Artykuły w czasopismach: „Splitting scheme”

1

Lai, Junjiang, i Zhencheng Fan. "Stability for discrete time waveform relaxation methods based on Euler schemes". AIMS Mathematics 8, nr 10 (2023): 23713–33. http://dx.doi.org/10.3934/math.20231206.

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<abstract>Stability properties of discrete time waveform relaxation (DWR) methods based on Euler schemes are analyzed by applying them to two dissipative systems. Some sufficient conditions for stability of the considered methods are obtained; at the same time two examples of instability are given. To investigate the influence of the splitting functions and underlying numerical methods on stability of DWR methods, DWR methods based on different splittings and different numerical schemes are considered. The obtained results show that the stabilities of waveform relaxation methods based on an implicit Euler scheme are better than those based on explicit Euler scheme, and that the stabilities of waveform relaxation methods based on the classical splittings such as Gauss-Jacobi and Gauss-Seidel splittings are worse than those based on the eigenvalue splitting presented in this paper. Finally, numerical examples that confirm the theoretical results are presented.</abstract>

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Ahmed, Nauman, Tahira S.S., M. Rafiq, M. A. Rehman, Mubasher Ali i M. O. Ahmad. "Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model". Open Mathematics 17, nr 1 (29.04.2019): 313–30. http://dx.doi.org/10.1515/math-2019-0027.

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Abstract In this work, we will introduce two novel positivity preserving operator splitting nonstandard finite difference (NSFD) schemes for the numerical solution of SEIR reaction diffusion epidemic model. In epidemic model of infection diseases, positivity is an important property of the continuous system because negative value of a subpopulation is meaningless. The proposed operator splitting NSFD schemes are dynamically consistent with the solution of the continuous model. First scheme is conditionally stable while second operator splitting scheme is unconditionally stable. The stability of the diffusive SEIR model is also verified numerically with the help of Routh-Hurwitz stability condition. Bifurcation value of transmission coefficient is also carried out with and without diffusion. The proposed operator splitting NSFD schemes are compared with the well-known operator splitting finite difference (FD) schemes.

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3

Lai, J. S., G. F. Lin i W. D. Guo. "Simulation of Hydraulic Shock Waves by Hybrid Flux-Splitting Schemes in Finite Volume Method". Journal of Mechanics 21, nr 2 (czerwiec 2005): 85–101. http://dx.doi.org/10.1017/s1727719100004561.

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AbstractIn the framework of the finite volume method, a robust and easily implemented hybrid flux-splitting finite-volume (HFF) scheme is proposed for simulating hydraulic shock waves in shallow water flows. The hybrid flux-splitting algorithm without Jacobian matrix operation is established by applying the advection upstream splitting method to estimate the cell-interface fluxes. The scheme is extended to be second-order accurate in space and time using the predictor-corrector approach with monotonic upstream scheme for conservation laws. The proposed HFF scheme and its second-order extension are verified through simulations of the 1D idealized dam-break problem, the 2D oblique hydraulic shock-wave problem, and the 2D dam-break experiments with channel contraction as well as wet/dry beds. Comparisons of the HFF and several well-known first-order upwind schemes are made to evaluate numerical performances. It is demonstrated that the HFF scheme captures the discontinuities accurately and produces no entropy-violating solutions. The HFF scheme and its second-order extension are proven to achieve the numerical benefits combining the efficiency of flux-vector splitting scheme and the accuracy of flux-difference splitting scheme for the simulation of hydraulic shock waves.

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4

Geiser, Jürgen. "Computing Exponential for Iterative Splitting Methods: Algorithms and Applications". Journal of Applied Mathematics 2011 (2011): 1–27. http://dx.doi.org/10.1155/2011/193781.

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Iterative splitting methods have a huge amount to compute matrix exponential. Here, the acceleration and recovering of higher-order schemes can be achieved. From a theoretical point of view, iterative splitting methods are at least alternating Picards fix-point iteration schemes. For practical applications, it is important to compute very fast matrix exponentials. In this paper, we concentrate on developing fast algorithms to solve the iterative splitting scheme. First, we reformulate the iterative splitting scheme into an integral notation of matrix exponential. In this notation, we consider fast approximation schemes to the integral formulations, also known as -functions. Second, the error analysis is explained and applied to the integral formulations. The novelty is to compute cheaply the decoupled exp-matrices and apply only cheap matrix-vector multiplications for the higher-order terms. In general, we discuss an elegant way of embedding recently survey on methods for computing matrix exponential with respect to iterative splitting schemes. We present numerical benchmark examples, that compared standard splitting schemes with the higher-order iterative schemes. A real-life application in contaminant transport as a two phase model is discussed and the fast computations of the operator splitting method is explained.

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5

Li, Wanling, i Gengjun Gao. "Research on Multi-product Order Splitting and Distribution Route Optimization Of "Multi-warehouse in One Place"". Frontiers in Business, Economics and Management 8, nr 3 (20.04.2023): 1–8. http://dx.doi.org/10.54097/fbem.v8i3.7449.

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In recent years, large online supermarkets have become a new trend in the development of e-commerce. Due to the limited storage capacity of a single warehouse, many large online supermarkets, such as Jingdong and Tmall, often adopt the warehouse layout of "one place and multiple warehouses" to quickly respond to customer needs, and the sorting and distribution tasks of orders are completed by the warehouse. At the same time, due to the change of people's lifestyle, customer demand presents the characteristics of "one order with multiple products" and "one order with multiple quantities", which makes the split fulfillment of orders become a common phenomenon. In this paper, under the condition that the warehouse is not out of stock in the layout of one place and many warehouses, aiming at the split execution problem of multi-category orders, the split order method is based on the combination of "minimum split order rate" and "principle of proximity". An order splitting optimization model considering both category and quantity splitting is established, and a set of initial order batch splitting schemes is formed to achieve the first optimization of multi-category order splitting. Secondly, the PLBH-LNS method is used to generate a better initial distribution scheme considering the customer preset time window limit and vehicle-mounted capacity constraint. Finally, with the goal of minimizing the total order performance cost, the solution idea of two-stage method is used for reference, based on the initial order splitting scheme and distribution scheme, the improved two-stage genetic algorithm is used to generate the optimal order allocation scheme and distribution scheme from the alternative schemes, and the global optimization of the splitting and distribution process is realized. The experimental results show that compared with the order splitting strategy using simple rules in practice, the PLBH-LNS method can reduce the average order splitting cost by 12.48%, which provides a new idea and effective auxiliary decision support for the order splitting problem of large online supermarkets.

