Numerical Grid Methods Research Articles

Ground state properties of the screened helium atom under harmonic confinement

In this contribution, we show electronic property results for the ground state of the helium atom screened by a Debye plasma under an isotropic harmonic confinement as a function of the screening length (λD) and the angular frequency (ω0). The associate Hamiltonian is solved by expanding the wave function in the Hylleraas basis to incorporate the electron correlation in a simple manner. Furthermore, in order to compare with other findings the Hartree–Fock approach is implemented using a numerical grid method. The main results show that the total energy reaches the ionization threshold by the action of the electron coupling with the plasma medium and the strength of the harmonic confinement. The electron–electron average distance shows the compression of the system as the Debye plasma and the harmonic interactions become strongest (low λD and large ω0 values). We found acceptable results for the electron correlation energy when separate interactions are considered. In the absence of both interactions, the findings are in excellent agreement with other theoretical results.

Shannon, Rényi entropies, and Fisher information calculations of the Li1+ and Be2+ ions screened by the ion-sphere plasma model

We show results for the Shannon entropy, Rényi entropy, Fisher information, and their relative relations in the position space for the Li1+ and Be2+ ions embedded in a strong plasma. This medium is simulated by the ion-sphere screening model as a function of the plasma electron density, N e . We solve the Hartree–Fock equations using a numerical grid method in a self-consisting manner and the simple Hylleraas-type correlated wave function is applied. Our findings show a reduction of the total energy for both ions as the electron density is increased reaching the ionization limit. We find interesting features in the description of the Shannon entropy and Fisher information where the correlation effects of the embedded ions are relevant. The Shannon-Fisher product results indicate that the electron-electron correlation decreases as N e increases and similar behavior is found for the Rényi-Fisher product. Finally, in the absence of the plasma medium, our results are in excellent agreement with those available in the literature.

Tsallis entropy: A comparative study for the 1s2-state of helium atom

The Tsallis entropies are reported, as a function of the Tsallis index q and the screening parameter μ, in position space, for the ground state of the helium atom. Three different potentials, namely Coulomb, Yukawa, and exponential cosine screened Coulomb potentials and their combinations are used in the calculations. The simple Hylleraas correlated wave function, with a three-parameter, is applied to focus on getting the most characteristic physical features of the electron-electron repulsion results. The values are calculated in one- and two-electron charge densities. Moreover, for a comparative study, the Hartree-Fock approximation, which implements a numerical grid method, is employed. Additionally, the modified Tsallis index, Qq, is proposed and calculated to predict the best value of q in the combined potentials, and it was found that 0.5≤q<1.0.

Grid methods for Bayes-optimal continuous-discrete filtering and utilizing a functional tensor train representation

Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.

The information theory of the helium atom in screened Coulomb potentials

AbstractIn this work, we show Hartree–Fock (HF) results for the Shannon entropy, Rényi entropy, Tsallis entropy, Onicescu information energy, and ground‐state properties of the helium atom in screened Coulomb potentials. We solve the HF equations through a numerical grid method where the screened functions model two plasma environments: the Debye–Hückel screening (DHS) and the exponential‐cosine‐screened‐Coulomb (ECSC) model potentials. For plasma conditions near to the energy ionization, for the Shannon entropy, we report an increase of the ∼80% and ∼60% for the DHS and ECSC potentials concerning the unscreened case. A similar situation is found for the Rényi and Tsallis entropies where an increase of ∼86% and ∼57% is reported in the presence of the DHS and ECSC potential functions. Our results allow us to obtain knowledge of the information theory parameters in the description of the electron localization for the helium atom screened by the DHS and ECSC potentials. Finally, our information theory findings in the absence of the screening functions are in excellent agreement with the available results found in the literature.

A Comparison of Different Wave Modelling Techniques in An Open-Source Hydrodynamic Framework

Modern design for marine and coastal activities places increasing focus on numerical simulations. Several numerical wave models have been developed in the past few decades with various techniques and assumptions. Those numerical models have their own advantages and disadvantages. The proper choice of the most useful numerical tool depends on the understanding of the validity and limitations of each model. In the past years, REEF3D has been developed into an open-source hydrodynamic numerical toolbox that consists of several modules based on the Navier–Stokes equations, the shallow water equations and the fully nonlinear potential theory. All modules share a common numerical basis which consists of rectilinear grids with an immersed boundary method, high-order finite differences and high-performance computing capabilities. The numerical wave tank of REEF3D utilises a relaxation method to generate waves at the inlet and dissipate them at the numerical beach. In combination with the choice of the numerical grid and discretisation methods, high accuracy and stability can be achieved for the calculation of free surface wave propagation and transformation. The comparison among those models provide an objective overview of the different wave modelling techniques in terms of their numerical performance as well as validity. The performance of the different modules is validated and compared using several benchmark cases. They range from simple propagations of regular waves to three-dimensional wave breaking over a changing bathymetry. The diversity of the test cases help with an educated choice of wave models for different scenarios.

