On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.

A numerical method for solving the dynamic three-dimensional Ericksen-Leslie equations for nematic liquid crystals subject to a strong magnetic field

A finite difference technique, based on a projection method, is developed for solving the dynamic three-dimensional Ericksen-Leslie equations for nematic liquid crystals subject to a strong magnetic field. The governing equations in this situation are derived using primitive variables and are solved using the ideas behind the GENSMAC methodology (Tome and McKee [32]; Tome et al. [34]). The resulting numerical technique is then validated by comparing the numerical solution against an analytic solution for steady three-dimensional flow between two-parallel plates subject to a strong magnetic field. The validated code is then employed to solve channel flow for which there is no analytic solution. (C) 2009 Elsevier B.V. All rights reserved.

Numerical and variational solutions of the dipolar Gross-Pitaevskii equation in reduced dimensions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Improved numerical approach for the time-independent Gross-Pitaevskii nonlinear Schrödinger equation

In the present work, we improve a numerical method, developed to solve the Gross-Pitaevkii nonlinear Schrödinger equation. A particular scaling is used in the equation, which permits us to evaluate the wave-function normalization after the numerical solution. We have a two-point boundary value problem, where the second point is taken at infinity. The differential equation is solved using the shooting method and Runge-Kutta integration method, requiring that the asymptotic constants, for the function and its derivative, be equal for large distances. In order to obtain fast convergence, the secant method is used. © 1999 The American Physical Society.

Generation of the Cape Ghir upwelling filament: a numerical study

[EN] Filaments are narrow, shallow structures of cool water originating from the coast. They are typical features of the four main eastern boundary upwelling systems (EBUS). In spite of their significant biological and chemical roles, through the offshore exportation of nutrient-rich waters, the physical processes that generate them are still not completely understood. This paper is a process-oriented study of filament generation mechanisms. Our goal is twofold: firstly, to obtain a numerical solution able to well represent the characteristics of the filament off Cape Ghir (30°38'N, northwestern Africa) in the Canary EBUS and secondly, to explain its formation by a simple mechanism based on the balance of potential vorticity. The first goal is achieved by the use of the ROMS model (Regional Ocean Modeling System) in embedded domains around Cape Ghir, with a horizontal resolution going up to 1.5 km for the finest domain. The latter gets its initial and boundary conditions from a parent solution and is forced by climatological, high-resolution atmospheric fields. The modeled filaments display spatial, temporal and physical characteristics in agreement with the available in situ and satellite observations. This model solution is used as a reference to compare the results with a set of process-oriented experiments. These experiments allow us to reach the second objective. Their respective solution serves to highlight the contribution of various processes in the filament generation. Since the study is focused on general processes present under climatological forcing conditions, inter-annual forcing is not necessary. The underlying idea for the filament generation is the balance of potential vorticity in the Canary EBUS: the upwelling jet is characterized by negative relative vorticity and flows southward along a narrow band of uniform potential vorticity. In the vicinity of the cape, an injection of relative vorticity induced by the wind breaks the existing vorticity balance. The upwelling jet is prevented from continuing its way southward and has to turn offshore to follow lines of equal potential vorticity. The model results highlight the essential role of wind, associated with the particular topography (coastline and bottom) around the cape. The mechanism presented here is general and thus can be applied to other EBUS.

(Table 2) Stratigraphic position of midpoints of reddish and beige layers of lithologic cycles in the Loulja-A Section and their astronomical ages after tuning to the ETP and Summer Insolation Curves of the La2004 solution