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Geiser, Jürgen. "Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications". International Journal of Differential Equations 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/314290.

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We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes.

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Wei, Weiping, Youlin Shang, Hongwei Jiao i Pujun Jia. "Shock stability of a novel flux splitting scheme". AIMS Mathematics 9, nr 3 (2024): 7511–28. http://dx.doi.org/10.3934/math.2024364.

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<abstract>This article introduced the HLL-CPS-T flux splitting scheme, which is characterized by low dissipation and robustness. A detailed theoretical analysis of the dissipation and shock stability of this scheme was provided. In comparison to Toro's TV flux splitting scheme, the HLL-CPS-T scheme not only exhibits accurate capture of contact discontinuity, but also demonstrates superior shock stability, as evidenced by its absence of 'carbuncle' phenomenon. Through an examination of the disturbance attenuation properties of physical quantities in the TV and HLL-CPS-T schemes, an inference was derived: The shock stability condition for an upwind method in the velocity perturbation was damped. Theoretical analysis was given to verify the reasonableness of this inference. Numerical experiments were carefully selected to test the robustness of the new splitting scheme.</abstract>

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8

RAY, M. P., B. P. PURANIK i U. V. BHANDARKAR. "DEVELOPMENT AND ASSESSMENT OF SEVERAL HIGH-RESOLUTION SCHEMES FOR COMPRESSIBLE EULER EQUATIONS". International Journal of Computational Methods 11, nr 01 (2.09.2013): 1350049. http://dx.doi.org/10.1142/s0219876213500497.

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High-resolution extensions to six Riemann solvers and three flux vector splitting schemes are developed within the framework of a reconstruction-evolution approach. Third-order spatial accuracy is achieved using two different piecewise parabolic reconstructions and a weighted essentially nonoscillatory scheme. A three-stage TVD Runge–Kutta time stepping is employed for temporal integration. The modular development of solvers provides an ease in selecting a reconstruction scheme and/or a Riemann solver/flux vector splitting scheme. The performances of these high-resolution solvers are compared for several one- and two-dimensional test cases. Based on a comprehensive assessment of the solutions obtained with all solvers, it is found that the use of the weighted essentially nonoscillatory reconstruction with the van Leer flux vector splitting scheme provides solutions for a variety of problems with acceptable accuracy.

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Chen, Jing-Bo. "High-order time discretizations in seismic modeling". GEOPHYSICS 72, nr 5 (wrzesień 2007): SM115—SM122. http://dx.doi.org/10.1190/1.2750424.

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Seismic modeling plays an important role in explor-ation geophysics. High-order modeling schemes are in demand for practical reasons. In this context, I present three kinds of high-order time discretizations: Lax-Wendroff methods, Nyström methods, and splitting methods. Lax-Wendroff methods are based on the Taylor expansion and the replacement of high-order temporal derivatives by spatial derivatives, Nyström methods are simplified Runge-Kutta algorithms, and splitting methods comprise substeps for one-step computation. Based on these methods, three schemes with third-order and fourth-order accuracy in time and pseudospectral discretizations in space are presented. I also compare their accuracy, stability, and computational complexity, and discuss advantages and shortcomings of these algorithms. Numerical experiments show that the fourth-order Lax-Wendroff scheme is more efficient for short-time simulations while the fourth-order Nyström scheme and the third-order splitting scheme are more efficient for long-term computations.

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10

Liu, Chein-Shan, Chih-Wen Chang i Chia-Cheng Tsai. "Numerical Simulations of Complex Helmholtz Equations Using Two-Block Splitting Iterative Schemes with Optimal Values of Parameters". AppliedMath 4, nr 4 (9.10.2024): 1256–77. http://dx.doi.org/10.3390/appliedmath4040068.

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For a two-block splitting iterative scheme to solve the complex linear equations system resulting from the complex Helmholtz equation, the iterative form using descent vector and residual vector is formulated. We propose splitting iterative schemes by considering the perpendicular property of consecutive residual vector. The two-block splitting iterative schemes are proven to have absolute convergence, and the residual is minimized at each iteration step. Single and double parameters in the two-block splitting iterative schemes are derived explicitly utilizing the orthogonality condition or the minimality conditions. Some simulations of complex Helmholtz equations are performed to exhibit the performance of the proposed two-block iterative schemes endowed with optimal values of parameters. The primary novelty and major contribution of this paper lies in using the orthogonality condition of residual vectors to optimize the iterative process. The proposed method might fill a gap in the current literature, where existing iterative methods either lack explicit parameter optimization or struggle with high wave numbers and large damping constants in the complex Helmholtz equation. The two-block splitting iterative scheme provides an efficient and convergent solution, even in challenging cases.

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11

Hureuski, A. N. "Using IIR filters to build high-order finite difference schemes for the unsteady Schrödinger equation". Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, nr 4 (7.01.2020): 413–24. http://dx.doi.org/10.29235/1561-2430-2019-55-4-413-424.

