Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross–Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre–Fourier spectral method with respect to the $(x,y)$ -variables as well as the Hermite spectral method in the $z$ -direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, a Sobolev-type inequality in a curved rectangle, and results on the asymptotical distribution of the nodes and weights associated with Gauß–Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.     

On the Convergence Behaviour of Variable Stepsize Multistep Methods for Singularly Perturbed Problems

In this note, we investigate the convergence behaviour of linear multistep discretizations for singularly perturbed systems, emphasising the features of variable stepsizes. We derive a convergence result for A()-stable linear multistep methods and specify a refined error estimate for backward differentiation formulas. Important ingredients in our convergence analysis are stability bounds for non-autonomous linear problems that are obtained by perturbation techniques.     

Phytotoxicity assessment of diclofenac and its phototransformation products

The occurrence of pharmaceuticals in the environment is an emerging issue. Several studies observed that the non-steroidal anti-inflammatory drug diclofenac is ubiquitously present in most of the surveyed surface waters, worldwide. Phototransformation of diclofenac was reported from laboratory assays as well as in natural water systems, raising the question of possible adverse effects of the phototransformation products of diclofenac to aquatic organisms. In this study the phytotoxicity of diclofenac exposed to natural sunlight was evaluated using synchronized cultures of the unicellular chlorophyte Scenedesmus vacuolatus. Diclofenac dissolved in ultra-pure water at 50 mg L⁻¹ was exposed to natural midsummer sunlight for a maximum of 145 h. Twice a day subsamples were taken for chromatography and parallel phytotoxicity assessment. Inhibition of algal reproduction of the initial diclofenac solution was in the mg L⁻¹ range indicating no specific toxicity of diclofenac towards S. vacuolatus. Fast degradation of diclofenac was observed with half lives between 3.3 and 6.4 h during the first and the third day of exposure, respectively. Phytotoxicity increased after 3.5 h of exposure of diclofenac to sunlight and showed a maximum of sixfold enhanced toxicity after 53 h of exposure to sunlight. Several phototransformation products were found during the experiment. The time courses of the relative concentration of three transformation products significantly correlated with enhanced phytotoxicity during the experiment. This indicates a high toxicity potential of phototransformation products of diclofenac at concentration levels that may come close to environmental concentrations of residual diclofenac after degradation. We conclude that toxicity assessment of phototransformation products should be included in the risk assessment of pharmaceuticals in the environment.     

How to confirm identified toxicants in effect-directed analysis

Due to the production and use of a multitude of chemicals in modern society, waters, sediments, soils and biota may be contaminated with numerous known and unknown chemicals that may cause adverse effects on ecosystems and human health. Effect-directed analysis (EDA), combining biotesting, fractionation and chemical analysis, helps to identify hazardous compounds in complex environmental mixtures. Confirmation of tentatively identified toxicants will help to avoid artefacts and to establish reliable cause–effect relationships. A tiered approach to confirmation is suggested in the present paper. The first tier focuses on the analytical confirmation of tentatively identified structures. If straightforward confirmation with neat standards for GC–MS or LC–MS is not available, it is suggested that a lines-of-evidence approach is used that combines spectral library information with computer-based structure generation and prediction of retention behaviour in different chromatographic systems using quantitative structure–retention relationships (QSRR). In the second tier, the identified toxicants need to be confirmed as being the cause of the measured effects. Candidate components of toxic fractions may be selected based, for example, on structural alerts. Quantitative effect confirmation is based on joint effect models. Joint effect prediction on the basis of full concentration–response plots and careful selection of the appropriate model are suggested as a means to improve confirmation quality. Confirmation according to the Toxicity Identification Evaluation (TIE) concept of the US EPA and novel tools of hazard identification help to confirm the relevance of identified compounds to populations and communities under realistic exposure conditions. Promising tools include bioavailability-directed extraction and dosing techniques, biomarker approaches and the concept of pollution-induced community tolerance (PICT). Figure Toxicity confirmation in EDA as a tiered approach     

An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime

In this paper, we are concerned with the derivation of a local error representation for exponential operator splitting methods when applied to evolutionary problems that involve critical parameters. Employing an abstract formulation of differential equations on function spaces, our framework includes Schrödinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients. We illustrate the general mechanism on the basis of the first-order Lie splitting and the second-order Strang splitting method. Further, we specify the local error representation for a fourth-order splitting scheme by Yoshida. From the given error estimate it is concluded that higher-order exponential operator splitting methods are favourable for the time-integration of linear Schrödinger equations in the semi-classical regime with critical parameter 0<ε?1, provided that the time stepsize h is sufficiently smaller than \(\sqrt[p]{\varepsilon}\), where p denotes the order of the splitting method.