Статті в журналах: "Schéma de Splitting"

1

Lai, Junjiang, and Zhencheng Fan. "Stability for discrete time waveform relaxation methods based on Euler schemes." AIMS Mathematics 8, no. 10 (2023): 23713–33. http://dx.doi.org/10.3934/math.20231206.

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<abstract><p>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.</p></abstract>

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2

Shin, Sang-Mook, In-Chul Kim, and Yong-Jig Kim. "Numerical Simulation of Free Surface Flows Using the Roe's Flux-difference Splitting Scheme." Journal of the Society of Naval Architects of Korea 47, no. 1 (2010): 11–19. http://dx.doi.org/10.3744/snak.2010.47.1.011.

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3

Liou, Meng-Sing, and Christopher J. Steffen. "A New Flux Splitting Scheme." Journal of Computational Physics 107, no. 1 (1993): 23–39. http://dx.doi.org/10.1006/jcph.1993.1122.

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4

Lai, J. S., G. F. Lin, and W. D. Guo. "Simulation of Hydraulic Shock Waves by Hybrid Flux-Splitting Schemes in Finite Volume Method." Journal of Mechanics 21, no. 2 (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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5

Tang, Yuru, Chen Chen, Min Liu, Pengfei Du, and H. Y. Fu. "Rate-Splitting-Based Generalized Multiple Access for Band-Limited Multi-User VLC." Photonics 10, no. 4 (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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6

Xia, Cheng Jun, Cui Qiong Chen, Kun Men, and Ji Xue Yan. "A Novel System Splitting Scheme Based on the Identification of Weak Connection." Advanced Materials Research 1008-1009 (August 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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7

Lee, Hyun Geun, Jaemin Shin, and June-Yub Lee. "A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional." Mathematics 7, no. 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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8

Bréhier, Charles-Edouard, Jianbo Cui, and Jialin Hong. "Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation." IMA Journal of Numerical Analysis 39, no. 4 (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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9

Li, Wanling, and 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, no. 3 (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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10

Ren, Yifei, and Zhiqiang Lu. "A flexible resource investment problem based on project splitting for aircraft moving assembly line." Assembly Automation 39, no. 4 (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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11

Yang, Jian, Fei Tang, Qing Fen Liao, and Yi Fei Wang. "Study on a Controlled Splitting Scheme Based on Layer Expanding Graph Algorithm." Applied Mechanics and Materials 577 (July 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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12

Areces, Carlos, Facundo Bustos, Martín Dominguez, and Jörg Hoffmann. "Optimizing Planning Domains by Automatic Action Schema Splitting." Proceedings of the International Conference on Automated Planning and Scheduling 24 (May 10, 2014): 11–19. http://dx.doi.org/10.1609/icaps.v24i1.13622.

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As modeling details can have a large impact on planner perormance, domain transformation has been a traditional subject of interest in the planning community not only betweenlanguages, but also within languages. Herein, we automatean intra-language transformation method that has as yet beenapplied only manually, and that has never been formally described: action schema splitting, which transforms an actionschema with a big interface (many parameters) into severalschemas with smaller interfaces, exponentially reducing thenumber of ground actions. We spell out this method, characterizing exactly the choice of splits preserving equivalence tothe original schema. Making that choice involves a trade-off between interface size and plan length, which we explore bydesigning automatic domain optimization methods. Our experiments show that these methods can substantially improveperformance on domains with large interfaces.

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13

Mingalev, I. V., O. V. Mingalev, O. I. Akhmetov, and Z. V. Suvorova. "Explicit Splitting Scheme for Maxwell’s Equations." Mathematical Models and Computer Simulations 11, no. 4 (2019): 551–63. http://dx.doi.org/10.1134/s2070048219040094.

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14

Avramidis, Athanassios N., and James R. Wilson. "A splitting scheme for control variates." Operations Research Letters 14, no. 4 (1993): 187–98. http://dx.doi.org/10.1016/0167-6377(93)90069-s.

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15

Einkemmer, L., A. Ostermann, and M. Residori. "A pseudo-spectral Strang splitting method for linear dispersive problems with transparent boundary conditions." Numerische Mathematik 150, no. 1 (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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16

Ahmed, Nauman, Tahira S.S., M. Rafiq, M. A. Rehman, Mubasher Ali, and M. O. Ahmad. "Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model." Open Mathematics 17, no. 1 (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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17

Huang, Lang-Yang, Zhi-Feng Weng, and 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, no. 02 (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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18

Wei, Weiping, Youlin Shang, Hongwei Jiao, and Pujun Jia. "Shock stability of a novel flux splitting scheme." AIMS Mathematics 9, no. 3 (2024): 7511–28. http://dx.doi.org/10.3934/math.2024364.

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<abstract><p>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.</p></abstract>

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19

Fan, Wenfei, Ziyan Han, Weilong Ren, et al. "Splitting Tuples of Mismatched Entities." Proceedings of the ACM on Management of Data 1, no. 4 (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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20

RAY, M. P., B. P. PURANIK, and U. V. BHANDARKAR. "DEVELOPMENT AND ASSESSMENT OF SEVERAL HIGH-RESOLUTION SCHEMES FOR COMPRESSIBLE EULER EQUATIONS." International Journal of Computational Methods 11, no. 01 (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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21

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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22

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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23

Viner, Kevin C., and Craig C. Epifanio. "An Analysis of Klemp–Wilhelmson Schemes as Applied to Large-Scale Wave Modes." Monthly Weather Review 136, no. 12 (2008): 4730–45. http://dx.doi.org/10.1175/2008mwr2200.1.

