Split-operator Approach Research Articles

Reflectionless Klein tunneling of Dirac fermions: comparison of split-operator and staggered-lattice discretization of the Dirac equation

Massless Dirac fermions in an electric field propagate along the field lines without backscattering, due to the combination of spin-momentum locking and spin conservation. This phenomenon, known as ‘Klein tunneling’, may be lost if the Dirac equation is discretized in space and time, because of scattering between multiple Dirac cones in the Brillouin zone. To avoid this, a staggered space-time lattice discretization has been developed in the literature, with one single Dirac cone in the Brillouin zone of the original square lattice. Here we show that the staggering doubles the size of the Brillouin zone, which actually contains two Dirac cones. We find that this fermion doubling causes a spurious breakdown of Klein tunneling, which can be avoided by an alternative single-cone discretization scheme based on a split-operator approach.

High-order harmonic generation by static coherent states method in single-electron atomic and molecular systems.

We solve the time-dependent Schrodinger equation using the coherent states as basis sets for computing high harmonic generation (HHG) in a full-dimensional single-electron "realistic" system. We apply the static coherent states (SCS) method to investigate HHG in the hydrogen molecular ion induced by a linearly polarized laser field. We show that SCS gives reasonable agreement compared to the three dimensional unitary split-operator approach. Next, we study isolated attosecond pulse generation in . To do so, we employ the well-known polarization gating technique, which combines two delayed counter-rotating circular laser pulses, and opens up a gate at the central portion of the superposed pulse. Our results suggest that the SCS method can be used for full-dimensional quantum simulation of higher dimensional systems such as the hydrogen molecule in the presence of an external laser field.

Sediment transport and bed evolution model for complex river systems.

This work presents the theoretical background, development, and preliminary evaluation of a one-dimensional unsteady sediment transport and bed evolution model for a looped river network. The sediment transport and bed evolution model employs a concept that differentiates sediment particles moving in the form of suspended sediment, and near bed, and bed sediment. Applying the active layer concept, the developed model utilizes the appropriate exchange mechanisms between the suspended sediment and active layer material, which are presented in great detail. The governing transport equations were solved using the split operator approach that resulted in two successive steps. The advection equations were solved using the characteristics method, whereas the diffusion equations were discretized using the Crank-Nicholson scheme. The obtained system was complemented with additional auxiliary equations in order to allow sediment transport and bed evolution simulation at nodes, thus enabling the same in a looped river network. The derived equations were applied to develop an open-channel flow, sediment transport, and bed evolution model that was subjected to a series of preliminary numerical tests.

Groundwater transport modeling with nonlinear sorption and intraparticle diffusion

Despite recent advances in the mechanistic understanding of sorption in groundwater systems, most contaminant transport models provide limited support for nonideal sorption processes such as nonlinear isotherms and/or diffusion-limited sorption. However, recent developments in the conceptualization of “dual mode” sorption for hydrophobic organic contaminants have provided more realistic and mechanistically sound alternatives to the commonly used Langmuir and Freundlich models. To support the inclusion of both nonlinear and diffusion-limited sorption processes in groundwater transport models, this paper presents two numerical algorithms based on the split operator approach. For the nonlinear equilibrium scenario, the commonly used two-step split operator algorithm has been modified to provide a more robust treatment of complex multi-parameter isotherms such as the Polanyi-partitioning model. For diffusion-limited sorption, a flexible three step split-operator procedure is presented to simulate intraparticle diffusion in multiple spherical particles with different sizes and nonlinear isotherms. Numerical experiments confirmed the accuracy of both algorithms for several candidate isotherms. However, the primary advantages of the algorithms are: (1) flexibility to accommodate any isotherm equation including “dual mode” and similar expressions, and (2) ease of adapting existing grid-based transport models of any dimensionality to include nonlinear sorption and/or intraparticle diffusion. Comparisons are developed for one-dimensional transport scenarios with different isotherms and particle configurations. Illustrative results highlight (1) the potential influence of isotherm model selection on solute transport predictions, and (2) the combined effects of intraparticle diffusion and nonlinear sorption on the plume transport and flushing for both single-particle and multi-particle scenarios.

