Extrapolation Scheme Research Articles

Micro-scale modelling of the urban wind speed for air pollution applications

Modelling wind speeds in urban areas have many applications e.g. in relation to assessment of wind energy, modelling air pollution, and building design and engineering. Models for extrapolating the urban wind speed exist, but little attention has been paid to the influence of the upwind terrain and the foundations for the extrapolation schemes. To analyse the influence of the upwind terrain and the foundations for the extrapolation of the urban wind speed, measurements from six urban and non-urban stations were explored, and a model for the urban wind speed with and without upwind influence was developed and validated. The agreement between the wind directions at the stations is found to be good, and the influence of atmospheric stability, horizontal temperature gradients, land-sea breeze, temperature, global radiation and Monin-Obukhov Length is found to be small, although future work should explore if this is valid for other urban areas. Moreover, the model is found to perform reasonably well, but the upwind influence is overestimated. Areas of model improvement are thus identified. The upwind terrain thus influences the modelling of the urban wind speed to a large extent, and the fundamental assumptions for the extrapolation scheme are fulfilled for this specific case.

Finite-Element Formulations for Systems With High-Temperature Superconductors

In this article, we consider finite-element models for high-temperature superconductors and compare two dual formulations, either magnetic-field conforming or magnetic-flux-density conforming. The electrical resistivity of superconductors is described by a power law and is strongly nonlinear. We compare the accuracy and the efficiency of the dual formulations by starting from simple considerations on the concavity/convexity of the constitutive law involved in each case. We then study the numerical behavior of each formulation in one-, two-, and three-dimensional problems and compare their results against benchmarks. We draw general recommendations for the choice of a formulation, an iteration scheme for treating the corresponding linearized constitutive law, and a time-stepping extrapolation scheme. This approach is extended to soft ferromagnetic materials with a saturation law. Since the outcome of our analysis shows that recommended formulations for treating ferromagnets are just the opposite of those for treating superconductors, we suggest a coupled formulation for systems where both types of materials are present. The coupled formulation is shown to be accurate and more efficient than single formulations applied indistinctly to all materials.

Complete Basis Set Extrapolation of Electronic Correlation Energies Using the Riemann Zeta Function.

In this article, we demonstrate the effectiveness of the method of complete basis set (CBS) extrapolation of correlation energies based on the application of the Riemann zeta function. Instead of fitting the results obtained with a systematic sequence of one-electron bases with a certain functional form, an analytic resummation of the missing contributions coming from higher angular momenta, l, is performed. The assumption that these contributions vanish asymptotically as an inverse power of l leads to an expression for the CBS limit given in terms of the zeta function. This result is turned into an extrapolation method that is very easy to use and requires no "empirical" parameters to be optimized. The performance of the method is assessed by comparing the results with very accurate reference data obtained with explicitly correlated theories and with results obtained with standard extrapolation schemes. On average, the errors of the zeta-function extrapolation are several times smaller compared with the conventional schemes employing the same sequence of bases. A recipe for the estimation of the residual extrapolation error is also proposed.

Direct characterisation of the annual performance of solar thermal and heat pump systems using a six-day whole system test

Systems that provide space heating and domestic hot water are characterized by dynamic operating conditions. For a reliable evaluation of the general system performance, dynamic system test procedures are a more reliable, more reproducible and faster way than using steady-state component test results in combination with annual system simulation or others extrapolation schemes. However, the utilization of dynamic whole system testing was so far hindered by the cost and time associated with the laboratory measurements, and by a lack of scientifically justified procedures for the determination of suitable test profiles.This paper presents an analysis of the suitability of using a six-day sequence for the direct characterization of a solar and heat pump system. The new six-day sequence was developed to perform a benchmark test and its representativeness was verified with one-hundred different solar and heat pump system configurations.It was found that some independent variables lead to a predictable deviation in the performance evaluation based on the six-day sequence compared to whole-year figures, that can be greatly reduced by simple correction factors. For the different configurations, the correction factors reduced the standard deviation of the energetic performance results to 2.5%.

