Galerkin/Runge-Kutta discretizations for semilinear parabolic equations

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems. Unlike any classical counterpart, this class offers arbitrarily high, optimal order convergence. In support of this claim, error estimates are proved, and computational results are presented. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

On implicit Runge-Kutta methods for parallel computations

Implicit Runge-Kutta methods which are well-suited for parallel computations are characterized. It is claimed that such methods are first of all, those for which the associated rational approximation to the exponential has distinct poles, and these are called multiply explicit (MIRK) methods. Also, because of the so-called order reduction phenomenon, there is reason to require that these poles be real. Then, it is proved that a necessary condition for a q-stage, real MIRK to be A sub 0-stable with maximal order q + 1 is that q = 1, 2, 3, or 5. Nevertheless, it is shown that for every positive integer q, there exists a q-stage, real MIRK which is I-stable with order q. Finally, some useful examples of algebraically stable MIRKs are given

Galerkin/Runge-Kutta discretizations for parabolic equations with time dependent coefficients

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

On Lipschitz continuity of nonlinear differential operators

In connection with approximations for nonlinear evolution equations, it is standard to assume that nonlinear terms are at least locally Lipschitz continuous. However, it is shown here that f = f(X,del sub u(X)) is Lipschitz continuous from the subspace W sup 1, infinity is a subset of L sub 2 into W sup 1,2, and maps W sup 2, infinity into W sup 1, infinity, if and only if f is affine with W sup 1, infinity coefficients. In fact, a local version of this claim is proved

Atmospheric potential oxygen: New observations and their implications for some atmospheric and oceanic models

Measurements of atmospheric O2/N2 ratios and CO2 concentrations can be combined into a tracer known as atmospheric potential oxygen (APO ≈ O2/N2 + CO2) that is conservative with respect to terrestrial biological activity. Consequently, APO reflects primarily ocean biogeochemistry and atmospheric circulation. Building on the work of Stephens et al. (1998), we present a set of APO observations for the years 1996-2003 with unprecedented spatial coverage. Combining data from the Princeton and Scripps air sampling programs, the data set includes new observations collected from ships in the low-latitude Pacific. The data show a smaller interhemispheric APO gradient than was observed in past studies, and different structure within the hemispheres. These differences appear to be due primarily to real changes in the APO field over time. The data also show a significant maximum in APO near the equator. Following the approach of Gruber et al. (2001), we compare these observations with predictions of APO generated from ocean O2 and CO2 flux fields and forward models of atmospheric transport. Our model predictions differ from those of earlier modeling studies, reflecting primarily the choice of atmospheric transport model (TM3 in this study). The model predictions show generally good agreement with the observations, matching the size of the interhemispheric gradient, the approximate amplitude and extent of the equatorial maximum, and the amplitude and phasing of the seasonal APO cycle at most stations. Room for improvement remains. The agreement in the interhemispheric gradient appears to be coincidental; over the last decade, the true APO gradient has evolved to a value that is consistent with our time-independent model. In addition, the equatorial maximum is somewhat more pronounced in the data than the model. This may be due to overly vigorous model transport, or insufficient spatial resolution in the air-sea fluxes used in our modeling effort. Finally, the seasonal cycles predicted by the model of atmospheric transport show evidence of an excessive seasonal rectifier in the Aleutian Islands and smaller problems elsewhere. Copyright 2006 by the American Geophysical Union

Assessing the rider's seat and horse's behavior: difficulties and perspectives

correct seat and position are the basis for a good performance in horseback riding. This study aimed to measure deviations from the correct seat, test a seat improvement program (dismounted exercises), and investigate whether horse behavior was affected by the rider's seat. Five experienced trainers defined 16 seat deviations and scored the occurrence in 20 riders in a dressage test. Half the riders then carried out an individual training program; after 9 weeks, riders were again scored. The study took no video or heart-rate recordings of horses and riders. Panel members did not agree on the deviations in the rider's seat; the study detected no differences¿with the exception of improvement of backward-tilted pelvis¿between the groups. Horse behavior, classified as ¿evasive,¿ increased; horse heart rate decreased in the experimental group. Heart rates of riders in both groups decreased. Seven of 9 riders in the experimental group had the impression that the exercises improved their riding performance. There is a clear need to develop a robust system that allows trainers to objectively evaluate the rider's sea

Spontaneous rotating vortex lattices in a pumped decaying condensate

Injection and decay of particles in an inhomogeneous quantum condensate can significantly change its behaviour. We model trapped, pumped, decaying condensates by a complex Gross-Pitaevskii equation and analyse the density and currents in the steady state. With homogeneous pumping, rotationally symmetric solutions are unstable. Stability may be restored by a finite pumping spot. However if the pumping spot is larger than the Thomas-Fermi cloud radius, then rotationally symmetric solutions are replaced by solutions with spontaneous arrays of vortices. These vortex arrays arise without any rotation of the trap, spontaneously breaking rotational symmetry.Comment: Updated title and introduction. 4 pages, 3 figure

Coherent Particle Transfer in an On-Demand Single-Electron Source

Coherent electron transfer from a localized state trapped in a quantum dot into a ballistic conductor, taking place in on-demand electron sources, in general may result in excitation of particle-hole pairs. We consider a simple model for these effects, involving a resonance level with time-dependent energy, and derive Floquet scattering matrix describing inelastic transitions of particles in the Fermi sea. We find that, as the resonance level is driven through the Fermi level, particle transfer may take place completely without particle-hole excitations for certain driving protocols. In particular, such noiseless transfer occurs when the level moves with constant rapidity, its energy changing linearly with time. A detection scheme for studying the coherence of particle transfer is proposed.Comment: 5 pages, 3 figures. Updated introduction, Fig. 1, and reference

Experimental pig-to-pig transmission dynamics for African swine fever virus, Georgia 2007/1 strain

African swine fever virus (ASFV) continues to cause outbreaks in domestic pigs and wild boar in Eastern European countries. To gain insights into its transmission dynamics, we estimated the pig-to-pig basic reproduction number (R 0) for the Georgia 2007/1 ASFV strain using a stochastic susceptible-exposed-infectious-recovered (SEIR) model with parameters estimated from transmission experiments. Models showed that R 0 is 2·8 [95% confidence interval (CI) 1·3–4·8] within a pen and 1·4 (95% CI 0·6–2·4) between pens. The results furthermore suggest that ASFV genome detection in oronasal samples is an effective diagnostic tool for early detection of infection. This study provides quantitative information on transmission parameters for ASFV in domestic pigs, which are required to more effectively assess the potential impact of strategies for the control of between-farm epidemic spread in European countries.ISSN:0950-2688ISSN:1469-440