Use Of Finite Difference Methods Research Articles

Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space

When a pulsed, few-cycle electromagnetic wave is focused by optics with f-number smaller than two, the frequency components it contains are focused to different regions of space, building up a complex electromagnetic field structure. Accurate numerical computation of this structure is essential for many applications such as the analysis, diagnostics, and control of high-intensity laser-matter interactions. However, straightforward use of finite-difference methods can impose unacceptably high demands on computational resources, owing to the necessity of resolving far-field and near-field zones at sufficiently high resolution to overcome numerical dispersion effects. Here, we present a procedure for fast computation of tight focusing by mapping a spherically curved far-field region to periodic space, where the field can be advanced by a dispersion-free spectral solver. In many cases of interest, the mapping reduces both run time and memory requirements by a factor of order 10, making it possible to carry out simulations on a desktop machine or a single node of a supercomputer. We provide an open-source C++ implementation with Python bindings and demonstrate its use for a desktop machine, where the routine provides the opportunity to use the resolution sufficient for handling the pulses with spectra spanning over several octaves. The described approach can facilitate the stability analysis of theoretical proposals, the studies based on statistical inferences, as well as the overall development and analysis of experiments with tightly-focused short laser pulses.

Improved particle position accuracy from off-axis holograms using a Chebyshev model.

Side scattered light from micrometer-sized particles is recorded using an off-axis digital holographic setup. From holograms, a volume is reconstructed with information about both intensity and phase. Finding particle positions is non-trivial, since poor axial resolution elongates particles in the reconstruction. To overcome this problem, the reconstructed wavefront around a particle is used to find the axial position. The method is based on the change in the sign of the curvature around the true particle position plane. The wavefront curvature is directly linked to the phase response in the reconstruction. In this paper we propose a new method of estimating the curvature based on a parametric model. The model is based on Chebyshev polynomials and is fit to the phase anomaly and compared to a plane wave in the reconstructed volume. From the model coefficients, it is possible to find particle locations. Simulated results show increased performance in the presence of noise, compared to the use of finite difference methods. The standard deviation is decreased from 3-39 μm to 6-10 μm for varying noise levels. Experimental results show a corresponding improvement where the standard deviation is decreased from 18 μm to 13 μm.

Finite difference methods for estimating marginal risk contributions in asset management

The decomposition of portfolio risks in terms of the underlying assets, which are extremely important for risk budgeting, asset allocation and risk monitoring, is well described by risk contributions. However, risk contributions cannot be calculated analytically for a considerable number of the risk models used in practice. We therefore study the use of finite difference methods for estimating risk contributions. We find that for practically relevant setups the additional estimation errors of the finite difference formulas are negligibly small. Since finite difference methods work for complex risk models and are independent of decisions about underlying distributions, we suggest the use of finite difference methods as the standard procedure for estimating risk contributions. As an application, we consider a general risk model that fits a kernel density estimation to the historical asset return distribution combined with a finite difference method in order to arrive at the risk contributions. It turns out that this general risk model combined with a finite difference method for calculating risk contributions works well in terms of estimation error.

JSE Local Volatility Finite Difference Model

The use of finite difference methods for solving PDEs on a computer goes back almost to the 1950s since its invention. In finance, these methods were introduced in the 1970’s after the derivation of the Black-Scholes model. The sophistication of these methods in finance has become a field of extensive research. These include, alternating finite difference (ADI) methods, adaptive grids, grids stretching and compact finite differences, to name but a few. The interested reader is referred to two important textbooks in literature that cover financial derivatives pricing with finite difference methods exclusively.The purpose of this document is to provide a base for the valuation methods used in the Can-Do product space. Thus we will not be re-documenting already well documented models and methodologies. This document will rather focus on the application of these methods to the JSE can-do products.