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H1‐norm and for pressure in L2‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.

1D study of the detonation phenomenon and its influence on the interior ballistics of the combustion light gas gun

Abstract One-dimensional simulations with a detailed hydrogen-oxygen reaction mechanism have been performed to investigate detonation phenomenon in a combustion light gas gun (CLGG). Convection fluxes of the Navier-Stokes equations are solved using the WAF (weighted average flux) scheme HLLC Riemann solver. A high resolution fifth-order WENO scheme for the variable extrapolation at the volume interface and a fourth-order Runge-Kutta scheme for the time advancement are used. Validation tests of the stoichiometric hydrogen-oxygen deflagration to detonation transition process shows good agreement between the computed results and the analytical and documented solutions, demonstrating the reliability on the detonation simulation of the current scheme. Simulation results of the interior ballistic process of the CLGG show that the flame propagation experiences three distinct stages. The blast detonation wave causes a high-pressure shock and hazardous oscillations in the chamber and makes the projectile accelerate with fluctuations, but has only a small improvement to the muzzle velocity.

Computational investigation of the valid valence state contribution in calculating the electronic stopping power of a proton in bulk Al within the linear response approach

The electronic stopping power is a fundamental quantity to many technological fields that use ion irradiation. Here we investigate the validity of using a fully ab initio computational scheme based on linear response time-dependent density functional theory to predict the random electronic stopping power (RESP) of a proton in bulk aluminum. We verify the power of using the extrapolation scheme to overcome the expected convergence issue of the RESP calculations. We show that the calculated RESP of valence electrons compares well with experimental data for low proton velocity only when at full convergence and including the exchange-correlation effect. We demonstrate that the inclusion of valence states only is sufficient for calculating the electronic stopping power up to the stopping maximum.

Convergence of the Crank-Nicolson extrapolation scheme for the Korteweg-de Vries equation

In this paper, the Crank-Nicolson extrapolation scheme is proposed for the Korteweg-de Vries equation, where a finite element method is applied for the spatial approximation, the time discretization is based on the Crank-Nicolson scheme for the linear term and the semi-implicit extrapolation scheme for the nonlinear term. Moreover, in numerical analysis, we split the error function into two parts, one from the spatial discretization and other one from the temporal discretization, by introducing a corresponding time-discrete system. Then, based on some regularity assumptions, we present optimal error estimate and prove that the scheme is almost unconditionally convergent, i.e., the scheme is convergent when the time step is less than or equal to a constant. Finally, numerical tests confirm the theoretical results of the presented method.

Moving overlapping grid methodology of spectral accuracy for incompressible flow solutions around rigid bodies in motion

The simulation of fluid flow around moving rigid bodies has proven to be a difficult task for traditional computational fluid dynamics solvers. Decomposing the global domain into overlapping subdomains and allowing each subdomain to move independently, permits solutions to many flow problems with complex moving geometries to be determined in a straightforward manner. The present development of the moving overlapping grid method is built within a spectral element method incompressible flow solver, and uses an Arbitrary Lagrangian-Eulerian formulation of the governing equations to prescribe subdomain motions. The method maintains global spectral spatial accuracy across the subdomains with the polynomial refinement. The global high-order temporal accuracy of the method is also maintained through subdomain coupling enforced by an explicit interface temporal extrapolation scheme. The method produces aerodynamic forces and vortex shedding around two- and three-dimensional moving rigid bodies that is in line with published experimental and computational data. Additionally, the method achieves near linear computational scaling to thousands of cores.