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High-order finite difference schemes for the time-dependent Schrödinger equation are investigated. Digital signal processing methods allowed proving the conservativeness of high-order finite difference schemes for the unsteady Schrödinger equation. The eighth-order scheme coefficients were found with the help of the proved theoretical results. The conditions for equivalence between the eighth-order finite difference scheme and the scheme in the form of a cascade of allpass first-order filters were found. The numerical analysis of the proposed scheme was made. It was shown that the high-order finite difference schemes gave better results on solving the linear Schrödinger equations comparing to the well-known fourthorder scheme on the six-point stencil, however, the high-order schemes in couple with the second-order splitting algorithm to the nonlinear Schrödinger equation do not lead to a radical improvement in the quality of numerical results. Practical issues implementing the proposed numerical technique are considered. The obtained results can be used to construct efficient solvers for linear and nonlinear Schrödinger-type equations by applying the splitting schemes of adequate accuracy order.

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12

Sukhinov, Alexander. "TWO-DIMENSIONAL SPLITTING SCHEMES FOR HYPERBOLIC EQUATIONS". Computational Mathematics and Information Technologies 1, nr 2 (2020): 71–86. http://dx.doi.org/10.23947/2587-8999-2020-1-2-71-86.

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The article considers splitting schemes in geometric directions that approximate the initial-boundary value problem for p-dimensional hyperbolic equation by chain of two-dimensional-one-dimensional problems. Two ways of constructing splitting schemes are considered with an operator factorized on the upper layer, algebraically equivalent to the alternating direction scheme, and additive schemes of total approximation. For the first scheme, the restrictions on the shape of the region G at p=3 can be weakened in comparison with schemes of alternating directions, which are a chain of three-point problems on the upper time layer, the region can be a connected union of cylindrical regions with generators parallel to the axis OX3. In the second case, for a three-dimensional equation of hyperbolic type, an additive scheme is constructed, which is a chain «two-dimensional problem – one-dimensional problem» and approximates the original problem in a summary sense (at integer time steps). For the numerical implementation of the constructed schemes – the numerical solution of two-dimensional elliptic problems – one can use fast direct methods based on the Fourier algorithm, cyclic reduction methods for three-point vector equations, combinations of these methods, and other methods. The proposed two-dimensional splitting schemes in a number of cases turn out to be more economical in terms of total time expenditures, including the time for performing computations and exchanges of information between processors, compared to traditional splitting schemes based on the use of three-point difference problems for multiprocessor computing systems, with different structures of connections between processors type «ruler», «matrix», «cube», with universal switching.

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13

Liou, Meng-Sing, i Christopher J. Steffen. "A New Flux Splitting Scheme". Journal of Computational Physics 107, nr 1 (lipiec 1993): 23–39. http://dx.doi.org/10.1006/jcph.1993.1122.

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Tiam Kapen, Pascalin, i Tchuen Ghislain. "A New Flux Splitting Scheme Based on Toro-Vazquez and HLL Schemes for the Euler Equations". Journal of Computational Methods in Physics 2014 (2.12.2014): 1–13. http://dx.doi.org/10.1155/2014/827034.

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This paper presents a new flux splitting scheme for the Euler equations. The proposed scheme termed TV-HLL is obtained by following the Toro-Vazquez splitting (Toro and Vázquez-Cendón, 2012) and using the HLL algorithm with modified wave speeds for the pressure system. Here, the intercell velocity for the advection system is taken as the arithmetic mean. The resulting scheme is more accurate when compared to the Toro-Vazquez schemes and also enjoys the property of recognition of contact discontinuities and shear waves. Accuracy, efficiency, and other essential features of the proposed scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. The accuracy of the scheme is shown in 1D test cases designed by Toro.

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Li, Qin, Dong Sun i Pengxin Liu. "Further Study on Errors in Metric Evaluation by Linear Upwind Schemes with Flux Splitting in Stationary Grids". Communications in Computational Physics 22, nr 1 (3.05.2017): 64–94. http://dx.doi.org/10.4208/cicp.oa-2016-0123.

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AbstractThe importance of eliminating errors in grid-metric evaluation for high-order difference schemes has been widely recognized in recent years, and it is known from the proof by Vinokur and Yee (NASA TM 209598, 2000) that when conservative derivations of grid metric are used by Thomas, Lombard and Neier (AIAA J., 1978, 17(10) and J. Spacecraft and rocket, 1990, 27(2)), errors caused by metric evaluation could be eliminated by linear schemes when flux splitting is not considered. According to the above achievement, central schemes without the use of flux splitting could fulfill the requirement of error elimination. Difficulties will arise for upwind schemes to attain the objective when the splitting is considered. In this study, further investigations are made on three aspects: Firstly, an idea of central scheme decomposition is introduced, and the procedure to derive the central scheme is proposed to evaluate grid metrics only. Secondly, the analysis has been made on the requirement of flux splitting to acquire free-stream preservation, and a Lax-Friedrichs-type splitting scheme is proposed as an example. Discussions about current study with that by Nonomura et al. (Computers and Fluids, 2015, 107) have been made. Thirdly, for half-node- or mixed-type schemes, interpolations should be used to derive variables at half nodes. The requirement to achieve metric identity on this situation is analyzed and an idea of directionally consistent interpolation is proposed, which is manifested to be indispensable to avoid violations of metric identity and to eliminate metric-caused errors thereafter. Two numerical problems are tested, i.e., the free-stream and vortex preservation on wavy, largely randomized and triangular-like grids. Numerical results validate aforementioned theoretical outcomes.

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Wilkins, Jesse L. M., i Anderson Norton. "The Splitting Loope". Journal for Research in Mathematics Education 42, nr 4 (lipiec 2011): 386–416. http://dx.doi.org/10.5951/jresematheduc.42.4.0386.