Kohn–Sham approach for fast hybrid density functional calculations in real-space numerical grid methods

Real-space methods have not been suitable for hybrid density functional calculations due to high cost coming from the nonlocality of Fock operator. Here we propose a practical approach for fast computation. The key is to use a strictly local Kohn–Sham potential that can be deduced from any hybrid functional using the optimized effective potential method. This new approach improved the computation speed of the global and range-separated hybrid methods up to 30 and 80 times, respectively. As a result, accurate prediction of the size-dependent excitonic spectra of Si quantum dots became feasible.

MODELLING OF MICROBIOLOGICAL METHOD OF OIL RECOVERY IMPROVEMENT

The problem of energy supply has been one of the most pressing issues for the humanity at all times. Oil is one of the most desired sources of energy. However, oil reserves are being depleted, and substantial amounts of subterranean residual oil cannot be extracted with the use of traditional methods, therefore requiring implementation of new technologies. The current paper suggests two mathematical models aimed to describe and predict implications of microbiological method of oil recovery improvement based on the currently available data, as well as to assess the efficiency of this method. The first model consists of partial differential equations that describe the process of expansion of microbial population through oil formation. The second model includes ordinary differential equations, describing growth and metabolism of bacterial cultures within the bottom-hole zone of the production well. The models were studied with the use of numerical grid and Euler methods. The modeling parameters were evaluated using the least squares method. The current paper includes the results of calculations performed on basis of the proposed models. Distribution profiles of the bacterial population and the metabolic products along the reservoir were obtained and studied; provided recommendations for practical implementation of the microbiological method.

Outstanding performance of configuration interaction singles and doubles using exact exchange Kohn-Sham orbitals in real-space numerical grid method.

To assess the performance of multi-configuration methods using exact exchange Kohn-Sham (KS) orbitals, we implemented configuration interaction singles and doubles (CISD) in a real-space numerical grid code. We obtained KS orbitals with the exchange-only optimized effective potential under the Krieger-Li-Iafrate (KLI) approximation. Thanks to the distinctive features of KLI orbitals against Hartree-Fock (HF), such as bound virtual orbitals with compact shapes and orbital energy gaps similar to excitation energies; KLI-CISD for small molecules shows much faster convergence as a function of simulation box size and active space (i.e., the number of virtual orbitals) than HF-CISD. The former also gives more accurate excitation energies with a few dominant configurations than the latter, even with many more configurations. The systematic control of basis set errors is straightforward in grid bases. Therefore, grid-based multi-configuration methods using exact exchange KS orbitals provide a promising new way to make accurate electronic structure calculations.

A new level set numerical wave tank with improved density interpolation for complex wave hydrodynamics

A new three-dimensional numerical wave tank is developed for the calculation of wave propagation and wave hydrodynamics by solving the incompressible Navier–Stokes equations. The free surface is modeled with the level set method based on a two-phase flow approximation, allowing for the simulation of complex phenomena such as wave breaking. The convection terms of the momentum and the level set equations are discretized with the finite difference version of the fifth-order WENO scheme. Time stepping is handled with the third-order TVD Runge–Kutta scheme. The equations are solved on a staggered Cartesian grid, with a ghost cell immersed boundary method for the treatment of irregular cells. Waves are generated at the inlet and dissipated at the numerical beach with the relaxation method. The choice of the numerical grid and discretization methods leads to excellent accuracy and stability for the challenging calculation of free surface waves. The performance of the numerical model is validated and verified through several benchmark cases: solitary wave interaction with a rectangular abutment, wave forces on a vertical cylinder, wave propagation over a submerged bar and plunging breaking waves on a sloping bed.

Configuration interaction singles based on the real-space numerical grid method: Kohn-Sham versus Hartree-Fock orbitals.