Numerical Modeling of Slip Flow and Heat Transfer Over Microcylinders With Lattice Boltzmann Method

In this paper, a lattice Boltzmann (LB) model is established to simulate the gaseous fluid flow and heat transfer in the slip regime under the curved boundary condition. A novel curved boundary treatment is proposed for the LB modeling, which is a combination of the nonequilibrium extrapolation scheme for the curved boundary and the counter-extrapolation method for the macroscopic variables on the curved gas–solid interface. The established numerical model can accurately predict the velocity slip and temperature jump of the microscale gas flow on the curved surface, which agrees well with the analytical solution for the microcylindrical Couette flow and heat transfer. Then, the slip flow and the heat transfer over the single microcylinder are numerically studied in this work. It shows that the velocity slip and the temperature jump are obviously influenced by the Knudsen number and the Reynolds number, and the local Nusselt number depends on which gas rarefaction effect (velocity slip or temperature jump) is dominant. An increase in the Prandtl number leads to a decrease in the temperature jump, which enhances the heat transfer on the microcylinder surface. The numerical simulation of the flow and heat transfer over two microcylinders in tandem configuration are carried out to investigate the wake interference effect. The results show that the slip flow and heat transfer characteristics of the downstream microcylinder are influenced by the wake region behind the upstream cylinder as the spacing is small.

Boundary scheme for lattice Boltzmann modeling of micro-scale gas flow in organic-rich pores considering surface diffusion**Project supported by the Strategic Program of Chinese Academy of Sciences (Grant No. XDB10030400), the Hundred Talent Program of Chinese Academy of Sciences (Grant No. Y323081C01), and the National Natural Science Foundation of China (Grant No. 51439008).

We propose a boundary scheme for addressing multi-mechanism flow in a porous medium in slip and early transition flow regimes, which is frequently encountered in shale gas reservoirs. Micro-gaseous flow in organic-rich shale involves a complex flow mechanism. A self-developed boundary scheme that combines the non-equilibrium extrapolation scheme and the combined diffusive reflection and bounce-back scheme (half-way DBB) to embed the Langmuir slip boundary into the single-relaxation-time lattice Boltzmann method (SRT-LBM) enables us to describe this process, namely, the coupling effect of micro-gaseous flow and surface diffusion in organic-rich nanoscale pores. The present LBM model comes with the careful consideration of the local Knudsen number, local pressure gradient, viscosity correction model, and regularization procedure to account for the rarefied gas flows in irregular pores. Its validity and accuracy are verified by several benchmarking cases, and the calculated results by this boundary scheme accord well with our analytical solutions. This boundary scheme shows a higher accuracy than the existing studies. Additionally, a subiteration strategy is presented to tackle the coupled micro-gaseous flow and surface diffusion, which necessitates the iteration process matching of these two mechanisms. The multi-mechanism flow in the self-developed irregular pores is also numerically investigated and analyzed over a wide range of parameters. The results indicate that the present model can effectively capture the coupling effect of micro-gaseous flow and surface diffusion in a tree-like porous medium.

Fully coupled (J > 0) time-dependent wave-packet calculations using hyperspherical coordinates for the H + O2 reaction on the CHIPR potential energy surface.

Quantum dynamics of the H + O2→ O + OH reaction has been extensively studied on the adiabatic ground state of CHIPR [A. J. C. Varandas, J. Chem. Phys., 2013, 138, 134117] potential energy surfaces by employing a coupled 3D time-dependent wavepacket approach in hyperspherical coordinates. Calculations have been performed for all non-zero J values for various initial rotational states of the diatom [O2(v = 0, j = 1-5)]. State-to-state and total integral cross sections are calculated using fully converged reaction probabilities, where initial state selected and Boltzmann averaged thermal rate constants are also subsequently calculated. Moreover, a comparison of various reaction attributes obtained by using the fully close coupled approach with the ones obtained from the J-shifting approximation and extrapolation scheme is presented along with other theoretical results and experimental observations.

Benchmarking DFT approaches for the calculation of polarizability inputs for refractive index predictions in organic polymers.