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Teaching experiments have generated several hypotheses concerning the construction of fraction schemes and operations and relationships among them. In particular, researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors propose that splitting constitutes a psychological structure similar to that of a mathematical group (Piaget, 1970b): a structure that introduces mutual reversibility of students' partitioning and iterating operations that the authors refer to as the splitting loope. Data consisted of 66 sixth–grade students' written performance on 20 tasks designed to provoke responses that would indicate particular fractions schemes and operations. Findings are consistent with hypotheses from related teaching experiments. In particular, they demonstrate–consistent with the notion of the splitting loope—that equipartitioning and the partitive unit fraction scheme mediate the construction of splitting from partitioning and iterating operations.

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17

YANG, JIE, YI-MIN LIU, XUE-QIN ZUO i ZHAN-JUN ZHANG. "TELEPORTING AND SPLITTING ARBITRARY SINGLE-QUBIT INFORMATION USING A CLASS OF THREE-QUBIT W STATES". International Journal of Quantum Information 07, nr 07 (październik 2009): 1349–56. http://dx.doi.org/10.1142/s0219749909005791.

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Using a class of three-qubit W states as quantum channel, we present a quantum teleportation (QT) scheme and a quantum information splitting (QIS) scheme, respectively. We compare our schemes with two similar schemes proposed recently. It is found that our QT scheme reduces the operation difficulty in contrast to Agrawal and Pati's QT scheme [Phys. Rev. A74 (2006) 062320], and our QIS scheme is more applicable than Zheng's QIS scheme [Phys. Rev. A74 (2006) 054303] for the latter is only a special case of the former in some given conditions.

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Chaoui, Slim, Omar Alruwaili, Chafaa Hamrouni, Aarif Alutaybi i Afif Masmoudi. "On the Performance of Coded Cooperative Communication with Multiple Energy-Harvesting Relays and Error-Prone Forwarding". Applied Sciences 13, nr 5 (24.02.2023): 2910. http://dx.doi.org/10.3390/app13052910.

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In this paper, we consider a coded cooperative communication network with multiple energy-harvesting (EH) relays. In order to adequately address the problem of error propagation due to the erroneous decoding at the relays, as in the case of conventional decode and forward (DF) relaying protocol, we propose coded cooperative schemes with hard information relaying (HIR) and soft information relaying (SIR) strategies. The performance of the relayed communication with EH relay depends crucially on the channel decoding capability at the relay, channel gains at the source–relay and relay–destination links, and ultimately on the power-splitting ratio of the relay EH receiver. The exact closed-form expression for the outage probability performance of the coded cooperative scheme with HIR strategy and relay selection (CC-HIR-RS) is derived for both cases, namely for constant and optimal power-splitting ratios. Concerning the coded cooperative scheme with SIR strategy, a Rayleigh Gaussian log likelihood ratio-based model is used to describe the soft estimated symbols at the output of the relay soft encoder. Directives are provided to determine the model parameters, and, accordingly, the signal-to-noise ratio (SNR) of the equivalent one-hop relaying channel is derived. A closed-form expression for the outage probability performance of the proposed coded cooperative scheme with SIR and relay selection (CC-SIR-RS) is derived. In addition, a fuzzy logic-based power-splitting scheme in EH relay applying SIR is proposed. The fading coefficients of the source–relay and relay–destination links and distance between source and relay node are considered as input parameters of the fuzzy logic system to obtain an appropriate power-splitting ratio that leads to a quasi-optimal SNR of the equivalent end-to-end channel. Monte Carlo simulations are presented to demonstrate the validity of the analytical results, and a comparison between the performance of the CC-HIR-RS scheme with constant and optimized power-splitting ratios and that of the CC-SIR-RS scheme with constant and fuzzy logic-based power-splitting ratios is provided.

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Park, Sang-Hun, i Tae-Young Lee. "High-Order Time-Integration Schemes with Explicit Time-Splitting Methods". Monthly Weather Review 137, nr 11 (1.11.2009): 4047–60. http://dx.doi.org/10.1175/2009mwr2885.1.

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Abstract New high-order time-integration schemes for fully elastic models are presented. The new schemes, formulated using the Richardson extrapolation that employs leapfrog-type schemes, can give a good performance for linear model problems and ensure overall stability when they are combined with a forward–backward scheme for fast waves. The new and existing schemes show differences in the order of accuracy. Thus, they can be useful for investigating the impacts of time-integration scheme accuracy on the performance of numerical models. The high-order schemes are found to play an important role in the improvement of high-resolution simulations, according to idealized tests. The new schemes are less efficient than other well-known schemes at moderate spatial resolutions. However, the new schemes can be more efficient than the existing schemes when the resolution becomes very high.

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Asante-Asamani, E. O., i Bruce A. Wade. "A Dimensional Splitting of ETD Schemes for Reaction-Diffusion Systems". Communications in Computational Physics 19, nr 5 (maj 2016): 1343–56. http://dx.doi.org/10.4208/cicp.scpde14.25s.

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AbstractNovel dimensional splitting techniques are developed for ETD Schemes which are second-order convergent and highly efficient. By using the ETD-Crank-Nicolson scheme we show that the proposed techniques can reduce the computational time for nonlinear reaction-diffusion systems by up to 70%. Numerical tests are performed to empirically validate the superior performance of the splitting methods.

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Chen, Lizhen, Jie Shen i Chuanju Xu. "Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations". East Asian Journal on Applied Mathematics 1, nr 3 (sierpień 2011): 215–34. http://dx.doi.org/10.4208/eajam.190411.240511a.

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AbstractWe propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

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Su, Chunmei, i Xiaofei Zhao. "On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential". ESAIM: Mathematical Modelling and Numerical Analysis 54, nr 5 (26.06.2020): 1491–508. http://dx.doi.org/10.1051/m2an/2020006.

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In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.