Ground-Motion Simulation in the Lower Tagus Valley Basin

Throughout history, the Lower Tagus Valley (LTV) region has been shaken by several earthquakes, including some with moderate to large magnitudes and with sources located inside the basin, for example the 1344 (M6.0) and 1909 (M6.0) earthquakes. Previous simulations (Bezzeghoud et al. Natural Hazard 69: 1229–1245, 2011) have revealed strong amplification of the amplitude waves in the region, because of the effect of the low-velocity sediments that fill the basin. The model used in those simulations was updated in this work by including new high-resolution geophysical and geotechnical data available for the area (seismic reflection, aeromagnetic, gravimetric, deep wells, standard penetration tests, and geological data). To contribute to improved assessment of seismic hazard in the LTV, we simulated propagation of seismic waves produced by moderate earthquakes in a 3D heterogeneous medium by using elastic finite-difference wave propagation code. The method, successfully used by Grandin et al. (Geophys J Int 171: 1144–1161, 2007), involves evaluation of the seismic potential of known faults in the area studied and three-dimensional seismic ground motion modelling by use of finite difference methods. On the basis of this methodology, we calculated the ground motion for the April 23, 1909, Benavente (Portugal) earthquake (Mw = 6.0) in dense grid points, then computed the synthetic isoseismic map of the area by use of appropriate relationships between seismic intensity (MMI) and peak ground velocity (PGV). The synthetic results, in contrast with available macroseismic and instrumental data, enable validation of the source models proposed for the area, identification of the sources of historical earthquakes, and could also indicate which areas are more exposed to seismic ground motion.

A GPGPU based program to solve the TDSE in intense laser fields through the finite difference approach

We present a General-purpose computing on graphics processing units (GPGPU) based computational program and framework for the electronic dynamics of atomic systems under intense laser fields. We present our results using the case of hydrogen, however the code is trivially extensible to tackle problems within the single-active electron (SAE) approximation. Building on our previous work, we introduce the first available GPGPU based implementation of the Taylor, Runge–Kutta and Lanczos based methods created with strong field ab-initio simulations specifically in mind; CLTDSE. The code makes use of finite difference methods and the OpenCL framework for GPU acceleration. The specific example system used is the classic test system; Hydrogen. After introducing the standard theory, and specific quantities which are calculated, the code, including installation and usage, is discussed in-depth. This is followed by some examples and a short benchmark between an 8 hardware thread (i.e. logical core) Intel Xeon CPU and an AMD 6970 GPU, where the parallel algorithm runs 10 times faster on the GPU than the CPU. Program summaryProgram title: CLTDSECatalogue identifier: AESM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3 and aboveNo. of lines in distributed program, including test data, etc.: 15988No. of bytes in distributed program, including test data, etc.: 233518Distribution format: tar.gzProgramming language: C99 and OpenCL C. C99 conformance is ensured through use of C99 and pedantic flags under GCC and Clang.Computer: Single compute node.Operating system: GNU/Linux. It should, in principle, work with little modification for other Unix-Like systems.Has the code been vectorised or parallelised?: OpenCL is a parallel language. Thus CLTDSE can use all cores on a processor or GPU. OpenCL supports using all available compute units (GPUs and CPUs etc.), although this is not currently implemented in CLTDSE.RAM: Negligible RAM and GPU global memory (20MiB)Classification: 2.5.External routines: An OpenCL library; the current major packages are APP by AMD (CPU and AMD GPU), the NVIDIA driver (GPU), and the Intel SDK for OpenCL Applications (CPU and Intel HD Graphics). libconfig for processing configuration files.Nature of problem:Describing the dynamics of electrons under intense laser fields in atoms or molecules.Solution method:Solving the discretised system through finite difference using the Lanczos, Runge–Kutta, and Taylor methods.Restrictions:The example problem which is implemented is under the single active electron approximation.Unusual features:Focused on GPGPU acceleration through OpenCL.Running time:The running time depends on the nature of the pulse. This can vary from seconds to tens of minutes.

Spatial Approximation of Nondivergent Type Parabolic PDEs with Unbounded Coefficients Related to Finance

We study the spatial discretisation of the Cauchy problem for a multidimensional linear parabolic PDE of second order, with nondivergent operator and unbounded time- and space-dependent coefficients. The equation free term and the initial data are also allowed to grow. Under a nondegeneracy assumption, we consider the PDE solvability in the framework of the variational approach and approximate in space the PDE problem's generalised solution, with the use of finite-difference methods. The rate of convergence is estimated.

