Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

不可压磁流体力学方程组的高精度数值解法

为数值求解低雷诺数下不可压流体在电磁场作用下的流动,提出一种四阶紧致差分方法.由二维原始变量的MHD方程组出发,推导出具有较少未知量的电流密度-涡量-流函数形式MHD方程组.建立了求解二维非定常不可压MHD方程组的电流密度-涡量-流函数形式的四阶精度紧致差分格式.为验证本文提出的高精度紧致差分方法的精确性和可靠性,对有...     

A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.     

Effect of instrumental particle sizing resolution on the modelling of aerosol radiative parameters

A more realistic estimation of the scattering and hemispheric backscattering coefficients, σ_sp and σ_bsp, and their respective optical cross section, C_sca and C_bk, of aerosol particles is presented on the basis of the exact resolution of the width of the size bins of the particle counter instruments when size distribution measurements are used, and, with the exact optical detector instruments ability. The scattering and hemispheric backscattering cross sections, C_sca and C_bk, of the particles are averaged over the full size bins of the particle counter instrument, while these quantities are usually estimated only on the value of the mean geometric diameter of each size bin. Six instruments, the APS, ASASP-X, DMPS, FSSP-100, ELPI, and SMPS frequently used in particle size distribution measurements are reviewed, for spherical sea-salt particles at a wavelength λ=0.55 μm. The comparison using the conventional geometric mean diameter versus the use of the full size bin leads to large amount of errors for the optical cross section with non-negligible effects on their respective optical coefficients. The maximal accuracy expected for these optical quantities depend on the particle diameter as well as on the channel width of the instruments, and are also function of the angular detector probe used to measure them.     

A second-order space–time accurate scheme for nonlinear diffusion equation with general capacity term

A fully implicit finite difference (FIFD) scheme with second-order space–time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays an important role in the analysis. It is established by choosing a new test function to bound the solution and its temporal and spatial difference quotients in suitable norms in the fixed point arguments, which is different from the traditional way. Based on these bounds, other fundamental properties of the scheme are rigorously analyzed consequently. It shows that the scheme is uniquely solvable, unconditionally stable, and convergent with second-order space–time accuracy in L^∞(L²) and L^∞(H¹) norms. The theoretical analysis adapts to both one- and multidimensional problems, and can be extended to schemes with first-order time accuracy. Numerical tests are provided to verify the theoretical results and highlight the high accuracy of the second-order space–time accurate scheme. The reasoning techniques can be extended to a broad family of discrete schemes for nonlinear problems with capacity terms.     

High‐order implicit staggered‐grid finite differences methods for the acoustic wave equation

Motivated by the idea that staggered‐grid methods give a greater stability and give energy conservation, this article presents a new family of high‐order implicit staggered‐grid finite difference methods with any order of accuracy to approximate partial differential equations involving second‐order derivatives. In particular, we numerically analyze our new methods for the solution of the one‐dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ‐point operator: the explicit formula is th‐order and the implicit formula is th‐order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.     

Bestimmung der Empfindlichkeit bei Tc-99-NMR-Messungen an einem 250-MHz-Spektrometer

By means of different concentrations the signal-to-noise ratios of Tc-99-NMR spectra were determined applying well measurable Tc-samples [tetrabutylammoniwn pertechnetate, TBA (TcO₄)J and a 250 MHz-spectrometer. The signal-to-noise ratios of the spectra were determined by using the integrated routines of the firm's software and accumulating different number of scans. By fittings of data of the signal-to-noise ratio dependence and by extrapolation the minimum Tc-concentralion could be empirically found out. Applying a duration of measurements of 12 hours about 10^?7 molar concentrations can be determined.     

Die Bestimmung der spezifischen Aktivität trägerarmer Radiojodidlösungen durch papierchromatographische Ermittlung der prozentualen Radioaktivitätsverteilung auf die Jodsubstitutionsproduke von Bromsulfan

Es wird eine neue Methode zur Bestimmung der spezifischen Aktivität trägerarmer Radiojodidlösungen zwischen 5 und 17 Ci/mg (185 bis 630 G Bq/mg) beschrieben, die wesentlich einfacher ist als die bekannten, zum Teil sehr komplizierten Mikroanalysenmethoden und im Prinzip auf eine einfache Trennung eines Radiojodierungsgemisches hinausläuft. Hierzu werden 1 bzw. 2 nmol Bromsulfan (BSP) mit einer exakt zu messenden Menge Na¹²⁵ J bzw. Na¹³¹ J unter Verwendung von Chloramin T radiojodiert. Da die Reaktionsgeschwindigkeitskonstanten der Mono- und Dijodierung des Bromsulfans nicht differierèn, ist es möglich, aus der papierchromatographisch ermittelten prozentualen Radioaktivitätsverteilung das molare Verhältnis der an ein BSP-Molekül gebundenen Jodalome zu berechnen. Bei bekannter BSP-Menge läßt sich daraus die an der Radiojodierung beteiligte Jodmasse ableiten.

Bei der Radiojodierung mit Chloramin T wird jedoch kein 100%iger Jodeinbau erreicht. Die gemessene Aktivitätsmenge muß deshalb durch einen Faktor korrigiert werden, der aus dem Ergebnis der papierchromalographischen Trennung des Reaktionsgemisches abgeleitet wird und der lediglich die auf Mono- und Dijod-BSP entfallenden Radioaktivitätsanteile, nicht dagegen das anorganische Radiojodid berücksichtigt. Die spezifische Aktivität der verwendeten Radiojodidlösung ergibt sich dann als Quotient der korrigierten Radioaktivitätsmenge und der berechneten Radiojodmasse.

Wir haben die erhaltenen Ergebnisse mit den nach einer Referenzmethode gefundenen Werten verglichen. Hierbei wurden die spezifischen Aktivitäten von Na¹²⁵ J aus T3-RIA-Standurdkurven mit ¹²⁵J-T3-Tracern bestimmt, die unter Verwen-dung dieses Na¹²⁵ J präpariert wurden. Es wurde eine Präzision mit einem mittleren Variationskoeffizienlen von ±4% festgestellt.     

Isotopenverdünnungsanalytische Multielementanalyse mit der Funkenquellen-Massenspektrographie

Es wird ein Verfahren zur Isotopenverdünnungsanalyse mit stabilen Isotopen unter Verwendung einer Funkenionenquelle und einem doppelfokussierenden Massenspektrographen beschrieben, welches die gleichzeitige Analyse mehrerer Elemente in pulverförmigen Feststoffen erlaubt.

Die Leistungsfähigkeit des Analysenverfahrens wild am Beispiel der Spurenanalyse geologischer Standardproben demonstriert.

Das Analysenverfahren erlaubt die Bestimmung von Spurenkonzentrationen bis in den ppb-Bereich mit Standardabweichungen zwischen 2 und 10%.