One-sided Difference Research Articles

Improved Block Interface Conditions for Aeroacoustic Problems with Grid Discontinuities

Aeroacoustic simulations are extremely sensitive to numerical errors. To achieve a high order of accuracy, optimized finite difference schemes, such as the well-known dispersion-relation-preservation schemes, and multiblock structured meshes are often used. Challenges appear in communicating the flow information near grid discontinuities between neighboring blocks, and sophisticated interface treatments are required. The characteristic interface condition has been successful in applications in previous publications, and yet limitations exist that are identified in the present study. This paper proposes a new interface treatment based on the analogy between finite difference and finite volume discretizations. One-sided finite difference schemes are used to calculate the spatial derivatives in the normal direction, and the residuals are reconstructed from a finite volume perspective. Applications to several benchmark cases show an encouraging improvement as compared to the previous characteristic interface treatments.

The use of high-order composite compact schemes for computing supersonic jet interaction with a surface

High-order composite compact schemes are applied to the simulation of viscous gas dynamics with strong discontinuities of flow variables. To perform shock-capturing computations of such problems, the dissipation of the basic operators is enhanced and the solutions obtained with these operators are locally replaced by those produced with the help of simple one-sided differences. Numerical results obtained for the shock interaction of a supersonic axisymmetric jet with a flat surface are presented.

High-order symplectic FDTD scheme for solving a time-dependent Schrödinger equation

Using the three-order symplectic integrators and fourth-order collocated spatial differences, a high-order symplectic finite-difference time-domain (SFDTD) scheme is proposed to solve the time-dependent Schrödinger equation. First, the high-order symplectic framework for discretizing a Schrödinger equation is described. Then the numerical stability and dispersion analyses are provided for the FDTD(2, 2), higher-order FDTD(2, 4) and SFDTD(3, 4) schemes. Next, to implement the Dirichlet boundary condition encountered in the quantum eigenvalue problem, the image theory and one-sided difference technique are manipulated particularly for high-order collocated differences. Finally, a detailed numerical study on 1D and 2D quantum eigenvalue problems is carried out. The simulation results of quantum wells and harmonic oscillators strongly confirm the advantages of the SFDTD scheme over the traditional FDTD method and other high-order approaches. The explicit SFDTD scheme, which is high-order-accurate and energy-conserving, is well suited for a long-term simulation and can save computer resources with large time step and coarse spatial grids.

Grid stabilization of high-order one-sided differencing II: Second-order wave equations

We demonstrate the stable boundary closure of difference methods of order up through 16 for the solution of wave equations in second order form. Our method combines the introduction of 1–2 judiciously placed subcell grid points near the boundary with minimal-stencil, one-sided difference operators of the same order as the interior scheme. The method is tested on a variety of problems including the scalar wave equation discretized on mapped grids and overlapping composite grids, as well as an integrable nonlinear system.

Adaptive parameter choice for one-sided finite difference schemes and its application in diabetes technology

In this paper we discuss the problem of approximation of the first derivative of a function at the endpoint of its definition interval. This problem is motivated by diabetes therapy management, where it is important to provide estimations of the future blood glucose trend from current and past measurements. A natural way to approach the problem is to use one-sided finite difference schemes for numerical differentiation, but, following this way, one should be aware that the values of the function to be differentiated are noisy and available only at given fixed points. Then (as we argue in the paper) the number of used point values is the only parameter to be employed for regularization of the above mentioned ill-posed problem of numerical differentiation. In this paper we present and theoretically justify an adaptive procedure for choosing such a parameter. We also demonstrate some illustrative tests, as well as the results of numerical experiments with simulated clinical data.

