High Order Finite Difference Approximations Research Articles

Real-space finite-difference method for conductance calculations

We present a general method for calculating coherent electronic transport in quantum wires and tunnel junctions. It is based upon a real space high order finite difference representation of the single particle Hamiltonian and wave functions. Landauer's formula is used to express the conductance as a scattering problem. Dividing space into a scattering region and left and right ideal electrode regions, this problem is solved by wave function matching (WFM) in the boundary zones connecting these regions. The method is tested on a model tunnel junction and applied to sodium atomic wires. In particular, we show that using a high order finite difference approximation of the kinetic energy operator leads to a high accuracy at moderate computational costs.

Stable and Accurate Artificial Dissipation

Stability for nonlinear convection problems using centered difference schemes require the addition of artificial dissipation. In this paper we present dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.

Characteristics Method Using Cubic–Spline Interpolation for Advection–Diffusion Equation

The characteristics method by using the cubic-spline interpolation is comparable to the Holly–Preissmann scheme in solving the advection portion of the advection–diffusion equation. In order to conduct a cubic-spline interpolation, an additional constraint must be specified at each endpoint. In general, four types of endpoint constraints are available, i.e., the first derivative, second derivative, quadratic, and not-a-knot constraints. The goal of this paper is to examine each type of endpoint constraints. Two hypothetical cases are used to conduct the investigation. Among the four types of constraints examined herein, the not-a-knot constraint and the first derivative constraint with high-order finite difference approximation yield the best results. However, as far as accuracy and simple implementation are concerned the not-a-knot constraint should be the best choice in solving the advection–diffusion equation

Car-Parrinello simulations with a real space method.

An approach for Car-Parrinello type molecular dynamics simulations using real space discretized wave functions is presented. The implementation uses a higher order finite difference approximation for the Laplacian in the kinetic energy operator and in the Poisson equation for the evaluation of the Hartree energy. Norm conserving pseudopotentials are employed to replace the nucleus-core interactions. Both nuclei and wave functions are propagated with a second-order Verlet propagator with a timestep comparable to standard plane wave implementations. The orthonormality constraints are fulfilled via Lagrange multipliers. Both periodic and nonperiodic boundary conditions can directly be imposed within the real space approach. In order to maintain energy conservation both forces on the nuclei and the wave functions must be sufficiently accurate. It is shown that the spurious energy dependence on the nuclear positions with respect to the grid points, which is an inevitable artifact of the discretization, is mirrored in the nuclear forces, and does not affect energy conservation. In contrast to the plane wave methodology, the Hartree energy is calculated via an iterative solution of the Poisson equation. As a result, the quality of the forces on the wave function crucially depends on its convergence, as well as on the accuracy of the approximation for the boundary values in nonperiodic calculations. The influence of these factors on the energy conservation is investigated. It is shown that Car-Parrinello simulations can be performed with real space wave functions, given a proper prescription to calculate the Hartree potential is used.

Application of High Order Finite Difference Approximation as Exponential Interpolation

Application of High Order Finite Difference Approximation as Exponential Interpolation

High Order Finite Difference Approximations of Electromagnetic Wave Propagation Close to Material Discontinuities

Electromagnetic wave propagation close to a material discontinuity is simulated by using summation by part operators of second, fourth and sixth order accuracy. The interface conditions at the discontinuity are imposed by the simultaneous approximation term procedure. Stability is shown and the order of accuracy is verified numerically.

Optimal Two-Dimensional Finite Difference Grids Providing Superconvergence

A novel multidimensional grid adaptation method based on minimization of the leading truncation error term of arbitrary second- and higher-order finite difference approximations is proposed. The method does not explicitly require the truncation error estimate, but increases the design order of approximation globally by one, so that the same finite difference operator is superconvergent on the optimal grid. If the differential operator and the metric coefficients are evaluated identically by some hybrid approximation, the single optimal grid generator can be used in the entire computational domain and does not depend on points where the hybrid discretization switches from one approximation to another. If one family of the coordinate lines is given a priori, then the analytical optimal mapping is constructed for any consistent second-order finite difference approximation of $\nabla f$ in two dimensions. The present approach can be extended directly to nonlinear partial differential equations and three dimensions. Numerical calculations show that the truncation error obtained on the two-dimensional optimal grid is both superconvergent and reduced by several orders of magnitude in comparison with the uniform grid results for all the test examples considered.