Interaction of acoustic waves with porous layer

Interaction of sound impulse with porous layer is investigated. The presence of a barrier shielded by the layer and the gap between the porous layer and the barrier is possible. Method of calculation in linear approximation of pressures and tensions in the porous layer, gap, and on the barrier has been proposed. The method serves to interpret data on acoustic waves interaction with porous layer obtained experimentally or with the use of finite-difference methods. Specifics of acoustic wave propagation into the porous layer and further reflection from the barrier has been studied. Comparison of calculation data with experimental data of other authors on impulse propagation through the porous plate submerged in water has been carried out.

New Computer Program for Three-Dimensional Relativistic Hydrodynamical Model

An effective computer program for three dimensional relativistic hydrodynamical model has been developed. It implements a new approach to the early hot phase of relativistic heavy-ion collisions. The computer program simulates time-space evolution of nuclear matter in terms of ideal-fluid dynamics. Equations of motions of hydrodynamics are solved making use of finite difference methods. Commonly-used algorithms of numerical relativistic hydrodynamics RHLLE and MUSTA-FORCE have been applied in simulations. To speed-up calculations, parallel processing has been made available for solving hydrodynamical equations. The test results of simulations for 3D, 2D and Bjorken expansion are reported in this paper. As a next step we plan to implement the hadronization algorithm by implementing the continuous particle emission for freeze-out and comparing it with Cooper-Frye formula.

Solving Multi‐Scale problems with heterogeneous finite difference methods

AbstractMulti‐scale differential equations are problems in which the variables can have different length scales. The direct numerical solution of differential equations with multiple scales is often difficult due to the work for resolving the smallest scale. We present here a strategy which allows the use of finite difference methods for the numerical solution of parabolic multi‐scale problems, based on a coupling of macroscopic and microscopic models for the original equation.

Errors-in-variables modeling in optical flow estimation

Gradient-based optical flow estimation methods typically do not take into account errors in the spatial derivative estimates. The presence of these errors causes an errors-in-variables (EIV) problem. Moreover, the use of finite difference methods to calculate these derivatives ensures that the errors are strongly correlated between pixels. Total least squares (TLS) has often been used to address this EIV problem. However, its application in this context is flawed as TLS implicitly assumes that the errors between neighborhood pixels are independent. In this paper, a new optical flow estimation method (EIVM) is formulated to properly treat the EIV problem in optical flow. EIVM is based on Sprent's (1966) procedure which allows the incorporation of a general EIV model in the estimation process. In EIVM, the neighborhood size acts as a smoothing parameter. Due to the weights in the EIVM objective function, the effect of changing the neighborhood size is more complex than in other local model methods such as Lucas and Kanade (1981). These weights, which are functions of the flow estimate, can alter the effective size and orientation of the neighborhood. In this paper, we also present a data-driven method for choosing the neighborhood size based on Stein's unbiased risk estimators (SURE).

Mathematical modeling of salt transport by coupled subsurface and surface water flows

A mathematical model is examined to describe the transport of salts by coupled flows of surface, soil, and subsoil waters for large-scale objects characterized by complex hydrogeologic conditions. Computational algorithms and computer programs developed for realization of the model are based on the use of finite-difference methods to distinguish between different physical processes and modeling regions. Results are presented from examples of calculations to illustrate characteristic features of the problem of mass transfer by coupled flows.

Structured and unstructured grid adaptation schemes for numerical modeling of field problems

The use of grid adaptation procedures for compressible flows and heat transfer is presented in this paper. Grid adaptation schemes are presented for structured and unstructured grids in the context of finite-difference and finite-element methods. For compressible flow problems with regular domains, the use of finite-difference methods has proven to be quite effective. For such analyses, a dynamic grid adaptation scheme is employed with a MacCormack predictor-corrector scheme to describe compressible flow features. The adaptation procedure uses a grid relocation stencil that is valid at both the interior and boundary points of the finite-difference grid. Linear combinations of spatial derivatives of specific flow variables, calculated with finite-element interpolation functions, are used as adaptation measures. The impact of the grid relocation procedure is demonstrated by simulating the effects of shock impingement on a three-dimensional unsymmetric axial corner. When the domain to be modeled is quite complex, unstructured finite-elementor finite-volume formulations are preferred. An unstructured mesh adaptation scheme with refinement and coarsening based on refinement indicators is used to describe inviscid and viscous flow features. This adaptation scheme uses quadrilateral and triangular elements to add grid density to regions of high gardients and remove elements where these gardients are low. Modifications of the mesh adaptation procedure enables its utilization in the analyses of heat transfer problems commonly associated with hypervelocity flows. The strategies used for mesh adaptation, computing refinement indicators, and time marching are described. The effectiveness of this procedure is reflected in good solution accuracy, reduction in the number of elements used, and computational efficiency.