Advection step in the split-operator approach applied to river modeling

The split-operator approach proposes to divide the numerical solution of the governing flow and constituent transport equations into a series of successive steps, one of them being the numerically most challenging advection step. This paper presents tests of characteristics and finite difference methods used in the numerical solution of one-dimensional linear and nonlinear advection equations that describe the transport and flow advection steps, thus demonstrating their accuracy, oscillations and damping features. The studied methods were incorporated into the split-operator approach and evaluated as a part of the complete solution, which led to the development of numerical models implementing an algorithm that can endure larger time steps without losing much accuracy. This algorithm was tested on both one- and two-dimensional examples, for complex computational domains that predominate in natural watercourses.

Ring polymer dynamics in curved spaces

We formulate an extension of the ring polymer dynamics approach to curved spaces using stereographic projection coordinates. We test the theory by simulating the particle in a ring, T(1), mapped by a stereographic projection using three potentials. Two of these are quadratic, and one is a nonconfining sinusoidal model. We propose a new class of algorithms for the integration of the ring polymer Hamilton equations in curved spaces. These are designed to improve the energy conservation of symplectic integrators based on the split operator approach. For manifolds, the position-position autocorrelation function can be formulated in numerous ways. We find that the position-position autocorrelation function computed from configurations in the Euclidean space R(2) that contains T(1) as a submanifold has the best statistical properties. The agreement with exact results obtained with vector space methods is excellent for all three potentials, for all values of time in the interval simulated, and for a relatively broad range of temperatures.

Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations

A detailed comparison of the time-dependent wave packet method using the split operator propagator and recently introduced Chebyshev real wave packet approach for calculating reactive scattering processes is reported. As examples, the state-to-state differential cross sections of the H + H(')D(v(0) = 0,j(0) = 1) --> H(')D + H/H(')H + D reaction, the state-to-state reaction probabilities of the (16)O + (35)O(2) (v(0) = 0,j(0) = 0) --> (17)O + (16)O(18)O/(18)O + (16)O(17)O reaction, the H + O(2) --> O + HO reaction, and the F + HD --> HF + D reaction are calculated, using an efficient reactant-coordinate-based method on an L-shape grid which allows the extraction of the state-to-state information of the two product channels simultaneously. These four reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these two propagation methods for describing reactive scattering dynamics. The results indicate that the Chebyshev real wave packet method is typically more accurate, particularly for reactions dominated by long-lived resonances. However, the split operator approach is often more cost effective, making it a method of choice for fast reactions. In addition, our results demonstrate accuracy of the reactant-coordinate-based method for extracting state-to-state information.

Two-Dimensional Nonequilibrium Noncohesive and Cohesive Sediment Transport Model

The purpose of this paper is to develop an unsteady 2D depth-averaged model for nonuniform sediment transport in alluvial channels. In this model, the orthogonal curvilinear coordinate system is adopted; the transport mechanisms of cohesive and noncohesive sediment are both embedded; the suspended load and bed load are treated separately. In addition, the processes of hydraulic sorting, armoring, and bed consolidation are also included in the model. The implicit two-step split-operator approach is used to solve the flow governing equations and the coupling approach with iterative method are used to solve the mass-conservation equation of suspended sediment, mass-conservation equation of active-layer sediment, and global mass-conservation equation for bed sediment simultaneously. Three sets of data, including suspension transport, degradation and aggradation cases for noncohesive sediment, and aggradation, degra- dation, and consolidation cases for cohesive sediment, have been demonstrated to show the rationality and accuracy of the model. Finally, the model is applied to evaluate the desilting efficiency for Ah Gong Diann Reservoir located in Taiwan to show its applicability.

Two-dimensional mixing of pollutants in streams with transverse line source

A numerical scheme based on the split-operator approach was developed for the solution of two-dimensional mixing equation with a transverse line-slug source in open channels. An exact solution of the advection equation was developed by adopting the Courant number equal to unity for the computational nodes lying on the flow surface. For the other computational nodes, the cubic spline interpolation scheme was utilized. The computation of the diffusion rate involved the alternate-direction implicit method. The scheme was illustrated by hypothesizing the transverse line injection of tracer at the flow surface, at mid-depth and at the channel bed; it was found that the numerical model is mass conservative. The scheme was validated by comparing its results with Fischer's analytical solution for one-dimensional mixing, the analytical solution by Lipsett and Beltaos for a continuous injection of pollutant as a steady transverse line source, and the laboratory data for slug injection.

Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport

Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1238-1255) developed exact first and second nonlocal moment equations for advective-dispersive transport in finite, randomly heterogeneous geologic media. The velocity and concentration in these equations are generally nonstationary due to trends in heterogeneity, conditioning on site data and the influence of forcing terms. Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) solved the Laplace transformed versions of these equations recursively to second order in the standard deviation σY of (natural) log hydraulic conductivity, and iteratively to higher-order, by finite elements followed by numerical inversion of the Laplace transform. They did the same for a space-localized version of the mean transport equation. Here we recount briefly their theory and algorithms; compare the numerical performance of the Laplace-transform finite element scheme with that of a high-accuracy ULTIMATE-QUICKEST algorithm coupled with an alternating split operator approach; and review some computational results due to Morales-Casique et al. (Adv. Water Res., 29 (2006), pp. 1399-1418) to shed light on the accuracy and computational efficiency of their recursive and iterative solutions in comparison to conditional Monte Carlo simulations in two spatial dimensions. AMS subject classifications: 65Z05, 65Z05, 68U20, 76S05

Polynomial-time quantum algorithm for the simulation of chemical dynamics

The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can be applied only to small systems. By contrast, we demonstrate that quantum computers could exactly simulate chemical reactions in polynomial time. Our algorithm uses the split-operator approach and explicitly simulates all electron-nuclear and interelectronic interactions in quadratic time. Surprisingly, this treatment is not only more accurate than the Born-Oppenheimer approximation but faster and more efficient as well, for all reactions with more than about four atoms. This is the case even though the entire electronic wave function is propagated on a grid with appropriately short time steps. Although the preparation and measurement of arbitrary states on a quantum computer is inefficient, here we demonstrate how to prepare states of chemical interest efficiently. We also show how to efficiently obtain chemically relevant observables, such as state-to-state transition probabilities and thermal reaction rates. Quantum computers using these techniques could outperform current classical computers with 100 qubits.

Attosecond-resolution quantum dynamics calculations for atoms and molecules in strong laser fields

A parallel quantum electron and nuclei wave packet computer code, LZH-DICP, has been developed to study laser-atom-molecule interaction in the nonperturbative regime with attosecond resolution. The nonlinear phenomena occurring in that regime can be studied with the code in a rigorous way by numerically solving the time-dependent Schrödinger equation of electrons and nuclei. Time propagation of the wave functions is performed using a split-operator approach, and based on a sine discrete variable representation. Photoelectron spectra for hydrogen and kinetic-energy spectra for molecular hydrogen ion in linearly polarized laser fields are calculated using a flux operator scheme, which testifies to the validity and the high efficiency of LZH-DICP.

Numerical Investigation of Wave–Current–Vegetation Interaction

A fully three-dimensional numerical model has been developed to simulate the wave–current–vegetation interaction phenomenon. Physical experiments have also been carried out to provide data for the verification of the model. The numerical model utilizes the split-operator approach, in which the advection, diffusion, and pressure propagation are solved separately. Vegetation is modeled as a sink of momentum. The unsteady fluid force on vegetation is split into a time-dependent inertia component and a drag component. The model has been applied to simulate vegetation under pure waves, pure current, as well as wave current. Compared to available experimental data, the model is capable of reproducing the turbulence and velocity profiles induced by vegetation–current interaction. The wave attenuation due to vegetation is simulated correctly with a proper value of drag coefficient. Both the physical experiments and numerical simulations show that the interaction of waves and current leads to a greater attenuation of waves in the presence of vegetation, which can be explained by the nonlinear nature of the resistance force induced by the vegetation.

Electron Transfer Dynamics in Solution Using Imaginary-time Split Operator Approach

An imaginary-time split operator approach is proposed to study electron transfer (ET) dynamics using Sumi-Marcus theory. The approach is applied to evaluate the intermolecular ET rate between oxazine 1 and N,N-dimethlaniline. By measuring the two average survival times of the donor state probability and the rate constant in long time limit, the full kinetics of the ET is revealed with a variety of sink functions. Several new properties for the ET have been found in numerical simulations.