Response to “Comment on ‘High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics’ ” [J. Chem. Phys. 115, 6794 (2001)

The comment of Mazziotti about the classification of the Lagrange distributed approximating functional method as a finite difference method is answered. Furthermore, the relations of high order finite difference approximation of the Laplacian of the Schrödinger equation to well known pseudospectral techniques such as the fast Fourier transform and discrete variable representations are clarified.

Construction and analysis of higher order finite difference schemes for the 1D wave equation

In this article, we propose to: 1. Establish most of the properties conjectured in [2] about the higher order finite difference approximation of the 1D Laplace operator. 2. Generalize to any order the fourth-order accurate scheme in space and time of Shubin and Bell [20] and Cohen [6]. For this new family of 2m–2m schemes, we establish, via elementary mathematics, various stability and dispersion results that are helpful to compare these schemes to the 2–2m schemes of Anne et al. [2].

High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics

The view of considering global Pseudospectral methods (Sinc and Fourier) as the infinite order limit of local finite difference methods, and vice versa, finite difference as a certain sum acceleration of the pseudospectral methods is exploited to investigate high order finite difference algorithms for solving the Schrödinger equation in molecular dynamics. A Morse type potential for iodine molecule is used to compare the eigenenergies obtained by a Sinc Pseudospectral method and a high order finite difference approximation of the action of the kinetic energy operator on the wave function. Two-dimensional and three-dimensional model potentials are employed to compare spectra obtained by fast Fourier transform techniques and variable order finite difference. It is shown that it is not needed to employ very high order approximations of finite differences to reach the numerical accuracy of pseudospectral techniques. This, in addition to the fact that for complex configuration geometries and high dimensionality, local methods require less memory and are faster than pseudospectral methods, put finite difference among the effective algorithms for solving the Schrödinger equation in realistic molecular systems.

Simulations of Acoustic Wave Phenomena Using High-Order Finite Difference Approximations

Numerical studies of hyperbolic initial boundary value problems (IBVP) in several space dimensions have been performed using high-order finite difference approximations. It is shown that for wave propagation problems, where the wavelengths are small compared to the domain and long time integrations are needed, high-order schemes are superior to low-order ones. In fact, in two dimensions an acoustic lens is simulated, leading to large scale computations where high-order methods and powerful parallel computers are necessary if an accurate solution is to be obtained.

Numerical simulation of transitional aiding mixed convective air flow in a bottom heated inclined rectangular duct

Buoyancy induced transient transitional flow structures and the associated heat transfer in mixed convective aiding flow of air in a bottom heated inclined rectangular duct were numerically investigated. The three-dimensional unsteady Navier-Stokes and energy equations were discretized by higher order finite-difference approximations and directly solved by the projection method. Computations were carried out for the Reynolds number ranging from 100 to 500, aspect ratio of the duct from 2 to 4, duct inclination from 0° to 60°C and buoyancy-to-inertia ratio high enough to cause flow transition. Attention was particularly paid to delineate the effects of the duct inclination on the flow transition and the associated temporal and spatial flow structures. It is of interest to note from the predictions that increasing the inclined angle measured from horizontal does not always stabilize the flow. In fact, the flow stability is highly dependent on all the governing parameters and is intimately related to the structural changes in the buoyancy induced longitudinal vortices.

EVOLUTION OF ROLL PATTERNS IN RAYLEIGH-BENARD CONVECTION IN A RECTANGULAR LAYER: A NUMERICAL STUDY

A three-dimensional unsteady numerical simulation was carried out to unravel the buoyancy-induced flow in a bottom-heated shallow rectangular box with perfect conducting side walls. The Navier-Stokes and energy equations were discretized by the higher order finite difference approximations and solved by the explicit projection method. Computations were specifically carried out for water Pr = 3.5 ) and air (Pr = 0.71) in a box of aspect ratios 10.6 X 5.3. The results for the water layer indicated that in raising the Rayleigh number (Ra) suddenly from zero to a value slightly above that for onset of convection, square cells were induced that later gradually merge to form parallel rolls along the short sides. Soft patterns were induced where Ra was abruptly raised from 0 to 4 Racr. The augmentation of wavelength in a parallel roll system was found to result from the disappearance of the rolls near the short sides and the expansion of the other rolls. In an air layer the increase in the roll size resulted fro...