The method of local linearization for the calculation of one-dimensional processes of two-phase flow in porous media

The authors consider processes of displacement of one fluid by another in a porous medium for which the corresponding solutions of the Buckley-Leverett problem [1] are nonself-similar because of the inhomogeneity of the initial and/or boundary conditions [2]. The traditional approach to the solution of these problems consists in the use of finite-difference methods or the method of characteristics in various modifications. However, finite-difference methods distort the solutions because of “smearing” of the discontinuities, and the method of characteristics (in the analytical variant for problems with step initial and boundary conditions or in the numerical variant for problems with conditions of the general type) becomes difficult to put into practice if the number of separated fronts (waves)is large.

An Adaptive Boundary Value Runge–Kutta Solver for First Order Boundary Value Problems

The use of finite difference methods which combine features of both Runge–Kutta processes and Gap schemes is proposed for developing adaptive codes for the solution of first order differential equations with two-point boundary conditions. The order conditions for the coefficients of these processes are given. A set of conditions, by which these order conditions can be reduced in number, are developed. An eighth order A-stable method which has second, fourth and sixth order A-stable methods embedded in it is given as an example. A variable order, variable step finite difference solver using embedded methods of this kind is described.

Dynamic effects of pair correlation functions and hydrodynamic interactions on reaction kinetics in liquids

The time-dependent reaction rate and cumulative reaction probability are solved numerically with the incorporation of an effective force and hydrodynamic interaction in the relative diffusion expression. The use of finite difference methods permits the solution no matter how complex the form of the pair correlation function (pcf) and the hydrodynamic interaction utilized. In particular, a Percus–Yevick pcf as well as pcf’s derived from Debye–Hückel theory are utilized to account for the initial distribution and interacting forces for the reactant (pair) molecules. Two measures of time-dependent reactivity, namely the cumulative reaction probability and the observed reaction rate, are found to exhibit significant effects of molecular interaction.

Relativistic nuclear fluid dynamics

By use of finite-difference methods we solve in three spatial dimensions the classical relativistic equations of motion for the collision of two heavy nuclei. These equations express the conservation of nucleon number, momentum, and energy, for a specified nuclear equation of state. For $^{20}\mathrm{Ne}$+$^{238}\mathrm{U}$ at laboratory bombarding energies per nucleon of 250 MeV and 2.1 GeV, we calculate the time evolution of the matter distribution for several impact parameters. Nearly central collisions deform and compress the system enormously, whereas in peripheral collisions the projectile is fragmented into a portion that proceeds roughly straight ahead at its original velocity and other portion that deposits its energy in the target. For a given impact parameter we construct from the velocity vectors at some large time the energy and angular distributions for the expanding matter. An integration of these results over impact parameter then gives us the double differential cross section $\frac{{d}^{2}\ensuremath{\sigma}}{d\mathrm{Ed}\ensuremath{\Omega}}$. For the 250-MeV case we compare calculated and experimental results in the form of proton energy spectra for four laboratory angles ranging from 30\ifmmode^\circ\else\textdegree\fi{} to 120\ifmmode^\circ\else\textdegree\fi{}. The calculations reproduce correctly the experimental slopes at each angle, as well as the overall decrease in the experimental cross section when going from forward to backward angles. However, at 30\ifmmode^\circ\else\textdegree\fi{} the calculated values are only one-half the experimental ones, whereas at 120\ifmmode^\circ\else\textdegree\fi{} they are twice as large. These comparisons, together with comparisons of calculations done by other workers, suggest that heavy nuclei are partially transparent to each other during collisions at high energy, but that the process is not solely a superposition of individual nucleon-nucleon collisions. Instead, coherent collective-field effects play some role.NUCLEAR REACTIONS High-energy heavy-ion $^{20}\mathrm{Ne}$+$^{238}\mathrm{U}$, $\frac{{E}_{\mathrm{bom}}}{20}=250$ MeV, 2.1 GeV. Calculated $\frac{{d}^{2}\ensuremath{\sigma}}{\mathrm{dEd}\ensuremath{\Omega}}$ for outgoing matter and compared with experimental data. Relativistic nuclear fluid dynamics, hydrodynamic model, nuclear equation of state, particle-in-cell finite-difference computing method.

Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids

It is shown how the equilibrium pair correlation function between spin-bearing molecules in liquids may be incorporated as an effective force in the relative diffusion expressions, and how one may solve for the resulting time correlation functions and spectral densities needed for studies of spin relaxation by translational diffusion. The use of finite difference methods permits the solution no matter how complex the form of the pair correlation function (pcf) utilized. In particular, a Percus–Yevick pcf as well as one corrected from computer dynamics, both for hard spheres, are utilized. Good agreement with the experiments of Harmon and Muller on dipolar relaxation in liquid ethane is obtained from this analysis. Effects of ionic interactions in electrolyte solutions upon dipolar relaxation are also obtained in terms of Debye–Hückel theory for the pcf. Analytic solutions are given which are appropriate for the proper boundary-value problem for the relative diffusion of molecules (i.e., a distance of minimum approach) that has usually been neglected in the spin relaxation theories. Other molecular dynamics aspects of spin relaxation by translational diffusion in liquids are briefly discussed.

Relativistic Hydrodynamic Theory of Heavy-Ion Collisions

By use of finite-difference methods we solve the classical relativistic equations of motion for the head-on collision of two heavy nuclei. For $^{16}\mathrm{O}$ projectiles incident onto various targets at laboratory bombarding energies per nucleon \ensuremath{\le}2.1 GeV, curved shock waves develop. The target and projectile are deformed and compressed into crescents of revolution. This is followed by rarefaction waves and an overall expansion of the matter into a moderately wide distribution of angles.

A programming package for the calculation of cross-sections and probabilities for charge-exchange processes

We present a General-purpose computing on graphics processing units (GPGPU) based computational program and framework for the electronic dynamics of atomic systems under intense laser fields. We present our results using the case of hydrogen, however the code is trivially extensible to tackle problems within the single-active electron (SAE) approximation. Building on our previous work, we introduce the first available GPGPU based implementation of the Taylor, Runge–Kutta and Lanczos based methods created with strong field ab-initio simulations specifically in mind; CLTDSE. The code makes use of finite difference methods and the OpenCL framework for GPU acceleration. The specific example system used is the classic test system; Hydrogen. After introducing the standard theory, and specific quantities which are calculated, the code, including installation and usage, is discussed in-depth. This is followed by some examples and a short benchmark between an 8 hardware thread (i.e. logical core) Intel Xeon CPU and an AMD 6970 GPU, where the parallel algorithm runs 10 times faster on the GPU than the CPU.Program title: CLTDSECatalogue identifier: AESM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3 and aboveNo. of lines in distributed program, including test data, etc.: 15 988No. of bytes in distributed program, including test data, etc.: 233 518Distribution format: tar.gzProgramming language: C99 and OpenCL C. C99 conformance is ensured through use of C99 and pedantic flags under GCC and Clang.Computer: Single compute node.Operating system: GNU/Linux. It should, in principle, work with little modification for other Unix-Like systems.Has the code been vectorised or parallelised?: OpenCL is a parallel language. Thus CLTDSE can use all cores on a processor or GPU. OpenCL supports using all available compute units (GPUs and CPUs etc.), although this is not currently implemented in CLTDSE.RAM: Negligible RAM and GPU global memory (20MiB)Classification: 2.5.External routines: An OpenCL library; the current major packages are APP by AMD (CPU and AMD GPU), the NVIDIA driver (GPU), and the Intel SDK for OpenCL Applications (CPU and Intel HD Graphics). libconfig for processing configuration files.Nature of problem:Describing the dynamics of electrons under intense laser fields in atoms or molecules.Solution method:Solving the discretised system through finite difference using the Lanczos, Runge–Kutta, and Taylor methods.Restrictions:The example problem which is implemented is under the single active electron approximation.Unusual features:Focused on GPGPU acceleration through OpenCL.Running time:The running time depends on the nature of the pulse. This can vary from seconds to tens of minutes.