Application of Richardson extrapolation to the numerical solution of partial differential equations

Richardson extrapolation is a methodology for improving the order of accuracy of numerical solutions that involve the use of a discretization size h. By combining the results from numerical solutions using a sequence of related discretization sizes, the leading order error terms can be methodically removed, resulting in higher order accurate results. Richardson extrapolation is commonly used within the numerical approximation of partial differential equations to improve certain predictive quantities such as the drag or lift of an airfoil, once these quantities are calculated on a sequence of meshes, but it is not widely used to determine the numerical solution of partial differential equations. Within this article, Richardson extrapolation is applied directly to the solution algorithm used within existing numerical solvers of partial differential equations to increase the order of accuracy of the numerical result without referring to the details of the methodology or its implementation within the numerical code. Only the order of accuracy of the existing solver and certain interpolations required to pass information between the mesh levels are needed to improve the order of accuracy and the overall solution accuracy. Using the proposed methodology, Richardson extrapolation is used to increase the order of accuracy of numerical solutions of the linear heat and wave equations and of the nonlinear St. Venant equations in one‐dimension. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Singlet exciton fission in anthracene at different temperatures

We have measured the magnetic field dependence of the prompt fluorescence of anthracene crystals at temperatures down to 10 K. Below 50 K the field dependence of prompt and delayed fluorescence is not symmetric anymore. This shows, that the prompt and the delayed fluorescence originate from different regions within the crystal.     

NMR studies of fluorocarbon diphosphines and oxy- and thio-bisphosphines

Both the diphosphine CH₃CF₃PPCH₃CF₃ and the thio-bisphosphine CH₃CF₃PSPCH₃CF₃ show ¹⁹F and ³¹P NMR spectra indicative of diastereomeric mixtures of d,l and meso isomers. The temperature-dependent reversible interconversions of the isomers have been attributed to the inversion at the phosphorus atom. The equilibrium constants for the isomerization reactions have been measured at various temperatures. The fluorine NMR spectra of the meso and d,l isomers of CD₃CF₃PSPCD₃CF₃ have been analyzed as M₃XX′M₃′ spin systems. Good agreement is found between the observed and calculated spectra. The five-bond fluorine-fluorine coupling constant of the low-field isomer is larger than that of the high-field isomer. The ¹⁹F NMR spectra of (CF₃)₂PP(CF₃)₂, (CF₃)₂POP(CF₃)₂, (CF₃)₂PSP(CF₃)₂, CH₃CF₃PPCH₃CF₃ and CH₃CF₃PSPCH₃CF₃ have been examined over a wide range of temperatures in an attempt to correlate the data to the geometries of these compounds.     

Transport of methylene chloride in poly(aryl-ether-ether-ketone) (PEEK)

The sorption of methylene chloride in neat PEEK was investigated as a function of temperature, sample thickness, surface treatment, and thermal history. The solubility of H₂CCl₂ in neat PEEK is 23 wt.% and is independent of thickness. Both surface treatment and thermal annealing strongly affect the rate of penetration; the as-Received material sorbs H₂CCl₂ more rapidly than abraded or annealed samples; however, the bulk solubility is independent of surface treatment. The sorption and desorption processes are considerably different, and the diffusion process is not simple Fickian Case I. The penetrant advances as a sharp front, suggesting a two-step, relaxation-controlled diffusion mechanism.     

Sixteen capillary electrophoresis applications for viral vaccine analysis

A broad range of CE applications from our organization is reviewed to give a flavor of the use of CE within the field of vaccine analyses. Applicability of CE for viral vaccine characterization, and release and stability testing of seasonal influenza virosomal vaccines, universal subunit influenza vaccines, Sabin inactivated polio vaccines (sIPV), and adenovirus vector vaccines were demonstrated. Diverse CZE, CE-SDS, CGE, and cIEF methods were developed, validated, and applied for virus, protein, posttranslational modifications, DNA, and excipient concentration determinations, as well as for the integrity and composition verifications, and identity testing (e.g., CZE for intact virus particles, CE-SDS application for hemagglutinin quantification and influenza strain identification, chloride or bromide determination in process samples). Results were supported by other methods such as RP-HPLC, dynamic light scattering (DLS), and zeta potential measurements. Overall, 16 CE methods are presented that were developed and applied, comprising six adenovirus methods, five viral protein methods, and methods for antibodies determination of glycans, host cell-DNA, excipient chloride, and process impurity bromide. These methods were applied to support in-process control, release, stability, process- and product characterization and development, and critical reagent testing. Thirteen methods were validated. Intact virus particles were analyzed at concentrations as low as 0.8 pmol/L. Overall, CE took viral vaccine testing beyond what was previously possible, improved process and product understanding, and, in total, safety, efficacy, and quality.     

Analytic study of 2D and 3D grid motion using modified Laplacian

The modified Laplacian has been used to move unstructured grids in response to changes in the surface grid for a variety of grid movement applications including store separation, aero‐elastic wing deformation and free surface flow simulations. However, the use of the modified Laplacian can result in elements with negative areas/volumes, because it has no inherent mechanism to prevent inversion of elements. In this paper, the use of a modified Laplacian is studied analytically for a two‐dimensional problem of deforming the inner circle of two concentric circles and for a three‐dimensional problem of deforming the inner sphere of two concentric spheres. By analysing the exact solution for this problem, the amount of translation and deformation of the inner circle that maintains a valid mesh is determined. A general grid movement theorem is presented which determines analytically the maximum allowable deformation before an invalid mesh results. Under certain circumstances, the inner circle and sphere can be expanded until it reaches the outer circle or sphere, while remaining a valid grid, and the inner circle and sphere can be rotated by an extreme amount before failure of the mesh occurs. By choosing the exponent to the modified Laplacian appropriately, extreme deformations for single frequency deformations is possible, although for practical applications where the grid movement has multiple frequencies, choosing the optimal exponent for the modified Laplacian may not be practical or provide much improvement. For grid movement simulations involving rigid body translation and rotation or uniform expansion, the modified Laplacian can yield excellent results, and the optimum choice of the modified Laplacian can be analytically determined for these types of motions, but when there are multiple frequencies in the deformation, the modified Laplacian does not allow much deformation before an invalid grid results. Copyright © 2006 John Wiley & Sons, Ltd.