Order In Δt Research Articles

Thermal defocus-free Hartmann Wavefront Sensors for monitoring aberrations in Advanced Virgo

Earth-based gravitational waves interferometric detectors are shot-noise limited in the high-frequency region of their sensitivity band. While enhancing the laser input power is the natural solution to improve on the shot noise limit, higher power also increases the optical aberration budget due to the laser absorption in the highly reflective coatings of mirrors, resulting in a drop of the sensitivity of the detector. Advanced Virgo exploits Hartmann Wavefront Sensors (HWSs) to locally measure the absorption-induced optical aberrations by monitoring the optical path length change in the core optics. Despite the very high sensitivity of Hartmann sensors, temperature fluctuations can cause a spurious curvature term to appear in the reconstructed wavefront due to the thermal expansion of the Hartmann plate, that could affect the accuracy of the aberration monitoring. We present the implementation and validation of a control loop to stabilize the Advanced Virgo HWS temperature at the order of ΔT⩽0.01 K, keeping the spurious curvature within the detector’s requirements on wavefront sensing accuracy.

Pyrazine Analogues on Clathrate Hydrates in Methane–Water Systems

This paper reports the change in equilibrium pressure–temperature (pT) boundaries in a methane (CH4)–water system by adding three pyrazine analogues: 2-methylpyrazine, 2,3-dimethylpyrazine, and 2,3,5-trimethylpyrazine. The 2-methylpyrazine, 2,3-dimethylpyrazine, and 2,3,5-trimethylpyrazine showed no effect on the clathrate hydrate structure that formed in the CH4–water system upon analysis via Raman spectroscopy. Raman spectra revealed that clathrate crystals that formed in the CH4–pyrazine analogue–water system were of a structure-I CH4 hydrate. Phase equilibrium pT boundaries in the three CH4–pyrazine analogue–water systems shifted to a higher p and lower T condition than that for the CH4–water system with no additive. The higher p/lower T shift confirmed that 2-methylpyrazine, 2,3-dimethylpyrazine, and 2,3,5-trimethylpyrazine act as thermodynamic inhibitors for hydrate formation. The order of ΔT, a lower T shift from the CH4–water system, was as follows: 2-methylpyrazine–water > 2,3-dimethylpyrazine > 2,3,5-trimethylpyrazine. This order is in agreement with the number of −CH3 groups in the molecules.

Comment on “On the numerical simulation of particle dynamics in the radiation belt. Part I: Implicit and semi‐implicit schemes” and “On the numerical simulation of particle dynamics in the radiation belt. Part II: Procedure based on the diagonalization of the diffusion tensor” by E. Camporeale et al.

Comment on “On the numerical simulation of particle dynamics in the radiation belt. Part I: Implicit and semi‐implicit schemes” and “On the numerical simulation of particle dynamics in the radiation belt. Part II: Procedure based on the diagonalization of the diffusion tensor” by E. Camporeale et al.

Ultrafast spectroscopy diagnostic to measure localized ion temperature and toroidal velocity fluctuations

A dual-channel high-efficiency, high-throughput custom spectroscopic system has been designed and implemented at DIII-D to measure localized ion thermal fluctuations associated with drift wave turbulence. A large-area prism-coupled transmission grating and high-throughput collection optics are employed to observe C VI emission centered near λ=529 nm. The diagnostic achieves 0.25 nm resolution over a 2.0 nm spectral band via eight discrete spectral channels. A turbulence-relevant time resolution of 1 μs is achieved using cooled high-speed avalanche photodiodes and ultralow-noise preamplifiers. The system sensitivity is designed to provide measurements of normalized ion temperature fluctuations on the order of δT(i)/T(i)≤1%.

Improvement in the Time Accuracy of Numerical Methods in Petroleum Engineering Problems—A New Combination

