Computational Inverse Problems for Partial Differential Equations (hybrid meeting)

Inverse problems in partial differential equations (PDEs) consist in reconstructing some part of a PDE such as a coefficient, a boundary condition, an initial condition, the shape of a domain, or a singularity from partial knowledge of solutions to the PDE. This has numerous applications in nondestructive testing, medical imaging, seismology, and optical imaging. Whereas classically mostly boundary or far field data of solutions to deterministic PDEs were considered, more recently also statistical properties of solutions to random PDEs have been studied. The study of numerical reconstruction methods of inverse problems in PDEs is at the interface of numerical analysis, PDE theory, functional analysis, statistics, optimization, and differential geometry. This workshop has mainly addressed five related topics of current interest: model reduction, control-based techniques in inverse problems, imaging with correlation data of waves, fractional diffusion, and model-based approaches using machine learning

A boundary element method formulation based on the Caputo derivative for the solution of the anomalous diffusion problem

This work presents a boundary element method formulation for the solution of the anomalous diffusion problem. By keeping the fractional time derivative as it appears in the governing differential equation of the problem, and by employing a Weighted Residuals Method approach with the steady state fundamental solution for anisotropic media playing the role of the weighting function, one obtains the boundary integral equation of the proposed formulation. The presence of a domain integral with the fractional time derivative as part of its integrand, and the evaluation of this fractional time derivative as a Caputo derivative, constitute the main feature of the formulation. The analyses of some examples, in which the numerical results are always compared with the corresponding analytical solutions, show the robustness of the formulation, as accurate results are obtained even for small values of the order of the time derivative

Transport dynamics of self-consistent, near-marginal drift-wave turbulence. II. Characterization of transport by means of passive scalars

From theoretical and modeling points of view, following Lagrangian trajectories is the most straightforward way to characterize the transport dynamics. In real plasmas, following Lagrangian trajectories is difficult or impossible. Using a blob of passive scalar (a tracer blob) allows a quasi-Lagrangian view of the dynamics. Using a simple two-dimensional electrostatic plasma turbulence model, this work demonstrates that the evolution of the tracers and the passive scalar field is equivalent between these two fluid transport viewpoints. When both the tracers and the passive scalar evolve in tandem and closely resemble stable distributions, namely, Gaussian distributions, the underlying turbulent transport character can be recovered from the temporal scaling of the second moments of both. This local transport approach corroborates the use of passive scalar as a turbulent transport measurement. The correspondence between the local transport character and the underlying transport is quantified for different transport regimes ranging from subdiffusive to superdiffusive. This correspondence is limited to the initial time periods of the spread of both the tracers and the passive scalar in the given transport regimes.This work was supported by U.S. DOE Contract No. DE-FG02-04ER54741 with the University of Alaska Fairbanks and in part by a grant of HPC resources from the Arctic Region Supercomputing Center at the University of Alaska Fairbanks. This research was also sponsored in part by DGICYT (Dirección General de Investigaciones Científicas y Tecnológicas) of Spain under Project No. ENE2015-68265

Investigation of the interaction between competing types of nondiffusive transport in drift wave turbulence

Radial transport in turbulence dominated tokamak plasmas has been observed to deviate from classical diffusion in certain regimes relevant for magnetic confinement fusion. These situations at least include near-marginal turbulence, where radial transport becomes superdiffusive and mediated by elongated radial structures (or avalanches) and transport across radially sheared poloidal flows, where radial subdiffusion often ensues. In this paper, the interaction between very different physical ingredients responsible for these two types of nondiffusive dynamics (namely, turbulent profile relaxation close to a local threshold and the interaction with radially sheared zonal flows) is studied in detail in the context of a simple two-dimensional electrostatic plasma fluid turbulence model based on the dissipative trapped electron mode. It is shown that, depending on the relative relevance of each of these ingredients, which can be tuned in various ways, a variety of nondiffusive radial transport behaviors can be found in the system. The results also illustrate the fact that the classical diffusion paradigm is often insufficient to describe turbulent transport in systems with self-generated flows and turbulent profile relaxations. Published by AIP Publishing.This work was supported by U.S. DOE under Contract No. DE-FG02-04ER54741 with the University of Alaska Fairbanks and in part by a grant of HPC resources from the Arctic Region Supercomputing Center at the University of Alaska Fairbanks. This research was also sponsored in part by DGICT (Direccion General de Investigaciones Cientıficas y Tecnologicas) of Spain under Project No. ENE2015-68265

Transport dynamics of self-consistent, near-marginal drift-wave turbulence. I. Investigation of the ability of external flows to tune the non-diffusive dynamics

The reduction of turbulent transport across sheared flow regions has been known for a long time in magnetically confined toroidal plasmas. However, details of the dynamics are still unclear, in particular, in what refers to the changes caused by the flow on the nature of radial transport itself. In Paper II, we have shown in a simplified model of drift wave turbulence that, when the background profile is allowed to evolve self-consistently with fluctuations, a variety of transport regimes ranging from superdiffusive to subdiffusive open up depending on the properties of the underlying turbulence [D. Ogata et al., Phys. Plasmas 24, 052307 (2017)]. In this paper, we show that externally applied sheared flows can, under the proper conditions, cause the transport dynamics to be diffusive or subdiffusive.This work was supported by U.S. DOE Contract No. DE-FG02-04ER54741 with the University of Alaska Fairbanks and in part by a grant of HPC resources from the Arctic Region Supercomputing Center at the University of Alaska Fairbanks. This research was also sponsored in part by DGICYT (Dirección General de Investigaciones Científicas y Tecnológicas) of Spain under Project No. ENE2015-68265

On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross-Pitaevskii equations

International audienceThe purpose of this paper is to discuss some recent developments concerning the numerical simulation of space and time fractional Schrödinger and Gross-Pitaevskii equations. In particular, we address some questions related to the discretization of the models (order of accuracy and fast implementation) and clarify some of their dynamical properties. Some numerical simulations illustrate these points

Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings