One-dimensional Scheme Research Articles

The Optical Path Method for the Problem of Oblique Incidence of a Plane Electromagnetic Wave on a Plane-Parallel Scatterer

A number of actual problems of integrated photonics are reduced to an oblique incidence of radiation on a plane-parallel scatterer. For such problems, an approximate method of integrating the Maxwell equations along the beam propagation direction is proposed. As a result, the original two-dimensional problem is reduced to a one-dimensional one, and recently proposed one-dimensional bicompact schemes are used to solve it. This approach provides a significant reduction of computational costs compared to traditional two-dimensional methods such as finite differences and finite elements. To verify the proposed method, calculations of test and applied problems with known exact reflection spectra are carried out.

Lie group symmetry analysis and invariant difference schemes of the two-dimensional shallow water equations in Lagrangian coordinates

The two-dimensional shallow water equations in Lagrangian coordinates are considered. Lie group classification for the class of the elliptic paraboloid bottom topography is performed.The transformations mapping the two-dimensional shallow water equations with a plane or rotation symmetric bottom into the gas dynamics equations of a polytropic gas with polytropic exponent γ=2 are represented. The group foliation of the two-dimensional shallow water equations in Lagrangian coordinates is discussed.New invariant conservative finite-difference schemes for the equations and their one-dimensional reductions are constructed. The schemes are derived either by extending the known one-dimensional schemes or by direct algebraic construction based on some assumptions on the form of the energy conservation law. Among the proposed schemes there are schemes possessing conservation laws of mass and energy.

Topological phase transitions and topological quantum states modulated by the counter-rotating wave terms in a one-dimensional superconducting microwave cavity lattice

We propose a one-dimensional lattice theory scheme based on superconducting microwave cavities, which includes two different types of microwave cavity unit cells. The coupling between unit cells is controlled by flux qubits to simulate and study their topological insulator characteristics. Specifically, a one-dimensional superconducting microwave cavity lattice scheme with a p-wave superconducting pairing term is achieved by mapping the counter-rotating wave terms to the p-wave superconducting pairing term. We found that the p-wave superconducting pairing term can modulate the topological quantum state of the system, allowing for the creation of topological quantum information transmission channels with four edge states. In addition, when the p-wave superconducting pairing term and the nearest-neighbor interaction exist, we find that the energy band undergoes fluctuations, inducing the generation of new energy bands, but the degeneracy of the edge states remains stable, which can achieve multiple topological quantum state transmission paths. However, when its regulatory value exceeds the threshold, the energy gap of the system will close, causing the edge states to annihilate in new energy bands. Furthermore, when considering the existence of defects in the system, we found that when the strength of the defects are small, the edge state produces small fluctuations, but it can be clearly distinguished, indicating its robustness. When the strength of the defect exceeds the threshold, the edge state and energy band cause irregular fluctuations, allowing the edge state to integrate into the energy band. Our research results have important theoretical value and practical significance, and can be applied in quantum optics and quantum information processing in the future.

Mathematical Aspects of <i>SU</i> (2) and <i>SO</i>(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form

Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

Topological phase transitions and topological quantum states modulated by the counter-rotating wave terms in a one-dimensional superconducting microwave cavity lattice

We propose a one-dimensional lattice theory scheme based on superconducting microwave cavities, which includes two different types of microwave cavity unit cells. The coupling between unit cells is controlled by flux qubits to simulate and study their topological insulator characteristics. Specifically, a one-dimensional superconducting microwave cavity lattice scheme with a p-wave superconducting pairing term is achieved by mapping the counter-rotating wave terms to the p-wave superconducting pairing term. We found that the p-wave superconducting pairing term can modulate the topological quantum state of the system, allowing for the creation of topological quantum information transmission channels with four edge states. In addition, when the p-wave superconducting pairing term and the nearest-neighbor interaction exist, we find that the energy band undergoes fluctuations, inducing the generation of new energy bands, but the degeneracy of the edge states remains stable, which can achieve multiple topological quantum state transmission paths. However, when its regulatory value exceeds the threshold, the energy gap of the system will close, causing the edge states to annihilate in new energy bands. Furthermore, when considering the existence of defects in the system, we found that when the strength of the defects are small, the edge state produces small fluctuations, but it can be clearly distinguished, indicating its robustness. When the strength of the defect exceeds the threshold, the edge state and energy band cause irregular fluctuations, allowing the edge state to integrate into the energy band. Our research results have important theoretical value and practical significance, and can be applied in quantum optics and quantum information processing in the future.