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Xu, Gang, Tianai Zhou, Xiu-Bo Chen i Xiaojun Wang. "Splitting an Arbitrary Three-Qubit State via a Five-Qubit Cluster State and a Bell State". Entropy 24, nr 3 (8.03.2022): 381. http://dx.doi.org/10.3390/e24030381.

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Quantum information splitting (QIS) provides an idea for transmitting the quantum state through a classical channel and a preshared quantum entanglement resource. This paper presents a new scheme for QIS based on a five-qubit cluster state and a Bell state. In this scheme, the sender transmits the unknown three-qubit secret state to two agents by the quantum channel with the Bell basis measurement three times and broadcasts the measurement results to the agents through the classical channel. The agent who restores the secret state can successfully recover the initial information to be transmitted through the appropriate unitary operation with the help of the other party. Firstly, our scheme’s process can be accurately realized by performing the applicable Bell basis measurement, single-qubit measurement, and local unitary operation instead of a multiparticle joint measurement. The splitting process of quantum information is realized through a convenient operation. Secondly, compared with some previous schemes, the efficiency of the total scheme has been improved in principle, and the qubit consumption is reduced. Finally, the security of the quantum information splitting scheme is analyzed from the perspectives of external attacks and participant attacks. It is proved that our scheme can effectively resist internal participant attacks and external eavesdropper attacks.

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Čiegis, Raimondas, Mečislovas Meilūnas i Olga Subač. "Trimačio parabolinio uždavinio su nelokalia kraštine sąlyga skaitinis sprendimas". Lietuvos matematikos rinkinys 44 (17.12.2004): 623–27. http://dx.doi.org/10.15388/lmr.2004.32209.

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Two finite difference schemes are used to solve the 3D parabolic problem with a non-local boundary condition. A new approximation of the initial condition is proposed for the explicit Euler scheme. Error estimates in the maximum norm are obtained and results of some numerical experiments are presented. The second scheme is based on implicit splitting method. An efficient realization algorithm of the LOD scheme is proposed.

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Mingalev, I. V., O. V. Mingalev, O. I. Akhmetov i Z. V. Suvorova. "Explicit Splitting Scheme for Maxwell’s Equations". Mathematical Models and Computer Simulations 11, nr 4 (lipiec 2019): 551–63. http://dx.doi.org/10.1134/s2070048219040094.

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Avramidis, Athanassios N., i James R. Wilson. "A splitting scheme for control variates". Operations Research Letters 14, nr 4 (listopad 1993): 187–98. http://dx.doi.org/10.1016/0167-6377(93)90069-s.

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Mani Aouadi, S., W. Mbarki i N. Zemzemi. "Stability Analysis of Decoupled Time-stepping Schemes for the Specialized Conduction System/myocardium Coupled Problem in Cardiology". Mathematical Modelling of Natural Phenomena 12, nr 5 (2017): 208–39. http://dx.doi.org/10.1051/mmnp/201712513.

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The Purkinje network is the rapid conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, we consider a problem that models the coupling between the Purkinje network and the myocardium. We first prove the stability of the space semi-discretized problem. Then we present four different strategies for solving the Purkinje/ myocardium coupling. The strategies are based on different time discretization of the coupling terms. The first scheme is fully coupled, where the coupling terms are considered implicit. The second and the third schemes are based on Gauss-Seidel time-splitting schemes where one coupling term is considered explicit and the other is implicit. The last is a Jacobi-like time-splitting scheme where both coupling terms are considered explicit. Our main result is the proof of the stability of the three considered schemes under the same restriction on the time step. Moreover, we show that the energy of the problem is slightly affected by the time-splitting schemes. We illustrate the theoretical result by different numerical simulations in 2D. We also conduct 3D simulations using physiologically detailed ionic models.

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Tang, Yuru, Chen Chen, Min Liu, Pengfei Du i H. Y. Fu. "Rate-Splitting-Based Generalized Multiple Access for Band-Limited Multi-User VLC". Photonics 10, nr 4 (13.04.2023): 446. http://dx.doi.org/10.3390/photonics10040446.

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In this paper, we propose a rate-splitting-based generalized multiple access (GMA) scheme for band-limited multi-user visible light communication (VLC) systems. By splitting and transmitting the input data of each user in a joint orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) manner, the proposed rate-splitting-based GMA scheme can obtain better bandwidth utilization than OMA and suffer less severe interference than NOMA. In order to achieve the maximum sum rate over typical low-pass VLC channels, the optimal rate-splitting-based GMA scheme was first obtained through theoretical analysis and computer simulations. Subsequently, the superiority of the optimal rate-splitting-based GMA scheme over both OMA and NOMA under various channel conditions, user separations, and error propagation levels was further verified by the theoretical, simulation, and experimental results. In particular, the experimental results showed that, when the error propagation ratio was increased from 0 to 0.2, the sum rate reduction ratio was significantly reduced from 31.4% to 7.5% by replacing NOMA with the obtained optimal rate-splitting-based GMA.

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29

Cheng, Chia-Hsin, i Yi Yan. "Indoor positioning system for wireless sensor networks based on two-stage fuzzy inference". International Journal of Distributed Sensor Networks 14, nr 5 (maj 2018): 155014771878064. http://dx.doi.org/10.1177/1550147718780649.

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Wireless indoor positioning systems are susceptible to environmental distortion and attenuation of the signal, which can affect positioning accuracy. In this article, we present a two-stage indoor positioning scheme using a fuzzy-based algorithm aimed at minimizing uncertainty in received signal strength indicator measures from reference nodes in wireless sensor networks. In the first stage, the indoor space is divided into several zones and a fuzzy-based indoor zone-positioning scheme is used to identify the zone in which the target node is located via zone splitting. In the second stage, adaptive trilateration is used to position the target node within the zone identified in the first stage. Simulation results demonstrate that the proposed two-stage fuzzy rectangular splitting outperforms non-fuzzy-based algorithms, including K-Nearest Neighbors–based localization, and traditional triangular splitting schemes. We also developed an expanded positioning scheme to facilitate the selection of a positioning map for large indoor spaces, thereby overcoming the limitations imposed by the size of the positioning area while maintaining high positioning resolution.