Convective dispersion is one of the most important phenomena that takes place during miscible and chemical displacements in porous media. This has application both in enhanced oil recovery and ground water modeling. The presence of the parabolic and hyperbolic terms in the convective dispersion equation makes it a difficult equation to solve. Solving the convective dispersion equation by finite difference schemes is always the first choice due to the simplicity of the technique. However, current techniques suffer from the lack of numerical accuracy, especially for a reasonable time step, Δt; an accuracy of θ(Δt) is used in most cases. In this article, a central difference model for the time term having the inherent accuracy of the order of Δt 2 is used for the first time along with the Barakat-Clark scheme to target the overall accuracy in time of the order of Δt 4. The effect of this new combination on the solution is analyzed with the help of a case study, in where a 2-D convective-dispersion equation is solved by using the Barakat-Clark scheme with given boundary conditions. The time term in the equation is approximated by a central difference scheme. A comparison is also presented among the DuFort-Frankel scheme, Barakat-Clark scheme with the forward time difference term (FTD) and the proposed scheme of Barakat-Clark with the central time difference term (CTD). The Barakat-Clark scheme is proved to be stable even for this new combination. These findings will be helpful in applying different numerical schemes in the areas of petroleum engineering and ground water modeling, for which larger time steps (while maintaining the same accuracy) can be very useful. The utilization of the central difference scheme for the time term in parallel with the Barakat-Clark scheme does not make any significant difference on the solution of the equation although the accuracy in time becomes approximately of the order Δt 4. It is also found that the DuFort-Frankel scheme gives the highest and most consistent accuracy in its stability range. This scheme is not found to be unconditionally stable as compared to what has been professed in literature especially for the equations having hyperbolic and parabolic terms. The use of central weighing scheme for the time term (proposed) in Barakat-Clark's finite difference approximation gives almost the identical results as compared to the original Barakat-Clark scheme. This new combination is proved to be unconditionally stable even for this accuracy in time. Finally it is recommended to extend the application of the proposed Barakat-Clark scheme in 3-D problems and higher order equations relating to other phenomena with an attempt to increase the spatial accuracy as well.

The inclusion of collisional effects in the splitting scheme

An algorithm is given for the numerical solution of the Boltzmann equation for a one-dimensional unmagnetized plasma with immobile ions, in which collisional effects are described by the Bhatnagar-Gross-Krook (BGK) model. The algorithm is a straightforward generalization of the splitting scheme, which solves the one-dimensional Vlasov-Poisson system. The accuracy of the splitting scheme to second order in Δt is preserved.

Velocity-controlled tracking of the sun

For solar concentrator systems as well as for radiometric measurement of the solar radiation the tracking of the sun is necessary. Conventional trackers periodically update the orientation of the device to the actual position of the sun. This approach, however, has two major disadvantages. First, the orientation of the tracking device changes by jerks, which can lead to considerable torque loads on gears and drives of large structures such as heliostats. Second, the aiming error of the device is proportional to the time step chosen for updating the position. Thus for a maximum error of 10 −4 (i.e., 1 cm/100 m), the worst case time is 1.4 s. Here we present an approach controlling the angular velocity of the tracking device. Based on the present orientation and the orientation the device should have δt later, we calculate the mean angular velocity for the device to have there correct orientation after δt. At that time the velocity is adjusted. This method is of second order in δt. Thus for the maximum pointing error of 10 −4 the worst case time step δt is 6.4 min. This approach has been successfully implemented to track the 51.8 m 2 glass heliostat of the solar furnace at the Paul Scherrer Institute. We use a standard servo unit to control the drives of the heliostat. The calculations of the orientation of the sun and the nominal velocity of the heliostat are performed on a PC. Other continuous motion tracking has been implemented for example by Centre Suisse d'Electronique et de Microtechnique (CSEM) for a sun tracker to carry an instrumentation panel.

Algorithms for mori chain dynamics : a unified treatment

Within the realm of stochastic dynamics we point out a natural distinction between ‘Mori chain’ and ‘non-Mori chain’ techniques, depending on the mathematical form of the memory functions used. A universal algorithm for Mori chain dynamics is developed which contains former, seemingly disparate algorithms as special cases. In the absence of systematic forces the algorithm is exact for time steps of any length Δt, otherwise it is of third order in Δt. Some suggestions are made regarding ‘non-Mori chain’ techniques.

The stochastic method for numerical simulations:: Higher order corrections

We derive discrete versions of stochastic differential equations governing the evolution of some random variable x( t) to arbitrary order in Δt, giving explicit formulae to second order. These are tested in the static case by examples where x takes values in the groups U(1) and SU(2).

The integration of the vlasov equation in configuration space

A convenient, fast, and accurate method of solving the one-dimensional Vlasov equation numerically in configuration space is described. It treats the convective terms in the x and v directions separately and produces a scheme of second order in Δt. The resulting free-streaming and accelerating equations are computed with Fourier interpolation and spline interpolation methods respectively. The numerical method is tested witth linear and nonlinear problems. The method is very accurate and efficient. A new method of smoothing the distribution function is given. It reduces the computational effort by artificially increasing the entropy of the system. As a result, the distribution function is smooth enough to be well represented on a given mesh. The methods can be generalized in a straightforward way to deal with more complicated cases such as problems with nonperiodic spatial boundary conditions, two- and three-dimensional problems with and without external magnetic and/or electric fields.