High-order Finite-difference Algorithm Research Articles

Hamiltonian Computational Chemistry: Geometrical Structures in Chemical Dynamics and Kinetics.

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon-Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes.

An approach to remove the numerical dispersion in elastic wave modeling using R-Cycle-GAN networks

Abstract The finite difference forward modeling has been widely used in geophysics exploration and petroleum fields. Because of its high efficiency and easy application for graphical processing units, it has been widely concerned by industry and academia. However, owing to many factors, the problem of numerical dispersion has been an important factor hindering this method. To overcome the numerical dispersion, this paper proposes a method for removing numerical dispersion using deep learning. Unlike the conventional optimized algorithms target to optimize the finite difference coefficients, our strategy is based on big data training to eliminate the dispersion data after seismic data modeling. We design a neural network architecture based on cycle-consistent generative adversarial networks (Cycle-GANs) and residual learning for elastic wave propagation. Under the premise of not significantly increasing the calculation time, we can obtain higher calculation accuracy. Compared with the high-order finite difference algorithm, the calculation time is the advantage of our proposed deep learning method. Tests prove the efficiency and stability of our proposed algorithm.

The effects of transaction costs and illiquidity on the prices of volatility derivatives

The prices of assets differ when considering transaction costs and their illiquidity, thus affecting their returns. As volatility derivatives such as variance swaps, gamma swaps, corridor variance swaps and volatility swaps depend on the expected values of these returns, we develop a high-order algorithm so that differences in the fair values of the volatility derivatives under different transaction costs and illiquidity models can be precisely observed. A high-order finite-difference algorithm is used to solve a framework of partial differential equations. A local mesh refinement strategy is applied at the singularities that appear in the pricing problem of some volatility derivatives so that nonuniform five-point stencil finite differences can be implemented in space. The time-stepping is dealt with by an efficient iterative scheme coupled with a Richardson extrapolation and Carathéodory–Fejér approximations. Consequently, the highly convergent solutions are used to analyze the effect of transaction costs and illiquidity on the fair values of volatility derivatives, and the results demonstrate an increase in these values when different transaction costs and liquidity models are implemented.

Shape reconstruction of large optical surface with high-order terms in fringe reflection technique

A fast and effective shape reconstruction method of large aspheric specular surfaces with high order terms is proposed in fringe reflection technique, which combines modal estimation with high-order finitedifference algorithm. The iterative equation with highorder truncation errors is derived for calculating the specular surface with large aperture based on high-order finite-difference algorithm. To achieve the wavefront estimation and improve convergence speed, the numerical orthogonal transformation method based on Zernike polynomials is implemented to obtain the initial iteration value. The reconstruction results of simulated surface identified the advantages of the proposed method. Furthermore, a freeform in illuminating system has been used to demonstrate the validity of the improved method in practical measurement. The results show that the proposed method has the advantages of making the reconstruction of different shape apertures accurate and rapid. In general, this method performs well in measuring large complex objects with high frequency information in practical measurement.

Two Kinds of Parabolic Equation algorithms in the Computational Electromagnetics

Slip-step Fourier Transform (SSFT) and Finite Difference (FD) algorithms are introduced to solve Parabolic Equation (PE). SSFT is applied to analyze the problem of radio propagation in vacuum. Based on Split-step Padé approximation, a high order FD algorithm is introduced to compute the Radar Cross Section (RCS). According to the results, the differences of the two kinds of PE method have been discussed.

A high-order finite-difference algorithm for direct computation of aerodynamic sound

A high-order finite-difference algorithm is proposed in the aim of performing LES calculations for CAA applications. The subgrid scale dissipation is performed by the explicit high-order numerical filter used for numerical stability purpose. A shock-capturing non-linear filter is also used to deal with compressible discontinuous flows. In order to tackle complex geometries, an overset-grid approach is used. High-order interpolations make possible the communication between overlapping domains. The whole algorithm is first validated on canonical flow problems to illustrate both its properties for shock-capturing as well as for accurate wave propagation. Then, the influence of the multi-domain approach on the high-order spatial accuracy is assessed. Finally, a rod-airfoil configuration is studied to highlight the potential of the proposed algorithm to deal with multi-scale aeroacoustic applications.