A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition

A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition

Stability and convergence of the locally one-dimensional scheme A. A. Samarskii, approximating the multidimensional integro-differential equation of convection-diffusion with inhomogeneous boundary conditions of the first kind

Изучена первая начально-краевая задача для многомерного (по пространственным переменным) интегро-дифференциального уравнения конвекции-диффузии. Для приближенного решения поставленной задачи предложена локально-одномерная схема А. А. Самарского с порядком аппроксимации $O(h^2+\tau)$. Исследование единственности и устойчивости решения проводится с помощью метода энергетических неравенств. Получены априорные оценки решения локально-одномерной разностной схемы, откуда следуют единственность решения, непрерывная и равномерная зависимость решения от входных данных, а также сходимость решения схемы к решению исходной дифференциальной задачи со скоростью, равной порядку аппроксимации разностной схемы. Для двумерной задачи построен алгоритм численного решения, проведены численные расчеты тестовых примеров, иллюстрирующие полученные в работе теоретические результаты.

Trans-level multi-scale simulation of porous catalytic systems: Bridging reaction kinetics and reactor performance

Multi-scale porous structures inside and/or between the catalyst pellets or particles are found in many chemical processes, where strong coupling of reaction and transport results in complex apparent reaction kinetics influential to the reactor performance. Traditional continuum-based porous media models and simulation methods can hardly describe such structures and their scale effects faithfully. A trans-level multi-scale discrete computational framework is hence proposed to address this complexity and implemented for an olefin catalytic cracking (OCC) process. The apparent reaction kinetics at the REV (representative elementary volume) scale is obtained by hard-sphere pseudo-particle modeling (HS-PPM), and coupled with computational fluid dynamics / discrete element method (CFD-DEM) for the reactor-level hydrodynamics via a one-dimensional (1D) finite difference scheme for particle-level diffusion. The mesoscales of the REVs and the flow networks between the particles are thus covered by the framework, which are previously described by simple average quantities in the continuum methods. The reactant conversion rate and target product selectivity obtained agree well with experimental results, while a continuum approach may give significantly different and unreasonable results. The multi-scale method is, therefore, demonstrated to be necessary and effective for bridging the intrinsic reaction kinetics with the performance of porous catalytic reactors.

Finite-difference method for solving a multidimensional pseudoparabolic equation with boundary conditions of the third kind

We study an initial-boundary value problem for a multidimensional pseudoparabolic equation with variable coefficients and boundary conditions of the third kind. The multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter. It is shown that as the small parameter tends to zero, the solution of the resulting modified problem converges to the solution of the original problem. For an approximate solution of the obtained problem, a locally one-dimensional difference scheme by A. A. Samarsky is constructed. An a priori estimate is obtained by the method of energy inequalities, from which the uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow. For a two-dimensional problem, an algorithm for the numerical solution of the initial-boundary value problem for a pseudoparabolic equation with conditions of the third kind is developed.

A novel method for work capacity calculation of centrifugal compressor impellers in energy storage systems

As the basis for centrifugal compressor design, the traditional one-dimensional scheme assumes that there is no flow prewhirl at the impeller inlet (i.e., C1u=0) when inlet guide vanes (IGVs) are absent. However, various experiments have proved that a natural prewhirl does exist at the impeller inlet even there is no IGV. In this study, a novel method is proposed to calculate the Euler work of impellers considering the natural prewhirl, and firstly applied to centrifugal compressors without IGVs. Stodola’s approach dealing with the slip velocity at the impeller outlet is adopted to the flow at the impeller inlet. A new formula for the Euler work calculation is deduced to evaluate the effect of natural prewhirl on centrifugal impellers. Then, this method is applied in two different model stages of industrial centrifugal compressors to verify against the experimental and numerical data. The results indicate that, compared to the traditional scheme assuming C1u=0, the novel method with modified formula considering the influence of C1u≠ 0 is more feasible for high-speed centrifugal compressors. With the aid of the novel calculation method proposed in this paper, the one-dimensional design accuracies of centrifugal compressor stages are improved by 7% compared with the traditional method.