A Third Order Accurate Fast Marching Method for the Eikonal Equation in Two Dimensions

In this paper, we develop a third order accurate fast marching method for the solution of the eikonal equation in two dimensions. There have been two obstacles to extending the fast marching method to higher orders of accuracy. The first obstacle is that using one-sided difference schemes is unstable for orders of accuracy higher than two. The second obstacle is that the points in the difference stencil are not available when the gradient is closely aligned with the grid. We overcome these obstacles by using a two-dimensional (2D) finite difference approximation to improve stability, and by locally rotating the grid 45 degrees (i.e., using derivatives along the diagonals) to ensure all the points needed in the difference stencil are available. We show that in smooth regions the full difference stencil is used for a suitably small enough grid size and that the difference scheme satisfies the von Neumann stability condition for the linearized eikonal equation. Our method reverts to first order accuracy near caustics without developing oscillations by using a simple switching scheme. The efficiency and high order of the method are demonstrated on a number of 2D test problems.

Can Variational Models for Correspondence Problems Benefit from Upwind Discretisations?

Optic flow and stereo reconstruction are important examples of correspondence problems in computer vision. Correspondence problems have been studied for almost 30 years, and energy-based methods such as variational approaches have become popular for solving this task. However, despite the long history of research in this field, only little attention has been paid to the numerical approximation of derivatives that naturally occur in variational approaches. In this paper we show that strategies from hyperbolic numerics can lead to a significant quality gain in computational results. Starting from a basic formulation of correspondence problems, we take on a novel perspective on the mathematical model. Switching the roles of known and unknown with respect to image data and displacement field, we use the arising hyperbolic colour equation as a basis for a refined numerical approach. For its discretisation, we propose to use one-sided differences in the correct direction identified via a smooth predictor solution. The one-sided differences that are first-order accurate are blended with higher-order central schemes. Thereby the blending mechanism interpolates between the following two situations: The one-sided method is employed at image edges which often coincide with edges in the displacement field. In smooth image parts the higher-order scheme is used. We apply our new scheme to several prototypes of variational models for optic flow and stereo reconstruction, where we achieve significant qualitative improvements compared to standard discretisations.

A comprehensive study for the plasmonic thin-film solar cell with periodic structure

A comprehensive study of the plasmonic thin-film solar cell with the periodic strip structure is presented in this paper. The finite-difference frequency-domain method is employed to discretize the inhomogeneous wave function for modeling the solar cell. In particular, the hybrid absorbing boundary condition and the one-sided difference scheme are adopted. The parameter extraction methods for the zeroth-order reflectance and the absorbed power density are also discussed, which is important for testing and optimizing the solar cell design. For the numerical results, the physics of the absorption peaks of the amorphous silicon thin-film solar cell are explained by electromagnetic theory; these peaks correspond to the waveguide mode, Floquet mode, surface plasmon resonance, and the constructively interference between adjacent metal strips. The work is therefore important for the theoretical study and optimized design of the plasmonic thin-film solar cell.

A modified Crank–Nicolson scheme with incremental unknowns for convection dominated diffusion equations

A modified Crank–Nicolson scheme based on one-sided difference approximations is proposed for solving time-dependent convection dominated diffusion equations in two-dimensional space. The modified scheme is consistent and unconditionally stable. A priori L 2 error estimate for the fully discrete modified scheme is derived. With the use of the incremental unknowns preconditioner at each time step, a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified Crank–Nicolson scheme.

Development of the North American Listening in Spatialized Noise–Sentences Test (NA LiSN-S): Sentence Equivalence, Normative Data, and Test–Retest Reliability Studies