We developed a program code of configuration interaction singles (CIS) based on a numerical grid method. We used Kohn-Sham (KS) as well as Hartree-Fock (HF) orbitals as a reference configuration and Lagrange-sinc functions as a basis set. Our calculations show that KS-CIS is more cost-effective and more accurate than HF-CIS. The former is due to the fact that the non-local HF exchange potential greatly reduces the sparsity of the Hamiltonian matrix in grid-based methods. The latter is because the energy gaps between KS occupied and virtual orbitals are already closer to vertical excitation energies and thus KS-CIS needs small corrections, whereas HF results in much larger energy gaps and more diffuse virtual orbitals. KS-CIS using the Lagrange-sinc basis set also shows a better or a similar accuracy to smaller orbital space compared to the standard HF-CIS using Gaussian basis sets. In particular, KS orbitals from an exact exchange potential by the Krieger-Li-Iafrate approximation lead to more accurate excitation energies than those from conventional (semi-) local exchange-correlation potentials.

Dynamic full-wave modeling of storm microseism propagation in an oceanic environment

The full-wave modeling of storm seismoacoustic field propagation in oceanic waveguides and at the ocean-continent border in the low-frequency range from 0.01 to 1 Hz using the numerical grid method is considered. The 2D model proposed by Press and Ewing for planar waveguides above the homogeneous elastic halfspace was chosen as a basis. The influence of a layer of deposits and features of the crust of the oceanic- and continent-type was taken into account in the more complex model. The effects of the carrier frequency, orientation of directivity diagram of the signal source, and steepness of the continental slope were investigated. A system of the second-order linear partial differential equations of hyperbolic type was solved in the process of modeling. The inertia properties and attenuation in the medium were taken into consideration. Fast-operating locally recursive nonlocally asynchronous parallel algorithms were used in the modeling, which were developed at the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences for modeling of plasma processes. At present, these algorithms are adapted for modeling of geophysical fields. A computing system consisting of two computers based on eight-core processors is used for computations. As a result, instantaneous pictures of microseism field propagation were constructed at the specified cross sections of the medium at sequential time moments, and the microseism’s frequency spectra and synthetic seismograms were obtained for some characteristic points of aqueous medium, bottom sediment, and land. The considered algorithms of full-wave numerical modeling and the computing system can be easily adapted to the 3D variant of anisotropic and inhomogeneous media specified by analytical expressions.

Double photoionization of excited lithium and beryllium

We present total, energy-sharing, and triple differential cross sections for one-photon, double ionization of lithium and beryllium starting from aligned, excited $P$ states. We employ a recently developed hybrid atomic orbital with numerical grid method based on the finite-element discrete-variable representation and exterior complex scaling. Comparisons with calculated results for the ground-state atoms, as well as analogous results for ground-state and excited helium, serve to highlight important selection rules and show some interesting effects that relate to differences between inter- and intrashell electron correlation.

Mesoscopic fluid flow simulation in double‐porosity rocks

A discrete Fourier transform algorithm is designed to simulate mesoscopic fluid flow(MFF) in double‐porosity rocks. Double‐porosity equations with MFF are derived. Results from pseudo‐spectral simulation show MFF transfers fast P waves and the first kind of slow P waves' energy to the second kind of slow P waves, and the Biot diffusive mode significantly attenuates the second kind of slow P waves' energy. We use a novel approach with a numerical grid method to solve the double‐porosity wave equations. We conclude that the wavefield's attenuation should be attributed to a two‐step mechanism: (1) MFF and (2) macroscopic Biot diffusion. Numerical estimations have shown that MFF in the double‐porosity model can produce a high attenuation (0.32DB/10 m, 1/Q = 0.1083) in the seismic band (50 Hz).

Voigt lineshape function as a solution of the parabolic partial differential equation

The goal of this paper is to show that the Voigt function may be found as a solution of a parabolic partial differential equation, like the heat conduction equation or other diffusion equations. A square of the Gaussian half-width of the Voigt function plays the role of ‘time’ and initial conditions are determined by a Lorentz function. Some questions concerning the practical application of the numerical grid methods for the calculation of the Voigt function are discussed. It is shown, that in some cases the offered calculation algorithm can be both faster and more accurate than other known algorithms.