Comparison of Three Convolution Neural Network Schemes to Retrieve Temperature and Humidity Profiles from the FY4A GIIRS Observations

FY4A/GIIRS (Geostationary Interferometric Infrared Sounder) is the first infrared hyperspectral atmospheric vertical sounder onboard a geostationary satellite. It can achieve observations of atmospheric temperature and humidity profiles with high vertical and temporal resolutions. Presently, convolutional neural network algorithms are relatively less used in the field of atmospheric profile retrieval, and different convolutional neural network approaches have different characteristics. The one-dimensional convolutional neural network scheme 1D-Net and two three-dimensional retrieval schemes U-Net 1 and U-Net 2 are used to achieve atmospheric temperature and humidity profiles under all skies based on GIIRS-observed brightness temperatures in this paper. After validation with test training data, the retrievals of different schemes derived from actual GIIRS observations and level 2 operational products were verified with ERA5 reanalysis data and radiosonde measurements in summer and winter respectively. The retrieved three-dimensional temperature and humidity fields from U-Net 1 and U-Net 2 are closer to the ERA5 reanalysis field in both distribution and value than the retrievals from the 1D-Net scheme and level 2 operational products. In particular, the inversion field of the U-Net 2 scheme is more continuous in space. Compared with radiosonde observations, the accuracy of the level 2 temperature product is the highest when the field of view is completely clear both in winter and summer month. The root mean square error (RMSE) of temperature retrieval of the two U-Net schemes is the second highest, and the RMSE and bias of the 1D-Net scheme are both large. Two U-Net schemes overestimate the temperature and humidity slightly in winter and underestimate it in summer in both clear and all sky cases. Under all sky conditions, the temperature retrieval RMSE and bias of the two U-Net schemes above 800 hPa are lower than those of the level 2 products, especially the U-Net 2 scheme with an RMSE of approximately 2.5 K. The U-Net 2 scheme bias is the smallest, with a value of approximately 0.5 K in winter. Since the level 2 product only provides the atmospheric temperature above the cloud top, it indicates that its temperature product accuracy is very low when the field of view is influenced by clouds. The humidity retrieval RMSEs of the two U-Net schemes is within 2 g/kg, better than that of the 1D-Net scheme. The retrieval accuracy of the U-Net 2 scheme is approximately 0.3 g/kg better than that of the U-Net 1 scheme below 600 hPa in winter. Level 2 does not provide humidity products. The summer humidity retrieval is worse than in winter. In general, among the three deep machine learning algorithms, 1D-Net has a large retrieval error, and the temperature and humidity from U-Net 2 have the highest accuracy. The retrieval speeds of the two U-Net schemes are nearly the same, and both are faster than that of scheme 1D-Net.

Grid Method for Solving an Initial-Boundary-value Problem for a General Multidimensional Equation of Parabolic Type

The problems concerning the physical process studies that lead to mathematical models based on a parabolic type equation are of great practical importance.When solving the parabolic equation, a transition from the one-dimensional equation to the multidimensional one gives a rise to some difficulties. The trouble is that amount of computations significantly increases in the transition from the one-dimensional problems to the multidimensional ones In this regard a task to build the economical difference schemes for a numerical solution of multidimensional problems is a challenge.A difference scheme that approximates the problem over time is economical, if it is unconditionally stable and the desirable number of arithmetic operations in the transition from layer to layer is proportional to the number of nodes in each time layer.The paper dwells on the construction of a locally one-dimensional (economical) difference scheme for the approximate solution of a parabolic type equation of a general form in a multidimensional domain, the main idea of which is to reduce a complex problem to a sequential solution of boundary value problems of a simpler structure. At the same time, there is a construction of the economical, unconditionally stable difference scheme for each of the intermediate problems. To have a numerical solution of the problem, there is a construction of the locally one-dimensional difference scheme of A.A. Samarsky. Using a method of energy inequalities allows us to obtain a priori estimates in the differential and difference interpretations, whence it follow that there are the uniqueness, stability, as well as the convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem. For a bi-dimensional problem, a numerical solution algorithm is constructed.