The Listening in Spatialized Noise-Sentences test (LiSN-S) was originally developed in Australia to assess auditory stream segregation skills in children with suspected central auditory processing disorder (CAPD). The software produces a three-dimensional auditory environment under headphones. A simple repetition-response protocol is utilized to determine speech reception thresholds (SRTs) for sentences presented from 0 degrees azimuth in competing speech. The competing speech (looped children's stories) is manipulated with respect to its location (0 degrees vs. +90 degrees and -90 degrees azimuth) and the vocal quality of the speaker(s) (same as, or different to, the speaker of the target stimulus). Performance is measured as two SRT and three advantage measures. The advantage measures represent the benefit in dB gained when either talker, spatial, or both talker and spatial cues combined are incorporated in the maskers. The objective of this research was to develop a version of the LiSN-S suitable for use in the United States and Canada. The original sentences and children's stories were reviewed for unfamiliar semantic items and rerecorded by native North American speakers. In a descriptive design, a sentence equivalence study was conducted to determine the relative intelligibility of the rerecorded sentences and adjust the amplitude of the sentences for equal intelligibility. Normative data and test-retest reliability data were then collected. Twenty-four children with normal hearing aged 8 years, 3 months, to 10 years, 0 months, took part in the sentence equivalence study. Seventy-two normal-hearing children aged 6 years, 2 months, to 11 years, 10 months, took part in the normative data study. Thirty-six children returned between two and three months after the initial assessment for retesting. Participants were recruited from sites in Cincinnati, Dallas, and Calgary. The sentence equivalence study showed that post-adjustment, sentence intelligibility increased by 18.7 percent for each 1 dB increase in signal-to-noise ratio. Analysis of the normative data revealed no significant differences on any performance measure as a consequence of data collection site or gender. Inter- and intra-participant variation was minimal. A trend of improved performance as a function of increasing age was found across performance measures, and cutoff scores, calculated as two standard deviations below the mean, were adjusted for age. Test-retest differences were not significant on any measure of the North American (NA) LiSN-S (p ranging from .080 to .862). Mean test-retest differences on the various NA LiSN-S performance measures ranged from 0.1 dB to 0.6 dB. One-sided critical difference scores calculated from the retest data ranged from 3 to 3.9 dB. These scores, which take into account mean practice effects and day-to-day fluctuations in performance, can be used to determine whether a child has improved on the NA LiSN-S on retest. The NA LiSN-S is a potentially valuable tool for assessing auditory stream segregation skills in children. The availability of one-sided critical difference scores makes the NA LiSN-S useful for monitoring listening performance over time and determining the effects of maturation, compensation (such as an assistive listening device), or remediation.

Multi-objective optimization in variably saturated fluid flow

A general purpose multi-objective inverse modeling strategy was developed and implemented to quantify fluid flow parameters in variably saturated porous materials. The strategy combines a robust and mass-conservative numerical simulator of the flow equation with an optimization algorithm and an experimental data set to estimate the parameters. The numerical simulator of the direct problem shows excellent agreement with a reference solution and conserves global mass with near perfection. An adaptive method was proposed in which the sensitivity matrix was calculated by one-sided finite difference approximation at the early stages of the optimization and the more accurate two-sided differentiation as the search approaches the minimum. A combined termination criterium was developed to stop the inverse code at the solution. The results of the multi-objective optimization were compared with those of single-objective minimization. While single-objective optimization generates reasonable results for either the fluid pressure head profile or the fluid content data, the proposed multi-objective optimization shows excellent agreement with both profiles simultaneously.

A two-step computational aeroacoustics method applied to high-speed flows

This study numerically investigates the generated noise and perturbation fields from an excited high-speed jet flow (air), which is modeled by the linearized Euler equations. The mean flow is prescribed by experimental-analytical approach. Harmonic hydrodynamic excitation is applied at the nozzle exit to mimic a previous experimental setup, and to enhance the periodic component of the perturbation field. We investigate the propagation of this excitation in the pressure (acoustic), velocity, and density waves, looking at details that were not addressed in the experiments. We integrate the governing equations using the finite-difference method and a special scheme that is second-order accurate in time and fourth-order accurate in space. Alternating one-sided differences are implemented to eliminate spatial biases in the calculations. We use an appropriate set of boundary conditions that minimizes spurious reflections, including the characteristics-based Thompson inflow, acoustic radiation, Tam and Webb outflow, and centerline conditions. We investigate the characteristics of these waves, including the spatial growth and damping, and the noise radiation pattern.