Acoustic wave propagation simulation in a poroelastic medium saturated by two immiscible fluids using a staggered finite-difference with a time partition method

Based on the three-phase theory proposed by Santos, acoustic wave propagation in a poroelastic medium saturated by two immiscible fluids was simulated using a staggered high-order finite-difference algorithm with a time partition method, which is firstly applied to such a three-phase medium. The partition method was used to solve the stiffness problem of the differential equations in the three-phase theory. Considering the effects of capillary pressure, reference pressure and coupling drag of two fluids in pores, three compressional waves and one shear wave predicted by Santos have been correctly simulated. Influences of the parameters, porosity, permeability and gas saturation on the velocities and amplitude of three compressional waves were discussed in detail. Also, a perfectly matched layer (PML) absorbing boundary condition was firstly implemented in the three-phase equations with a staggered-grid high-order finite-difference. Comparisons between the proposed PML method and a commonly used damping method were made to validate the efficiency of the proposed boundary absorption scheme. It was shown that the PML works more efficiently than the damping method in this complex medium. Additionally, the three-phase theory is reduced to the Biot’s theory when there is only one fluid left in the pores, which is shown in Appendix. This reduction makes clear that three-phase equation systems are identical to the typical Biot’s equations if the fluid saturation for either of the two fluids in the pores approaches to zero.

High-order compact finite-difference methods on general overset grids

This work investigates the coupling of a very high-order finite-difference algorithm for the solution of conservation laws on general curvilinear meshes with overset-grid techniques originally developed to address complex geometric configurations. The solver portion of the algorithm is based on Padé-type compact finite-differences of up to sixth-order, with up to 10th-order filters employed to remove spurious waves generated by grid non-uniformities, boundary conditions and flow non-linearities. The overset-grid approach is utilized as both a domain-decomposition paradigm for implementation of the algorithm on massively parallel machines and as a means for handling geometric complexity in the computational domain. Two key features have been implemented in the current work; the ability of the high-order algorithm to accommodate holes cut in grids by the overset-grid approach, and the use of high-order interpolation at non-coincident grid overlaps. Several high-order/high-accuracy interpolation methods were considered, and a high-order, explicit, non-optimized Lagrangian method was found to be the most accurate and robust for this application. Several two-dimensional benchmark problems were examined to validate the interpolation methods and the overall algorithm. These included grid-to-grid interpolation of analytic test functions, the inviscid convection of a vortex, laminar flow over single- and double-cylinder configurations, and the scattering of acoustic waves from one- and three-cylinder configurations. The employment of the overset-grid techniques, coupled with high-order interpolation at overset boundaries, was found to be an effective way of employing the high-order algorithm for more complex geometries than was previously possible.

Comment on “High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics” [J. Chem. Phys. 111, 10827 (1999)

The spectral difference methods [D. A. Mazziotti, Chem. Phys. Lett. 299, 473 (1999)] for solving differential equations in chemical physics combine the useful features of matrix sparsity and rapid convergence. In their recent article [J. Chem. Phys. 111, 10827 (1999)] Guantes and Farantos incorrectly classify the Lagrange distributed approximating functional (LDAF) method in the category of finite differences. This comment clarifies the connections among higher-order finite difference, Lagrange distributed approximating functionals, and other spectral difference methods.

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.

High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics. II. Periodic variables

Variable high order finite difference methods are applied to calculate the action of molecular Hamiltonians on the wave function using centered equi-spaced stencils, mixed centered and one-sided stencils, and periodic Chebyshev and Legendre grids for the angular variables. Results from one-dimensional model Hamiltonians and the three-dimensional spectroscopic potential of SO2 demonstrate that as the order of finite difference approximations of the derivatives increases the accuracy of pseudospectral methods is approached in a regular manner. The high order limit of finite differences to Fourier and general orthogonal polynomial discrete variable representation methods is analytically and numerically investigated.

A High-Order Finite-Difference Algorithm for the Analysis of Standing Acoustic Waves of Finite but Moderate Amplitude

A numerical formulation for modelling standing acoustic waves of finite but moderate amplitude is presented. A thermoviscous fluid contained in a one-dimensional tube with rigid walls is considered. The fluid is initially at rest and is excited by means of a harmonic piston. A second-order wave equation for viscous and homogeneous fluids is used. The perturbation method is employed. The numerical simulation is carried out by a multi-time-step, implicit, six-point finite-difference scheme of high order in the time domain. Displacement and pressure waveforms and distributions are presented. Numerical results are validated by comparison with an analytical model. The numerical scheme is illustrated with several examples.

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.

Performance study of an adaptive dual antenna handset for indoor communications

The focus of the paper is the design and evaluation of adaptive diversity for mobile handsets. Usually, diversity principles are optimised for base stations. However, for mobile handsets new concepts must be developed to include the size and power consumption constraints. A new modelling approach is introduced, which combines indoor radio channel simulations with circuit design parameters. This enables the inclusion of key system parameters, such as the speed of the user, the scanning speed of the antenna beams and the number of phase shifts. The radio channel simulations are based on a high-order finite-difference algorithm using the Berenger absorbing boundary condition to truncate the computational domain. The algorithm is found to be efficient and accurate enough to simulate very large structures. The analysis has resulted in an optimal design of an adaptive dual antenna handset, which combines received signals (fixed beam) while scanning the environment at the same time (scan beams). A prototype is evaluated with the numerical modelling tools and a measurement set-up. The performance is close to that of a perfect equal gain combiner.