Summation by Parts, Projections, and Stability. II

In this paper we prove strict stability of high-order finite difference approximations of parabolic and symmetric hyperbolic systems of partial differential equations on bounded, curvilinear domains in two space dimensions. The boundary need not be smooth. We also show how to derive strict stability estimates for inhomogeneous boundary conditions.

Summation by Parts, Projections, and Stability. I

We have derived stability results for high-order finite difference approximations of mixed hyperbolic-parabolic initial-boundary value problems (IBVP). The results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection. By rearranging the analytic equations slightly, we can prove strict stability for hyperbolic-parabolic IBVP. Furthermore, we generalize our technique so as to yield stability on nonsmooth domains in two space dimensions. Using the same procedure, one can prove stability in higher dimensions as well.

Polynomial interpolation schemes for internal derivative distributions on structured grids

Computational fluid dynamics problems which use a streamfunction formulation require differentiation to obtain the velocity field. If the velocity components are required at internal points on the grid (for example, for particle trajectory calculations) this necessitates some form of interpolation. Similarly, N-body computations which include self-gravity and cannot afford the luxury of an O( N 2) 1 r 2 calculation, require interpolation to obtain the force components at locations internal to an overlaid grid. These two independent examples reduce to the same generalized problem. Given a function distribution on a structured grid, which interpolation schemes give the most suitable solutions in terms of accuracy, computational efficiency and smoothness properties? The best numerical schemes are those which use higher order finite difference approximations for the derivatives (but not vast templates), perform interpolation directly over the derivative (and not the function), and have an odd power as the leading term in the polynomial expansion for the derivative. These methods include forms of bicubic and quintic-cubic interpolation but although generating accurate and realistic internal derivative estimates, come at a higher computational price. For simulations where the interpolation routines need to be called frequently, the lower order schemes such as cubic-linear and biquadratic interpolation, although less accurate, are more computationally efficient and may be more appropriate.

Summation by parts, projections, and stability. I

We have derived stability results for high-order finite difference approximations of mixed hyperbolic-parabolic initial-boundary value problems (IBVP). The results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection. By rearranging the analytic equations slightly, we can prove strict stability for hyperbolic-parabolic IBVP. Furthermore, we generalize our technique so as to yield stability on nonsmooth domains in two space dimensions. Using the same procedure, one can prove stability in higher dimensions as well.

Summation by parts, projections, and stability. II

In this paper we prove strict stability of high-order finite difference approximations of parabolic and symmetric hyperbolic systems of partial differential equations on bounded, curvilinear domains in two space dimensions. The boundary need not be smooth. We also show how to derive strict stability estimates for inhomogeneous boundary conditions.

Formulation of high order finite difference approximations to the 3-D neutron diffusion equation

A high order finite difference approximation (FDA) for the efficient solution of 3-D equations containing the Laplacian operator has been formulated for homogeneous mediums. The method was developed for equal spacing and can have either a 4th or 6th order truncation error, depending on the number of points involved in the computational molecule and on 2 adjustable parameters evaluated from the numerical solution. The model is well suited for numerical computation and can easily be implemented in the FDA codes.

Analysis of high order finite difference approximations to the 3-D neutron diffusion equation

In the previous paper a high order correct finite difference approximation FDA to the Laplacian operator in 3-D was obtained. In this paper, the effectiveness of this new FDA will be demonstrated through the use of the one-group diffusion equation in a cubical reactor. The computational performances of the 6th, 4th and 2nd order approximations are compared to the analytical solution. From the computer results of sample problems, considerable (94%) savings can be obtained using the higher order formulas, especially the 27-point relation as compared to the 2nd order approximation.