An integrated high-resolution stratigraphy and orbital tuning is presented for the Loulja sections located in the Bou Regreg area on the Atlantic side of Morocco. The sections constitute the upward continuation of the upper Messinian Ain el Beida section and contain a well-exposed, continuous record of the interval straddling the Miocene-Pliocene (M-P) boundary. The older Loulja-A section, which covers the interval from ~5.59 to 5.12 Ma, reveals a dominantly precession-controlled color cyclicity that allows for a straightforward orbital tuning of the boundary interval and for detailed cyclostratigraphic correlations to the Mediterranean; the high-resolution and high-quality benthic isotope record allows us to trace the dominantly obliquity-controlled glacial history. Our results reveal that the M-P boundary coincides with a minor, partly precession-related shift to lighter "interglacial" values in d18O. This shift and hence the M-P boundary may not correlate with isotope stage TG5, as previously thought, but with an extra (weak) obliquity-controlled cycle between TG7 and TG5. Consequently, the M-P boundary and basal Pliocene flooding of the Mediterranean following the Messinian salinity crisis are not associated with a major deglaciation and glacio-eustatic sea level rise, indicating that other factors, such as tectonics, must have played a fundamental role. On the other hand, the onset of the Upper Evaporites in the Mediterranean marked by hyposaline conditions coincides with the major deglaciation step between marine isotope stage TG12 and TG11, suggesting that the associated sea level rise is at least partly responsible for the apparent onset of intermittently restricted marine conditions following the main desiccation phase. Finally, the Loulja-A section would represent an excellent auxiliary boundary stratotype for the M-P boundary as formally defined at the base of the Trubi marls in the Eraclea Minoa section on Sicily.

Numerical analysis of air flow past the 2415-3S airfoil for an unmanned aerial vehicle with internal propulsion system

This paper deals with the prediction of pressure and velocity fields on the 2415-3S airfoil which will be used for and unmanned aerial vehicle with internal propulsion system and in this way analyze the air flow through an internal duct of the airfoil using computational fluid dynamics. The main objective is to evaluate the effect of the internal air flow past the airfoil and how this affects the aerodynamic performance by means of lift and drag forces. For this purpose, three different designs of the internal duct were studied; starting from the base 2415-3S airfoil developed in previous investigation, basing on the hypothesis of decreasing the flow separation produced when the propulsive airflow merges the external flow, and in this way obtaining the best configuration. For that purpose, an exhaustive study of the mesh sensitivity was performed. It was used a non-structured mesh since the computational domain is tridimensional and complex. The selected mesh contains approximately 12.5 million elements. Both the computational domain and the numerical solution were made with commercial CAD and CFD software respectively. Air, incompressible and steady was analyzed. The boundary conditions are in concordance with experimental setup in the AF 6109 wind tunnel. The k-ε model is utilized to describe the turbulent flow process as followed in references. Results allowed obtaining pressure and velocity contours as well as lift and drag coefficients and also the location of separation and reattachment regions in some cases for zero degrees of angle of attack on the internal and external surfaces of the airfoil. Finally, the selection of the configuration with the best aerodynamic performance was made, selecting the option without curved baffles.

Difference schemes for numerical solutions of lagging models of heat conduction

Non-Fourier models of heat conduction are increasingly being considered in the modeling of microscale heat transfer in engineering and biomedical heat transfer problems. The dual-phase-lagging model, incorporating time lags in the heat flux and the temperature gradient, and some of its particular cases and approximations, result in heat conduction modeling equations in the form of delayed or hyperbolic partial differential equations. In this work, the application of difference schemes for the numerical solution of lagging models of heat conduction is considered. Numerical schemes for some DPL approximations are developed, characterizing their properties of convergence and stability. Examples of numerical computations are included.

High performance numerical modeling of ultra-short laser pulse propagation based on multithreaded parallel hardware

The focus of this study is development of parallelised version of severely sequential and iterative numerical algorithms based on multi-threaded parallel platform such as a graphics processing unit. This requires design and development of a platform-specific numerical solution that can benefit from the parallel capabilities of the chosen platform. Graphics processing unit was chosen as a parallel platform for design and development of a numerical solution for a specific physical model in non-linear optics. This problem appears in describing ultra-short pulse propagation in bulk transparent media that has recently been subject to several theoretical and numerical studies. The mathematical model describing this phenomenon is a challenging and complex problem and its numerical modeling limited on current modern workstations. Numerical modeling of this problem requires a parallelisation of an essentially serial algorithms and elimination of numerical bottlenecks. The main challenge to overcome is parallelisation of the globally non-local mathematical model. This thesis presents a numerical solution for elimination of numerical bottleneck associated with the non-local nature of the mathematical model. The accuracy and performance of the parallel code is identified by back-to-back testing with a similar serial version.