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30

Peng, Chunling, Guozhong Wang, Fangwei Li i Huaping Liu. "Joint Resource Allocation for SWIPT-Based Two-Way Relay Networks". Energies 13, nr 22 (18.11.2020): 6024. http://dx.doi.org/10.3390/en13226024.

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This paper considers simultaneous wireless information and power transfer (SWIPT) in a decode-and-forward two-way relay (DF-TWR) network, where a power splitting protocol is employed at the relay for energy harvesting. The goal is to jointly optimize power allocation (PA) at the source nodes, power splitting (PS) at the relay node, and time allocation (TA) of each duration to minimize the system outage probability. In particular, we propose a static joint resource allocation (JRA) scheme and a dynamic JRA scheme with statistical channel properties and instantaneous channel characteristics, respectively. With the derived closed-form expression of the outage probability, a successive alternating optimization algorithm is proposed to tackle the static JRA problem. For the dynamic JRA scheme, a suboptimal closed-form solution is derived based on a multistep optimization and relaxation method. We present a comprehensive set of simulation results to evaluate the proposed schemes and compare their performances with those of existing resource allocation schemes.

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31

Skamarock, William C. "Positive-Definite and Monotonic Limiters for Unrestricted-Time-Step Transport Schemes". Monthly Weather Review 134, nr 8 (1.08.2006): 2241–50. http://dx.doi.org/10.1175/mwr3170.1.

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Abstract General positive-definite and monotonic limiters are described for use with unrestricted-Courant-number flux-form transport schemes. These limiters are tested using a time-split multidimensional transport scheme. The importance of minimizing the splitting errors associated with the time-split operator and of the consistency between the transport scheme and the discrete continuity equation is demonstrated.

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32

Chen, Yaming, Songhe Song i Huajun Zhu. "Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation". Advances in Applied Mathematics and Mechanics 6, nr 4 (sierpień 2014): 494–514. http://dx.doi.org/10.4208/aamm.2013.m278.

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AbstractIn this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

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33

Lee, Hyun Geun, Jaemin Shin i June-Yub Lee. "A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional". Mathematics 7, nr 12 (16.12.2019): 1242. http://dx.doi.org/10.3390/math7121242.

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Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

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34

Bréhier, Charles-Edouard, Jianbo Cui i Jialin Hong. "Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation". IMA Journal of Numerical Analysis 39, nr 4 (30.07.2018): 2096–134. http://dx.doi.org/10.1093/imanum/dry052.

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Abstract This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

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35

Čiegis, Raimondas, i Remigijus Čiegis. "Numerical algorithms for one parabolic-elliptic problem". Lietuvos matematikos rinkinys 43 (22.12.2003): 581–85. http://dx.doi.org/10.15388/lmr.2003.32532.

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In this paper we solve numerically a parabolic-elliptic problem. Two finite difference schemes are proposed. The first scheme is a modification of the backward Euler algorithm and it requires to solve an elliptic problem at each time step. The spectral estimates of the obtained matrix are presented. The second scheme is a modification of the stability-correction scheme. This scheme is used as a classical splitting scheme in the parabolic region of the problem definition and as a new iterative algorithm in the elliptic part of the problem. We prove the convergence of the proposed scheme.

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36

Xia, Cheng Jun, Cui Qiong Chen, Kun Men i Ji Xue Yan. "A Novel System Splitting Scheme Based on the Identification of Weak Connection". Advanced Materials Research 1008-1009 (sierpień 2014): 473–79. http://dx.doi.org/10.4028/www.scientific.net/amr.1008-1009.473.

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In this paper, a new system splitting scheme based on the identification of weak connection is proposed. This paper firstly claims that if two nodes in a transmission line are related to two different generator groups or irrelated to any generator group, the transmission line is regarded as weak connection. And matrix R is presented to reflect the relevance between load nodes and generator groups after faults are cleared. The candidate strategy space is composed of these transmission lines of weak connection. Secondly, the procedure of searching the splitting surfaces is listed to illustrate how the splitting scheme works. Finally IEEE 39-bus system is used to verify the feasibility of this splitting scheme.

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37

Fan, Wenfei, Ziyan Han, Weilong Ren, Ding Wang, Yaoshu Wang, Min Xie i Mengyi Yan. "Splitting Tuples of Mismatched Entities". Proceedings of the ACM on Management of Data 1, nr 4 (8.12.2023): 1–29. http://dx.doi.org/10.1145/3626763.

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There has been a host of work on entity resolution (ER), to identify tuples that refer to the same entity. This paper studies the inverse of ER, to identify tuples to which distinct real-world entities are matched by mistake, and split such tuples into a set of tuples, one for each entity. We formulate the tuple splitting problem. We propose a scheme to decide what tuples to split and what tuples to correct without splitting, fix errors/assign attribute values to the split tuples, and impute missing values. The scheme introduces a class of rules, which embed predicates for aligning entities across relations and knowledge graphs G, assessing correlation between attributes, and extracting data from G. It unifies logic deduction, correlation models, and data extraction by chasing the data with the rules. We train machine learning models to assess attribute correlation and predict missing values. We develop algorithms for the tuple splitting scheme. Using real-life data, we empirically verify that the scheme is efficient and accurate, with F-measure 0.92 on average.

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38

Volosova, N. K., K. A. Volosov, A. K. Volosova, M. I. Karlov, D. F. Pastuhov i Yu F. Pastuhov. "Explicit Difference Scheme N-fold Splitting For the Vortex Equation in a Viscous Incompressible Fluid". Вестник Пермского университета. Математика. Механика. Информатика, nr 4 (63) (2023): 12–21. http://dx.doi.org/10.17072/1993-0550-2023-4-12-21.

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This work is the first to consider the possibility of N-fold (n=100,200) splitting of an explicit difference scheme for the vortex equation in the system of equations of a hydrodynamic problem in a rectangular cavity with a viscous incompressible fluid and with the Reynolds number Re=1000. The algorithm proposed in the work allows us to significantly increase the maximum time step per iteration of the general problem and reduce the total calculation time by tens to hundreds of times. The splitting algorithm for the vortex equation explicit difference scheme is effective if the time spent by the program on the splitting cycle is many times less than the general problem on one iterationsolving time. It is shown numerically that the solution without splitting qualitatively coincides with the solution of the split circuit (match to five significant figures). In this case, the solution to the problem without splitting is not completely steady (the first five significant digits are constant in time after 400000 iterations). It is shown numerically that two-layer and three-layer explicit difference schemes have steady-state solutions with fields matching in 11-12 significant signs at each node of the computational grid (velocity, vortex, stream function) after 21000 iterations.

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39

Einkemmer, L., A. Ostermann i M. Residori. "A pseudo-spectral Strang splitting method for linear dispersive problems with transparent boundary conditions". Numerische Mathematik 150, nr 1 (29.11.2021): 105–35. http://dx.doi.org/10.1007/s00211-021-01252-1.

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AbstractThe present work proposes a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient subject to transparent boundary conditions. For its spatial discretization, a dual Petrov–Galerkin method is considered which gives spectral accuracy. The main difficulty in constructing a second-order splitting scheme in such a situation lies in the compatibility condition at the boundaries of the sub-problems. In particular, the presence of an inflow boundary condition in the advection part results in order reduction. To overcome this issue a modified Strang splitting scheme is introduced that retains second-order accuracy. For this numerical scheme a stability analysis is conducted. In addition, numerical results are shown to support the theoretical derivations.

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40

Zbunjak i Kuzle. "System Integrity Protection Scheme (SIPS) Development and an Optimal Bus-Splitting Scheme Supported by Phasor Measurement Units (PMUs)". Energies 12, nr 17 (3.09.2019): 3404. http://dx.doi.org/10.3390/en12173404.

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System integrity protection schemes (SIPS) are schemes that can, under potentially hazardous conditions, prevent a complete blackout of endangered parts of an electrical power system (EPS). The main objective of SIPS is to monitor the state of the power transmission network in real time and to react in emergency cases. This paper explores the use of phasor measurement unit (PMU) technology for the development of SIPS as a part of wide-area monitoring, protection, and control (WAMPAC) systems. A new SIPS development method is described using the experience from the real-time operation. The developed optimal bus-splitting scheme identifies potential actions that can eliminate or reduce power system overloads and protect the integrity of the power system. An optimal bus-splitting scheme based on a DC power flow model and PMU measurements is given as an example and is explained and tested on an IEEE 14 bus test system. Conducted simulations indicate that the described SIPS methodology supported by the PMU measurements can mitigate potential overloads of the observed network part.

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41

Huang, Lang-Yang, Zhi-Feng Weng i Chao-Ying Lin. "Compact splitting symplectic scheme for the fourth-order dispersive Schrödinger equation with Cubic-Quintic nonlinear term". International Journal of Modeling, Simulation, and Scientific Computing 10, nr 02 (kwiecień 2019): 1950007. http://dx.doi.org/10.1142/s1793962319500077.

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Combining symplectic algorithm, splitting technique and compact method, a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schrödinger equation with cubic-quintic nonlinear term. The scheme has fourth-order accuracy in space and second-order accuracy in time. The discrete charge conservation law and stability of the scheme are analyzed. Numerical examples are given to confirm the theoretical results.

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42

Lin, F. B., i F. Sotiropoulos. "Assessment of Artificial Dissipation Models for Three-Dimensional Incompressible Flow Solutions". Journal of Fluids Engineering 119, nr 2 (1.06.1997): 331–40. http://dx.doi.org/10.1115/1.2819138.

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Various approaches for constructing artificial dissipation terms for three-dimensional artificial compressibility algorithms are presented and evaluated. Two, second-order accurate, central-differencing schemes, with explicitly added scalar and matrix-valued fourth-difference artificial dissipation, respectively, and a third-order accurate flux-difference splitting upwind scheme are implemented in a multigrid time-stepping procedure and applied to calculate laminar flow through a strongly curved duct. Extensive grid-refinement studies are carried out to investigate the grid sensitivity of each discretization approach. The calculations indicate that even the finest mesh employed, consisting of over 700,000 grid nodes, is not sufficient to establish grid independent solutions. However, all three schemes appear to converge toward the same solution as the grid spacing approaches zero. The matrix-valued dissipation scheme introduces the least amount of artificial dissipation and should be expected to yield the most accurate solutions on a given mesh. The flux-difference splitting upwind scheme, on the other hand, is more dissipative and, thus, particularly sensitive to grid resolution, but exhibits the best overall convergence characteristics on grids with large aspect ratios.

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43

Yang, Jian, Fei Tang, Qing Fen Liao i Yi Fei Wang. "Study on a Controlled Splitting Scheme Based on Layer Expanding Graph Algorithm". Applied Mechanics and Materials 577 (lipiec 2014): 974–77. http://dx.doi.org/10.4028/www.scientific.net/amm.577.974.

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Optimal controlled splitting is an emergency strategy to split the power system into several sub-regions based on global electrical information before the collapse of the system which is subject to severe disturbances. How to seek the optimal splitting sections rapidly and accurately is a key problem in controlled splitting field. A controlled splitting scheme based on layer expanding graph algorithm is presented in this paper. Firstly, source nodes of island regions extend out for the formation of the island regions. Secondly, island regions can be combined to make up synchronous sub-regions on the basis of the clustering of generators. At last, the optimal sections can be determined according to the initial and improved adjustment of splitting sections. Moreover, the scheme proposed can be adapt to the change of operation mode of the power system. The accuracy and effectiveness of the scheme is shown by the simulation results of CEPRI 36-bus system.