Simplified Modeling and Analysis of Surface Temperature Distribution in Electrically Heated Catalyst for Diesel Urea-SCR Systems

Impending emission regulations of diesel engines for construction machineries would regulate nitrogen oxide emissions strictly in cold operating conditions. The urea-based selective catalytic reduction (urea-SCR) system coupled with the electrically heated catalyst (EHC) has been considered as a potential measure to meet the strict emission regulations by promoting evaporation and thermal decomposition of urea–water solution in cold operating conditions. Analyzing the thermal conditions in the EHC is crucial for the optimized operation and control of EHC-based urea-SCR systems under various engine operating conditions. In the current study, we introduce a simple one-dimensional analysis scheme to characterize the surface temperature distribution in the EHC based on energy conservation and the theories of forced internal convection. Since the EHC has a complicated internal structure with fine flow cells inside it, a flow cell in the EHC is extracted for the one-dimensional heat transfer analysis. EHC operation parameters such as exhaust gas flow rate and supplied electric power to the EHC are scaled to be applied for the flow cell analysis. The adequacy of the analysis scheme is then validated by surface temperature measurement results at the EHC outlet. The validation results showed over 95% prediction accuracy of the 1D analysis scheme in the operating conditions of a heavy-duty diesel engine. Based on proven reliability, the effects of geometric and operation parameters on the surface temperature distribution in the EHC were analyzed and discussed using the analysis results.

Разностные методы решения интегро-дифференциального уравнения конвекции-диффузии с граничными условиями третьего рода

An initial-boundary value problem for a multidimensional integro-differential convection-diffusion equation with variable coefficients and boundary conditions of the third kind is investigated. For the numerical solution of the multidimensional problem, a locally one-dimensional difference scheme is constructed. It is shown that the approximation error of the scheme is 𝑂(|ℎ|2+𝜏), where |ℎ|2=ℎ12+ℎ22+...+ℎ𝑝2. Using the method of energy inequalities, we obtain an a priori estimate in the 𝐿2 - norm, which implies uniqueness, stability with respect to the right-hand side and initial data, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem at a rate equal to the approximation error. Numerical calculations are carried out on a test example, illustrating the theoretical calculations obtained in this work.

Конечно-разностные методы решения нелокальной краевой задачи для многомерного параболического уравнения с граничными условиями интегрального вида

The article considers a non-local boundary value problem for a multidimensional parabolic equation with integral boundary conditions. To solve the problem, we obtain an a priori estimate in differential form, which implies the uniqueness and stability of the solution with respect to the right-hand side and initial data on the layer in the $L_2$-norm. For the numerical solution of a nonlocal boundary value problem, a locally one-dimensional (economical) difference scheme by A.A. Samarskii with the order of approximation $O(h^2+\tau)$, the main idea of which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Using the method of energy inequalities, a priori estimates are obtained, which imply uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the $L_2$-norm at a rate equal to the order of approximation of the difference scheme. An algorithm for the numerical solution is constructed.

Improved Representation of Horizontal Variability and Turbulence in Mesoscale Simulations of an Extended Cold-Air Pool Event

Abstract Cold-air pools (CAPs), or stable atmospheric boundary layers that form within topographic basins, are associated with poor air quality, hazardous weather, and low wind energy output. Accurate prediction of CAP dynamics presents a challenge for mesoscale forecast models in part because CAPs occur in regions of complex terrain, where traditional turbulence parameterizations may not be appropriate. This study examines the effects of the planetary boundary layer (PBL) scheme and horizontal diffusion treatment on CAP prediction in the Weather Research and Forecasting (WRF) Model. Model runs with a one-dimensional (1D) PBL scheme and Smagorinsky-like horizontal diffusion are compared with runs that use a new three-dimensional (3D) PBL scheme to calculate turbulent fluxes. Simulations are completed in a nested configuration with 3-km/750-m horizontal grid spacing over a 10-day case study in the Columbia River basin, and results are compared with observations from the Second Wind Forecast Improvement Project. Using event-averaged error metrics, potential temperature and wind speed errors are shown to decrease both with increased horizontal grid resolution and with improved treatment of horizontal diffusion over steep terrain. The 3D PBL scheme further reduces errors relative to a standard 1D PBL approach. Error reduction is accentuated during CAP erosion, when turbulent mixing plays a more dominant role in the dynamics. Last, the 3D PBL scheme is shown to reduce near-surface overestimates of turbulence kinetic energy during the CAP event. The sensitivity of turbulence predictions to the master length-scale formulation in the 3D PBL parameterization is also explored. Significance Statement In this article, we demonstrate how a new framework for modeling atmospheric turbulence improves cold pool predictions, using a case study from January 2017 in the Columbia River basin (U.S. Pacific Northwest). Cold pools are regions of cold, stagnant air that form within valleys or basins, and improved forecasts could help to mitigate the risks they pose to air quality, transportation, and wind energy production. For the chosen case study, our tests show a reduction in temperature and wind speed errors by up to a factor of 2–3 relative to standard model options. These results strongly motivate continued development of the framework as well as its application to other complex weather events.