Quantum effect of intramolecular high-frequency vibrational modes on diffusion-controlled electron transfer rate: From the weak to the strong electronic coupling regions

The Sumi-Marcus theory is extended by introducing two approaches to investigate electron transfer reactions from weak-to-strong electronic coupling regime. One of these approaches is the quantum R-matrix theory, useful for dealing with the intramolecular vibrational motions in the whole electronic coupling domain. The other is the split operator approach that is employed to solve the reaction-diffusion equation. The approaches are then applied to electron transfer in the Marcus inverted regime to investigate the nuclear tunneling effect on the long time rate and the survival probabilities. The numerical results illustrate that the adiabatic suppression obtained from the R-matrix approach is much smaller than that from the Landau-Zener theory whereas it cannot be predicted by the perturbation theory. The jointed effects of the electronic coupling and solvent relaxation time on the rates are also explored.

Qprop: A Schrödinger-solver for intense laser–atom interaction

The Qprop package is presented. Qprop has been developed to study laser–atom interaction in the nonperturbative regime where nonlinear phenomena such as above-threshold ionization, high order harmonic generation, and dynamic stabilization are known to occur. In the nonrelativistic regime and within the single active electron approximation, these phenomena can be studied with Qprop in the most rigorous way by solving the time-dependent Schrödinger equation in three spatial dimensions. Because Qprop is optimized for the study of quantum systems that are spherically symmetric in their initial, unperturbed configuration, all wavefunctions are expanded in spherical harmonics. Time-propagation of the wavefunctions is performed using a split-operator approach. Photoelectron spectra are calculated employing a window-operator technique. Besides the solution of the time-dependent Schrödinger equation in single active electron approximation, Qprop allows to study many-electron systems via the solution of the time-dependent Kohn–Sham equations. Program summary Program title: QPROP Catalogue number:ADXB Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXB Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Computer on which program has been tested:PC Pentium IV, Athlon Operating system:Linux Program language used:C++ Memory required to execute with typical data:Memory requirements depend on the number of propagated orbitals and on the size of the orbitals. For instance, time-propagation of a hydrogenic wavefunction in the perturbative regime requires about 64 KB RAM (4 radial orbitals with 1000 grid points). Propagation in the strongly nonperturbative regime providing energy spectra up to high energies may need 60 radial orbitals, each with 30000 grid points, i.e. about 30 MB. Examples are given in the article. No. of bits in a word:Real and complex valued numbers of double precision are used No. of lines in distributed program, including test data, etc.:69 995 No. of bytes in distributed program, including test data, etc.: 2 927 567 Peripheral used:Disk for input–output, terminal for interaction with the user CPU time required to execute test data:Execution time depends on the size of the propagated orbitals and the number of time-steps Distribution format:tar.gz Nature of the physical problem:Atoms put into the strong field of modern lasers display a wealth of novel phenomena that are not accessible to conventional perturbation theory where the external field is considered small as compared to inneratomic forces. Hence, the full ab initio solution of the time-dependent Schrödinger equation is desirable but in full dimensionality only feasible for no more than two (active) electrons. If many-electron effects come into play or effective ground state potentials are needed, (time-dependent) density functional theory may be employed. Qprop aims at providing tools for (i) the time-propagation of the wavefunction according to the time-dependent Schrödinger equation, (ii) the time-propagation of Kohn–Sham orbitals according to the time-dependent Kohn–Sham equations, and (iii) the energy-analysis of the final one-electron wavefunction (or the Kohn–Sham orbitals). Method of solution:An expansion of the wavefunction in spherical harmonics leads to a coupled set of equations for the radial wavefunctions. These radial wavefunctions are propagated using a split-operator technique and the Crank–Nicolson approximation for the short-time propagator. The initial ground state is obtained via imaginary time-propagation for spherically symmetric (but otherwise arbitrary) effective potentials. Excited states can be obtained through the combination of imaginary time-propagation and orthogonalization. For the Kohn–Sham scheme a multipole expansion of the effective potential is employed. Wavefunctions can be analyzed using the window-operator technique, facilitating the calculation of electron spectra, either angular-resolved or integrated Restrictions onto the complexity of the problem:The coupling of the atom to the external field is treated in dipole approximation. The time-dependent Schrödinger solver is restricted to the treatment of a single active electron. As concerns the time-dependent density functional mode of Qprop, the Hartree-potential (accounting for the classical electron–electron repulsion) is expanded up to the quadrupole. Only the monopole term of the Krieger–Li–Iafrate exchange potential is currently implemented. As in any nontrivial optimization problem, convergence to the optimal many-electron state (i.e. the ground state) is not automatically guaranteed External routines/libraries used:The program uses the well established libraries blas, lapack, and f2c