In a previous study, we introduced a new computational protocol to accurately predict the index of refraction (RI) of organic polymers using a combination of first-principles and data modeling. This protocol is based on the Lorentz-Lorenz equation and involves the calculation of static polarizabilities and number densities of oligomer sequences, which are extrapolated to the polymer limit. We chose to compute the polarizabilities within the density functional theory (DFT) framework using the PBE0/def2-TZVP-D3 model chemistry. While this ad hoc choice proved remarkably successful, it is also relatively expensive from a computational perspective. It represents the bottleneck step in the overall RI modeling protocol, thus limiting its utility for virtual high-throughput screening studies, in which efficiency is essential. For polymers that exhibit late-onset extensivity, the employed linear extrapolation scheme can require demanding calculations on long-oligomer sequences, thus becoming another bottleneck. In the work presented here, we benchmark DFT model chemistries to identify approaches that optimize the balance between accuracy and efficiency for this application domain. We compare results for conjugated and non-conjugated polymers, augment our original extrapolation approach with a non-linear option, analyze how the polarizability errors propagate into the RI predictions, and offer guidance for method selection.

Accelerating Nonnegative Matrix Factorization Algorithms Using Extrapolation.

We propose a general framework to accelerate significantly the algorithms for nonnegative matrix factorization (NMF). This framework is inspired from the extrapolation scheme used to accelerate gradient methods in convex optimization and from the method of parallel tangents. However, the use of extrapolation in the context of the exact coordinate descent algorithms tackling the nonconvex NMF problems is novel. We illustrate the performance of this approach on two state-of-the-art NMF algorithms: accelerated hierarchical alternating least squares and alternating nonnegative least squares, using synthetic, image, and document data sets.

Numerical Investigation of Coalescing Filtration Process

AbstractCoalescing filtration is a mechanical process which is employed to remove dispersed aerosol particles from a gas stream. This kind of filtration is a depth‐filtration process and is widely used in process industries to remove particulate matter from exhaust gases or in compressed‐air applications to filter oil particles introduced during the compression process. Fibrous filters are often used due to low costs, high capture efficiency and low pressure drop. The droplets are first captured on the fibers, then coalesce, and eventually drain out. The performance of a filter medium is judged based on its capture efficiency and pressure‐drop characteristics. Estimating these parameters without setting up experimental investigations of each filter medium would be beneficial for choosing optimal filters. In the present study, numerical simulations are used to predict the behavior of a fully‐wetting fibrous filter using an Euler‐Euler formulation in ANSYS CFX. Different theoretical models to predict single fibre capture efficiencies are compared and a novel extrapolation scheme to predict the net capture of oil particles is presented. Pressure‐drop studies are carried out to find the pressure drop of a viscous fluid flowing through an array of fibers. A steady‐state multiphase simulation with the influence of gravity is set up, and the physical relevance of the results is examined.

Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models

We analyze the problem of extracting the correlation length from infinite matrix product states (MPS) and corner transfer matrix (CTM) simulations. When the correlation length is calculated directly from the transfer matrix, it is typically significantly underestimated for finite bond dimensions used in numerical simulation. This is true even when one considers ground states at a distance from the critical point. We introduce extrapolation procedure to overcome this problem. To that end we quantify how much the dominant part of the MPS/CTM transfer matrix spectrum deviates from being continuous. The latter is necessary to capture the exact asymptotics of the correlation function where the exponential decay is typically modified by an additional algebraic term. By extrapolating such a refinement parameter to zero, we show that we are able to recover the exact value of the correlation length with high accuracy. In a generic setting, our method reduces the error by a factor of $\sim 100$ as compared to the results without extrapolation and a factor of $\sim 10$ as compared to simple extrapolation schemes using bond dimension. We test our approach in a number of solvable models both in 1d and 2d. Subsequently, we apply it to 1d XXZ spin-$\frac32$ and the Bose-Hubbard models in a massive regime in the vicinity of BKT critical point. We then fit the scaling form of the correlation length and extract the position of the critical point and obtain results comparable or better than those of other state-of-the-art numerical methods. Finally, we show how the algebraic part of the correlation function asymptotics can be directly recovered from the scaling of the dominant form factor within our approach. Our method provides the means for detailed studies of phase diagrams of quantum models in 1d and, through the finite correlation length scaling of projected entangled pair states, also in 2d.