A High Accuracy Local One-Dimensional Explicit Compact Scheme for the 2D Acoustic Wave Equation

In this paper, we develop a highly accurate and efficient finite difference scheme for solving the two-dimensional (2D) wave equation. Based on the local one-dimensional (LOD) method and Padé difference approximation, a fourth-order accuracy explicit compact difference scheme is proposed. Then, the Fourier analysis method is used to analyze the stability of the scheme, which shows that the new scheme is conditionally stable and the Courant-Friedrichs-Lewy (CFL) condition is superior to most existing methods of equivalent order of accuracy in the literature. Finally, numerical experiments demonstrate the high accuracy, stability, and efficiency of the proposed method.

Provably stable flux reconstruction high-order methods on curvilinear elements

Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, the curvilinear metric split-form for a linear physical flux is examined as it enables the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and non-conservative forms are inherently different–even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders of convergence. The scheme loses the orders of accuracy at the equivalent correction parameter value c as that of the one-dimensional ESFR scheme.

High order entropy stable and positivity-preserving discontinuous Galerkin method for the nonlocal electron heat transport model

The nonlocal electron heat transport model in laser heated plasmas plays a crucial role in inertial confinement fusion (ICF), and it is important to solve it numerically in an accurate and robust way. In this paper, we first develop an one-dimensional high-order entropy stable discontinuous Galerkin method for the nonlocal electron heat transport model. We further design our DG scheme to have the positivity-preserving property by using the scaling positivity-preserving limiter, which is shown, by a computer-aided proof, to have no extra time step constraint than that required by L2 stability. Next, we extend our one-dimensional scheme to two-dimensions on rectangular meshes and tensor product polynomial spaces. Numerical examples are given to verify the high-order accuracy and positivity-preserving properties of our scheme. By comparing the local and nonlocal electron heat transport models, we also observe more physical phenomena such as the flux reduction and the preheat effect from the nonlocal model.

Simulation and detection of the topological properties of phonon-photon in frequency-tunable optomechanical lattice

We propose a one-dimensional lattice theory scheme based on a coupled optomechanical system consisting of multiple cavity field modes and mechanical modes, where their frequencies can be tuned. In this system, by manipulating parameters to obtain collective dynamical evolution of the system, we study topological properties and topological quantum channels in the system. Firstly, the topological insulator properties and topological quantum channels of the system are studied by modulating the periodic coupling parameters of the system and analyzing the characteristics of the energy spectrum and edge states of the system. It is found that edge state distributions can exhibit flipping processes, which can be applied to quantum information processing. Secondly, based on the scattering theory of topological insulators and the relationship between input and output, the variation characteristics of the steady-state average photon number of the cavity field and the winding number of the reflection coefficient phase are analyzed. It is found that the dissipation of the cavity field has a certain influence on the locality of the distribution of the average photon number in the lattice, and it also indirectly explains the locality of the edge states of the system, and the topological invariants are detected by the winding number. In addition, considering the effect of disordered defects on topological properties, we further analyze their effects on the energy spectrum of the system, the winding number of the reflection coefficient phase and the average photon number of the cavity field. It is found that two defects in the system cause different physical effects, and when their values are small, the edge states of the system are robust to it, which also shows that the system has the characteristics of topological protection. However, when disorder and perturbation are larger than the energy gap, the topological properties of the system will be annihilated, so that the edge states will be indistinguishable, and the topological invariants will change at the same time. The research results of this system can be generalized to other types of models and can be applied to quantum communication and quantum information processing, which will have certain constructive suggestions for the development of future quantum technology.