Coupled Surface–Subsurface Solute Transport Model for Irrigation Borders and Basins. I. Model Development

Surface fertigation is widely practiced in irrigated crop production systems. Lack of design and management tools limits the effectiveness of surface fertigation practices. The availability of a process-based coupled surface–subsurface hydraulic and solute transport model can lead to improved surface fertigation management. This paper presents the development of a coupled surface–subsurface solute transport model. A hydraulic model described in a previous paper by the writers provided the hydrodynamic basis for the solute transport model presented here. A numerical solution of the area averaged advection–dispersion equation, based on the split-operator approach, forms the surface solute transport component of the coupled model. The subsurface transport process is simulated using HYDRUS-1D, which also solves the one-dimensional advection–dispersion equation. A driver program is used for the internal coupling of the surface and subsurface transport models. Solute fluxes calculated using the surface transpor...

Three-dimensional model for multi-component reactive transport with variable density groundwater flow

PHWAT is a new model that couples a geochemical reaction model (PHREEQC-2) with a density-dependent groundwater flow and solute transport model (SEAWAT) using the split-operator approach. PHWAT was developed to simulate multi-component reactive transport in variable density groundwater flow. Fluid density in PHWAT depends not on only the concentration of a single species as in SEAWAT, but also the concentrations of other dissolved chemicals that can be subject to reactive processes. Simulation results of PHWAT and PHREEQC-2 were compared in their predictions of effluent concentration from a column experiment. Both models produced identical results, showing that PHWAT has correctly coupled the sub-packages. PHWAT was then applied to the simulation of a tank experiment in which seawater intrusion was accompanied by cation exchange. The density dependence of the intrusion and the snow-plough effect in the breakthrough curves were reflected in the model simulations, which were in good agreement with the measured breakthrough data. Comparison simulations that, in turn, excluded density effects and reactions allowed us to quantify the marked effect of ignoring these processes. Next, we explored numerical issues involved in the practical application of PHWAT using the example of a dense plume flowing into a tank containing fresh water. It was shown that PHWAT could model physically unstable flow and that numerical instabilities were suppressed. Physical instability developed in the model in accordance with the increase of the modified Rayleigh number for density-dependent flow, in agreement with previous research.

Quantum Propagator for Arbitrary Potentials on General Grids

The point-to-point Feynman quantum propagator ⟨q1|exp(−iHt/ℏ)|q2⟩ has an analyticform only for quadratic potentials. We apply the splitoperator approach to obtain a propagator matrix for arbitrarypotentials for non-uniform grids, which are particularly usefulfor real physical potentials with both rapid-varying and smoothregions. We exemplify our method with the wave propagation functionand the extraction of an eigenvalue and aneigenfunction of a Morse system modelling diatomic molecules.

A split operator approach to reactive transport with the forward particle tracking Eulerian Lagrangian localized adjoint method

A forward particle tracking Eulerian Lagrangian localized adjoint method (ELLAM) is applied to the multicomponent reactive transport problem using a split operator approach. Two split operator algorithms are compared, the Strang algorithm and the sequential non-iterative algorithm (SNIA). The reaction equations are integrated using a coupled predictor corrector algorithm with adaptive time stepping. Reaction time steps are adjusted at the inflow boundary to reflect the actual time of transport inside the solution domain. Results show that split operator ELLAM formulations are competitive with direct or fully coupled ELLAM solutions for reactive transport problems. The SNIA algorithm is more accurate than the Strang splitting algorithm when large time steps are used. The reaction algorithm employed dominates computational effort in runs with large time step sizes. To illustrate the use of the method in practical problems, the model is fitted to aerobic aniline degradation data from laboratory scale column experiments. Model inversion is achieved using non-linear regression with a shuffled complex evolution optimization algorithm and parameter uncertainty is assessed using a Bayesian uncertainty analysis procedure.