A direct comparison between volume and surface tracking methods with a boundary-fitted coordinate transformation and third-order upwinding

A Volume Tracking (VT) and a Front Tracking (FT) algorithm are implemented and compared for locating the interface between two immiscible, incompressible, Newtonian fluids in a tube with a periodically varying, circular cross-section. Initially, the fluids are stationary and stratified in an axisymmetric arrangement so that one is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). A constant pressure gradient sets them in motion. With both VT and FT, a boundary-fitted coordinate transformation is applied and appropriate modifications are made to adopt either method in this geometry. The surface tension force is approximated using the continuous surface force method. All terms appearing in the continuity and momentum equations are approximated using centered finite differences in space and one-sided forward finite differences in time. In each time step, the incompressibility condition is enforced by a transformed Poisson equation, which is linear in pressure. This equation is solved by either direct LU decomposition or a Multigrid iterative solver. When the two fluids have the same density, the former method is about 3.5 times faster, but when they do not, the Multigrid solver is as much as 10 times faster than the LU decomposition. When the interface does not break and the Reynolds number remains small, the accuracy and rates of convergence of VT and FT are comparable. The well-known failure of centered finite differences arises as the Reynolds number increases and leads to non-physical oscillations in the interface and failure of both methods to converge with mesh refinement. These problems are resolved and computations with Reynolds as large as 500 converged by approximating the convective terms in the momentum equations by third-order upwind differences using Lagrangian Polynomials. When the volume of the core fluid or the Weber number decrease, increasing the importance of interfacial tension and leading to breakup of the interface forming a drop of core fluid, the FT method converges faster with mesh refinement than the VT method and upwinding may be required. Finally, examining the generation of spurious currents around a stationary “bubble” in the tube for Ohnesorge numbers between 0.1 and 10 it is found that the maximum velocity remains approximately the same in spite mesh refinements when VT is applied, whereas it is of the same order of magnitude for the coarsest mesh and monotonically decreases with mesh refinement when FT is applied.

On the spurious solutions in the high-order finite difference methods for eigenvalue problems

In this paper, the origins of spurious solutions occurring in the high-order finite difference methods are studied. Based on a uniform mesh, spurious modes are found in the high-order one-sided finite difference discretizations of many eigenvalue problems. Spurious modes are classified as spectral pollution and non-spectral pollution. The latter can be partially avoided by mesh refinement, while the former persists when the mesh is refined. Through numerical studies of some prototype eigenvalue problems, such as those of the Helmholtz and beam equations, we show that perfect central differentiation schemes do not produce any spurious modes. Nevertheless, high-order central difference schemes encounter difficulties in implementing complex boundary conditions. We further show that the central difference schemes will produce spurious modes as well, if asymmetric approximation is involved in boundary treatments. In general, central difference schemes are less likely to produce spurious modes than one-sided difference schemes.

Grid stabilization of high-order one-sided differencing I: First-order hyperbolic systems

We construct stable, maximal order boundary closures for high order central difference methods. The stability is achieved by adding a small number of additional subcell nodes near the boundaries at experimentally determined locations. We find that methods up through 8th order can be stabilized by the addition of a single node, up through 16th order by the addition of two nodes, and up through 22nd order with three extra nodes. We also consider the application of the technique to dispersion relation preserving methods, and we construct and test artificial dissipation operators.

Numerical stability for finite difference approximations of Einstein’s equations

We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in numerical relativity and, more generally, in Hamiltonian formulations of field theories. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy–Ortiz–Reula formulation of Einstein’s equations. We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems.