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44

Zhou, Zhongguo, i Lin Li. "The high accuracy conserved splitting domain decomposition scheme for solving the parabolic equations". Applied Mathematics and Nonlinear Sciences 3, nr 2 (31.12.2018): 583–92. http://dx.doi.org/10.2478/amns.2018.2.00045.

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AbstractIn this paper, the high accuracy mass-conserved splitting domain decomposition method for solving the parabolic equations is proposed. In our scheme, the time extrapolation and local multi-point weighted average schemes are used to approximate the interface fluxes on interfaces of sub-domains, while the interior solutions are computed by one dimension high-order implicit schemes in sub-domains. The important feature is that the developed scheme keeps mass conservation and are of second-order convergent in time and fourth-order convergent in space. Numerical experiments confirm the convergence.

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45

Ren, Yifei, i Zhiqiang Lu. "A flexible resource investment problem based on project splitting for aircraft moving assembly line". Assembly Automation 39, nr 4 (2.09.2019): 532–47. http://dx.doi.org/10.1108/aa-09-2018-0126.

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Purpose In response to the station design and flexible resources allocation of the aircraft moving assembly line, a new problem named flexible resource investment problem based on project splitting (FRIP_PS), which minimizes total cost of resources with a given deadline are proposed in this paper. Design/methodology/approach First, a corresponding mathematical model considering project splitting is constructed, which needs to be simultaneously determined together with job scheduling to acquire the optimized project scheduling scheme and resource configurations. Then, an integrated nested optimization algorithm including project splitting policy and job scheduling policy is designed in this paper. In the first stage of the algorithm, a heuristic algorithm designed to get the project splitting scheme and then in the second stage a genetic algorithm with local prospective scheduling strategy is adopted to solve the flexible resource investment problem. Findings The heuristic algorithm of project splitting gets better project splitting results through the job shift selection strategy and meanwhile guides the algorithm of the second stage. Furthermore, the genetic algorithm solves resources allocation and job schedule through evaluation rules which can effectively solve the delayed execution of jobs because of improper allocation of flexible resources. Originality/value This paper represents a new extension of the resource investment problem based on aircraft moving assembly line. An effective integrated nested optimization algorithm is proposed to specify station splitting scheme, job scheduling scheme and resources allocation in the assembly lines, which is significant for practical engineering applications.

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46

Tang, Shuting, Xiuqin Deng i Rui Zhan. "The general tensor regular splitting iterative method for multilinear PageRank problem". AIMS Mathematics 9, nr 1 (2023): 1443–71. http://dx.doi.org/10.3934/math.2024071.

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<abstract>The paper presents an iterative scheme called the general tensor regular splitting iterative (GTRS) method for solving the multilinear PageRank problem, which is based on a (weak) regular splitting technique and further accelerates the iterative process by introducing a parameter. The method yields familiar iterative schemes through the use of specific splitting strategies, including fixed-point, inner-outer, Jacobi, Gauss-Seidel and successive overrelaxation methods. The paper analyzes the convergence of these solvers in detail. Numerical results are provided to demonstrate the effectiveness of the proposed method in solving the multilinear PageRank problem.</abstract>

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47

Ponomareva, Karina A. "Business splitting: compliance with the principle of tax certainty in law enforcement practice". Law Enforcement Review 4, nr 2 (30.06.2020): 41–48. http://dx.doi.org/10.24147/2542-1514.2020.4(2).41-48.

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The subject. The problems of business splitting, when several new business entities are created on the basis of an existing organization in order to maintain a preferential special tax regime, are considered in the article. The aim of this paper is to find out criteria of unjustified tax benefit in the cases concerning business splitting. The methodology. The author uses methods of theoretical analysis, particularly the theory of integrative legal consciousness, as well as legal methods, including formal legal method and analysis of recent judicial practice. The main results, scope of application. The problems of assessing the circumstances of cases involving the application of a business splitting scheme by the taxpayer are inextricably linked to the assessment of the validity of the tax benefit. According to the author, splitting schemes should not be considered as tax evasion, but as an abuse of law. In addition, in order to substantiate the conclusion that a taxpayer has applied a business splitting scheme, the tax authority must have evidence that will indicate that the taxpayer has committed deliberate concerted actions together with persons under its control, aimed not so much at dividing the business as at obtaining an unjustified tax benefit as a result of using such a scheme. Judicial practice is quite ambiguous. Conclusions. The author comes to the conclusion that еhe key concept subject to criticism is the blurred criteria for obtaining tax benefits for taxpayers and the definition of the edge when it passes into the category of unjustified tax benefit.

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48

Mingalev, I., Z. Suvorova, O. Ahmetov i O. Mingalev. "The explicit splitting scheme for Maxwell's equations". Matematicheskoe modelirovanie 30, nr 12 (grudzień 2018): 17–38. http://dx.doi.org/10.31857/s023408790001934-1.

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Vabishchevich, P. N., M. V. Vasil’eva i A. E. Kolesov. "Splitting scheme for poroelasticity and thermoelasticity problems". Computational Mathematics and Mathematical Physics 54, nr 8 (sierpień 2014): 1305–15. http://dx.doi.org/10.1134/s0965542514080132.

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50

Lian, Yong-Sheng, i Ruquan Wang. "An implicit kinetic flux vector splitting scheme". Communications in Nonlinear Science and Numerical Simulation 2, nr 3 (wrzesień 1997): 186–90. http://dx.doi.org/10.1016/s1007-5704(97)90025-3.

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