Finite basis representations with nondirect product basis functions having structure similar to that of spherical harmonics

The currently most efficient finite basis representation (FBR) method [Corey et al., in Numerical Grid Methods and Their Applications to Schrodinger Equation, NATO ASI Series C, edited by C. Cerjan (Kluwer Academic, New York, 1993), Vol. 412, p. 1; Bramley et al., J. Chem. Phys. 100, 6175 (1994)] designed specifically to deal with nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc., employs very special l-independent grids and results in a symmetric FBR. While highly efficient, this method is not general enough. For instance, it cannot deal with nondirect product bases of the above structure efficiently if the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are discrete variable representation (DVR) functions of the infinite type. The optimal-generalized FBR(DVR) method [V. Szalay, J. Chem. Phys. 105, 6940 (1996)] is designed to deal with general, i.e., direct and/or nondirect product, bases and grids. This robust method, however, is too general, and its direct application can result in inefficient computer codes [Czako et al., J. Chem. Phys. 122, 024101 (2005)]. It is shown here how the optimal-generalized FBR method can be simplified in the case of nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc. As a result the commonly used symmetric FBR is recovered and simplified nonsymmetric FBRs utilizing very special l-dependent grids are obtained. The nonsymmetric FBRs are more general than the symmetric FBR in that they can be employed efficiently even when the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are DVR functions of the infinite type. Arithmetic operation counts and a simple numerical example presented show unambiguously that setting up the Hamiltonian matrix requires significantly less computer time when using one of the proposed nonsymmetric FBRs than that in the symmetric FBR. Therefore, application of this nonsymmetric FBR is more efficient than that of the symmetric FBR when one wants to diagonalize the Hamiltonian matrix either by a direct or via a basis-set contraction method. Enormous decrease of computer time can be achieved, with respect to a direct application of the optimal-generalized FBR, by employing one of the simplified nonsymmetric FBRs as is demonstrated in noniterative calculations of the low-lying vibrational energy levels of the H3+ molecular ion. The arithmetic operation counts of the Hamiltonian matrix vector products and the properties of a recently developed diagonalization method [Andreozzi et al., J. Phys. A Math. Gen. 35, L61 (2002)] suggest that the nonsymmetric FBR applied along with this particular diagonalization method is suitable to large scale iterative calculations. Whether or not the nonsymmetric FBR is competitive with the symmetric FBR in large-scale iterative calculations still has to be investigated numerically.

Calculation of temperature fields in the ground with water-ice phase transitions in the temperature spectrum

An algorithm of integration of the nonlinear heat-conduction equation for a semi-bounded medium with a uniform initial distribution of temperature and its constant value on the surface in the case of nonlinear dependence of thermophysical characteristics on temperature is described. This algorithm can be used to test numerical grid methods of calculation of temperature fields in freezing and thawing of grounds. The procedure of correction of a step-by-step mechanism of temperature variation in numerical calculation in the zone of intense phase transitions by a grid method was suggested and tested.

An analysis of the indirect boundary element method for seismic modelling

An indirect boundary element method has been used to compute seismic wavefields in layered media. The advantages of this technique over numerical grid methods are that it satisfies radiation conditions, incorporates free surface effects, solves viscoelastic problems and can handle arbitrary source types. The technique is more appropriate for models with high aspect ratios and a limited number of layers. The computational cost increases with the number of layers and frequency content. Examples are presented in this study to illustrate some of the features of the technique. Seismograms are computed for SH and P–SV line sources in 2-D earth models. Finite-difference and the Cagniard–de Hoop methods are used to test the accuracy of the technique. Even though the computation time for the method is several times greater than finite differences, the technique becomes cost effective as the number of computed shot-gathers increases. The technique also allows computation of constant offset gathers at a small additional cost which is very expensive to do by using finite differences.

Numerical grid methods for quantum-mechanical scattering problems

We show how the finite-element method can be implemented using a discrete variable representation to provide an efficient means for directly solving the time-independent Schr\"odinger equation on a multidimensional numerical grid. For collision problems, an exterior complex scaling transformation obviates the need for explicit imposition of asymptotic boundary conditions, making the method particularly useful for studying three-body breakup. The method is illustrated by studying an analytically solvable two-dimensional (2D) breakup problem as well as a 2D model problem with exponential potentials.

Atomic calculations with an augmented Fourier basis

We present a scheme for Hartree–Fock and related calculations on atoms using a Fourier basis augmented by an exponential function. All multiple sums in the variational energy expression are written as single sums over combinations of multiplications and convolutions. The computational cost of the method is thus practically linear in the number of basis functions. The results compare favorably with the most precise published results of numerical grid methods and variational calculations. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 189–197, 1998