Multi-threaded parallel numerical modelling of femtosecond pulse propagation in laser machining

Femtosecond laser microfabrication has emerged over the last decade as a 3D flexible technology in photonics. Numerical simulations provide an important insight into spatial and temporal beam and pulse shaping during the course of extremely intricate nonlinear propagation (see e.g. [1,2]). Electromagnetics of such propagation is typically described in the form of the generalized Non-Linear Schrdinger Equation (NLSE) coupled with Drude model for plasma [3]. In this paper we consider a multi-threaded parallel numerical solution for a specific model which describes femtosecond laser pulse propagation in transparent media [4, 5]. However our approach can be extended to similar models. The numerical code is implemented in NVIDIA Graphics Processing Unit (GPU) which provides an effitient hardware platform for multi-threded computing. We compare the performance of the described below parallel code implementated for GPU using CUDA programming interface [3] with a serial CPU version used in our previous papers [4,5]. © 2011 IEEE.

Numerical Simulation Of Ultrasonic Wave Propagation In Elastically Anisotropic Media

The ultrasonic non-destructive testing of components may encounter considerable difficulties to interpret some inspections results mainly in anisotropic crystalline structures. A numerical method for the simulation of elastic wave propagation in homogeneous elastically anisotropic media, based on the general finite element approach, is used to help this interpretation. The successful modeling of elastic field associated with NDE is based on the generation of a realistic pulsed ultrasonic wave, which is launched from a piezoelectric transducer into the material under inspection. The values of elastic constants are great interest information that provide the application of equations analytical models, until small and medium complexity problems through programs of numerical analysis as finite elements and/or boundary elements. The aim of this work is the comparison between the results of numerical solution of an ultrasonic wave, which is obtained from transient excitation pulse that can be specified by either force or displacement variation across the aperture of the transducer, and the results obtained from a experiment that was realized in an aluminum block in the IEN Ultrasonic Laboratory. The wave propagation can be simulated using all the characteristics of the material used in the experiment evaluation associated to boundary conditions and from these results, the comparison can be made.

The parallel solution of early-exercise Asian options with stochastic volatility

This paper describes an parallel semi-Lagrangian finite difference approach to the pricing of early exercise Asian Options on assets with a stochastic volatility. A multigrid procedure is described for the fast iterative solution of the discrete linear complementarity problems that result. The accuracy and performance of this approach is improved considerably by a strike-price related analytic transformation of asset prices. Asian options are contingent claims with payoffs that depend on the average price of an asset over some time interval. The payoff may depend on this average and a fixed strike price (Fixed Strike Asians) or it may depend on the average and the asset price (Floating Strike Asians). The option may also permit early exercise (American contract) or confine the holder to a fixed exercise date (European contract). The Fixed Strike Asian with early exercise is considered here where continuous arithmetic averaging has been used. Pricing such an option where the asset price has a stochastic volatility leads to the requirement to solve a tri-variate partial differential inequation in the three state variables of asset price, average price and volatility (or equivalently, variance). The similarity transformations [6] used with Floating Strike Asian options to reduce the dimensionality of the problem are not applicable to Fixed Strikes and so the numerical solution of a tri-variate problem is necessary. The computational challenge is to provide accurate solutions sufficiently quickly to support realtime trading activities at a reasonable cost in terms of hardware requirements.

Wavelets And Adaptive Grids For The Discontinuous Galerkin Method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method. © Springer 2005.

A Study on Unconditionally Stable FDTD Methods for the Modeling of Metamaterials

We assess the performance of three unconditionally stable finite-difference time-domain (FDTD) methods for the modeling of doubly dispersive metamaterials: 1) locally one-dimensional FDTD; 2) locally one-dimensional FDTD with Strang splitting; and (3) alternating direction implicit FDTD. We use both double-negative media and zero-index media as benchmarks.

A coupled boundary element/differential equation method formulation for plate-beam interaction analysis

In this work, a new boundary element formulation for the analysis of plate-beam interaction is presented. This formulation uses a three nodal value boundary elements and each beam element is replaced by its actions on the plate, i.e., a distributed load and end of element forces. From the solution of the differential equation of a beam with linearly distributed load the plate-beam interaction tractions can be written as a function of the nodal values of the beam. With this transformation a final system of equation in the nodal values of displacements of plate boundary and beam nodes is obtained and from it, all unknowns of the plate-beam system are obtained. Many examples are analyzed and the results show an excellent agreement with those from the analytical solution and other numerical methods. (C) 2009 Elsevier Ltd. All rights reserved.