A LOW-ORDER-SINGULARITY ELECTRIC-FIELD INTEGRAL EQUATION SOLVABLE WITH PULSE BASIS FUNCTIONS AND POINT MATCHING

Abstract : Unlike the magnetic-field integral equation, the conventional form of the electric-field integral equation (EFIE) cannot be solved accurately with the method of moments (MOM) using pulse basis functions and point matching. The highly singular kernel of the EFIE, rather than the current derivatives, precludes the use of the pulse-basis function point-matching MOM. A new form of the EFIE has been derived whose kernel has no greater singularity than that of the free-space Green's function. This new low-order singularity form of the EFIE, the LEFIE, has been solved numerically for perfectly electrically conducting bodies of revolution (BORs) using pulse basis functions and point-matching. Derivatives of the current are approximated with finite differences using a quadratic Lagrangian interpolation polynomIal. This simple solution of the LEFIE is contingent, however, on the vanishing of a linear integral that appears when the original EFIE is transformed to obtain the LEFIE. This generally restricts the applicability of the LEFIE to smooth closed scatterers. Bistatic scattering calculations performed for a prolate spheroid demonstrate that results comparable in accuracy to the conventional EFIE can be obtained with the LEFIE using pulse basis functions and point matching provided a higher density of points is used close to the ends of the BOR generating curve to compensate for the use of one-sided finite difference approximations of the first and second derivatives of the current.

Characteristic Interface Conditions for Multiblock High-Order Computation on Singular Structured Grid

A structured grid with a body usually has a certain point where an abrupt change in the slope of grid line exists. The grid metrics are discontinuous at the point because of the discrepancy between the left- and the right-hand limits of the gradients, which leads to grid singularity. It may cause serious numerical oscillations especially when high-order finite difference schemes are applied to solving conservation-form governing equations in generalized coordinates. In this paper, it is handled by decomposing a computational domain into blocks along the singular lines and imposing interface conditions at the block interfaces for communication between the blocks. A set of high-order finite difference schemes is used in each block: central differences on the interior nodes and one-sided differences on the near-interface nodes. The differencing stencils do not cross the block interfaces and each block is isolated without the singularity, which results in no oscillations. For the communication between the isolated blocks, the interface conditions are newly derived from the characteristic relations of the compressible Euler or Navier-Stokes equations. The exactness and the feasibility of the interface conditions are investigated for the high-order multi-block computation on structured grid containing singular points.

Two-timescale simultaneous perturbation stochastic approximation using deterministic perturbation sequences

Simultaneous perturbation stochastic approximation (SPSA) algorithms have been found to be very effective for high-dimensional simulation optimization problems. The main idea is to estimate the gradient using simulation output performance measures at only two settings of the N -dimensional parameter vector being optimized rather than at the N + 1 or 2 N settings required by the usual one-sided or symmetric difference estimates, respectively. The two settings of the parameter vector are obtained by simultaneously changing the parameter vector in each component direction using random perturbations. In this article, in order to enhance the convergence of these algorithms, we consider deterministic sequences of perturbations for two-timescale SPSA algorithms. Two constructions for the perturbation sequences are considered: complete lexicographical cycles and much shorter sequences based on normalized Hadamard matrices. Recently, one-simulation versions of SPSA have been proposed, and we also investigate these algorithms using deterministic sequences. Rigorous convergence analyses for all proposed algorithms are presented in detail. Extensive numerical experiments on a network of M / G /1 queues with feedback indicate that the deterministic sequence SPSA algorithms perform significantly better than the corresponding randomized algorithms.

Efficient Multiscale Methods for Optimal In Situ Bioremediation Design

This paper presents a multiscale derivative method for solving a successive approximation linear quadratic regulator model for optimal in situ bioremediation design. An efficient one-sided forward divided difference numerical derivatives calculation was implemented as the first stage of the method, which only required assembling the right-hand-side vector of the linear systems of equations of the simulation model and performing backward substitution. The derivative calculation was reduced from O( N 3 ) to nearly O(N 2 ), where N is the number of non-Dirichlet state variables. A V-cycle multiscale derivatives approximation was implemented as the second stage, which used coarser mesh derivatives to interpolate finer mesh derivatives. Implementing the numerical derivatives method in a case study with over 1,600 state variables caused a reduction of more than two-thirds in computing time over the previous analytical derivatives method without loss of accuracy. Using the V-cycle multiscale derivatives approximation further reduced computing time by 29%, resulting in an overall 77% reduction compared to the previous analytical derivatives method. The reduction will be even greater for applications with more state variables, enabling the solution of much larger-scale problems than was previously possible.