Highly accurate simplified lattice Boltzmann method

In this paper, we present a highly accurate simplified lattice Boltzmann method (HSLBM) which can achieve the third-order of accuracy in space. By introducing virtual streaming nodes and decoupling the streaming distance from the mesh spacing, HSLBM effectively combines the local second-order simplified and highly stable lattice Boltzmann method (SHSLBM) and the overall high order scheme. The correlation between the streaming distance and the mesh spacing is accomplished by a high-order Lagrange interpolation algorithm. Through a series of tests, it is found that using 5 interpolation points and setting streaming distance as 1/5 of the mesh spacing can give optimal results. In general, HSLBM improves the accuracy of SHSLBM from the second order to the third order, while maintaining its merits like low memory cost, convenient implementation of physical boundary conditions, and good numerical stability. To match the accuracy of the flow solver, a linear extrapolation scheme with the third-order of accuracy is also proposed to determine the boundary values of the non-equilibrium distribution functions. Various benchmark tests are performed to demonstrate the robustness of HSLBM in simulating two- and three-dimensional incompressible viscous flows as well as its flexibility in problems with curved boundaries and on body-fitted meshes.

Optimal convergence analysis of Crank–Nicolson extrapolation scheme for the three-dimensional incompressible magnetohydrodynamics

This paper considers a Crank–Nicolson extrapolation scheme based on mixed finite element method to solve the three-dimensional incompressible magnetohydrodynamics (MHD) equations. We prove that the fully discrete scheme is almost unconditionally stable and convergent, i.e., stable and convergent when the time step is less than or equal to a constant. By a new negative norm technique, the optimal error estimates in L2-norm are derived. Meanwhile, the numerical investigations provide a sufficient support for the theoretical analysis.

An Efficient Lossless Compression Algorithm for Trajectories of Atom Positions and Volumetric Data.

We present our newly developed and highly efficient lossless compression algorithm for trajectories of atom positions and volumetric data. The algorithm is designed as a two-step approach. In the first step, efficient polynomial extrapolation schemes reduce the information entropy of the data by exploiting both spatial and temporal continuity. The second step processes the data by a series of transformations (Burrows-Wheeler, move-to-front, run length encoding) and finally compresses the stream with multitable canonical Huffman coding. Our approach reaches a compression ratio of around 15:1 for typical position trajectories in the XYZ format. For volumetric data trajectories in Gaussian Cube format (such as electron density), even a compression ratio of around 35:1 is yielded, which is by far the smallest size of all formats compared here. At the same time, compression and decompression are still reasonably fast for everyday use. The precision of the data can be selected by the user. For storage of the compressed data, we introduce the BQB file format, which is very robust, flexible, and efficient. In contrast to most archiving formats, it allows fast random access to individual trajectory frames. Our method is implemented in C++ and provided as free software under the GNU LGPL license. It has been included in the TRAVIS program package but is also available as stand-alone tool and as a library ("libbqb") for use in other projects.

An efficient pricing algorithm for American options with double stochastic volatilities and double jumps

The purpose of the paper is to provide an efficient pricing algorithm for American options with stochastic volatilities and jumps. This paper extends the double Heston model with double exponential jumps and derives the characteristic function of the model by Feynman–Kac theorem. With the obtained characteristic function, this paper also extends the Fourier-cosine expansion method for pricing Bermudan options to the model. Based on the COS method, this paper approximates American options by using Richardson extrapolation schemes on a series of Bermudan options and provides a pricing algorithm for American put options. Numerical results show that the proposed pricing algorithm is efficient, especially for short-term American put options.