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Abstract The use of Klemp–Wilhelmson (KW) time splitting for large-scale and global modeling is assessed through a series of von Neumann accuracy and stability analyses. Two variations of the KW splitting are evaluated in particular: the original acoustic-mode splitting of Klemp and Wilhelmson (KW78) and a modified splitting due to Skamarock and Klemp (SK92) in which the buoyancy and vertical stratification terms are treated as fast-mode terms. The large-scale case of interest is the problem of Rossby wave propagation on a resting background state. The results show that the original KW78 splitting is surprisingly inaccurate when applied to large-scale wave modes. The source of this inaccuracy is traced to the compressible vertical adjustment—and more precisely, to the splitting of the hydrostatic balance terms between the small and large time steps. The splitting errors can be reduced somewhat through implicit biasing, but large biasing coefficients are needed for acceptable error levels—and even then the time steps are limited to moderate values. The errors in the KW78 splitting are shown to be largely absent from the SK92 scheme. Two versions of the SK92 splitting are considered in particular: the original leapfrog splitting (SK92-LF) of Skamarock and Klemp and the third-order Runge–Kutta splitting (SK92-RK) proposed by Wicker and Skamarock. The mixed cubic (on the large time step) and quadratic (on the small step) behavior of the SK92-RK scheme is described in detail and is compared with the strictly quadratic behavior of the SK92-LF method.

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24

Iwase, Yukari, Osamu Tomita, Masanobu Higashi, and Ryu Abe. "Enhanced oxygen evolution on visible light responsive TaON photocatalysts co-loaded with highly active Ru species for IO3− reduction and Co species for water oxidation." Sustainable Energy & Fuels 1, no. 4 (2017): 748–54. http://dx.doi.org/10.1039/c7se00110j.

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Loading an appropriate cocatalyst significantly enhances the activity of semiconductor photocatalysts in both conventional one-step water splitting and Z-scheme-type water splitting with a redox couple.

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25

Shim, Yeonggyu, and Wonjae Shin. "Energy Rate Maximization with Sum-Rate Constraint for SWIPT in Multiple-Access Channels." Electronics 8, no. 12 (2019): 1525. http://dx.doi.org/10.3390/electronics8121525.

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This paper considers simultaneous wireless information and power transfer (SWIPT) systems in the two-user Gaussian multiple access channel (G-MAC). In SWIPT systems, for a transmit signal each transmitter consists of an information-carrying signal and energy-carrying signal. By controlling a different set of the power for the information transmission and power for the energy transmission under a total power constraint, the information sum-rate and energy transmission rate can be achieved. As the information carrying-to-transmit power ratio at transmitters and the information sum-rate increases, however, the energy transmission rate decreases. In other words, there is a fundamental trade-off between the information sum-rate and the energy transmission rate according to the power-splitting ratio at each transmitter. Motivated by this, this paper proposes an optimal power-splitting scheme that maximizes the energy transmission rate subject to a minimum sum-rate constraint. In particular, a closed-form expression of the power-splitting coefficient is presented for the two-user G-MAC under a minimum sum-rate constraint. Numerical results show that the energy rate of the proposed optimal power-splitting scheme is greater than that of the fixed power-splitting scheme.

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26

Shuai, Chunjiang, Kaimin Teng, and Hongen Jia. "On the Error Estimates of a New Operator Splitting Scheme for the Navier-Stokes Equations with Coriolis Force." Mathematical Problems in Engineering 2012 (2012): 1–23. http://dx.doi.org/10.1155/2012/105735.

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An operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations with Coriolis force. Under some mild regularity assumptions on the continuous solution, error estimates and the stability analysis for the velocity and the pressure of the new operator splitting scheme are obtained. Some numerical results are presented to verify the theoretical predictions.

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27

Woźnicki, Zbigniew I. "Matrix splitting principles." International Journal of Mathematics and Mathematical Sciences 28, no. 5 (2001): 251–84. http://dx.doi.org/10.1155/s0161171201007062.

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The systematic analysis of convergence conditions, used in comparison theorems proven for different matrix splittings, is presented. The central idea of this analysis is the scheme of condition implications derived from the properties of regular splittings of a monotone matrixA=M1−N1=M2−N2. An equivalence of some conditions as well as an autonomous character of the conditionsM1−1≥M2−1≥0andA−1N2≥A−1N1≥0are pointed out. The secondary goal is to discuss some essential topics related with existing comparison theorems.

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28

Zhou, Zhongguo, and Dong Liang. "The Mass-Preserving S-DDM Scheme for Two-Dimensional Parabolic Equations." Communications in Computational Physics 19, no. 2 (2016): 411–41. http://dx.doi.org/10.4208/cicp.070814.190615a.

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AbstractIn the paper, we develop and analyze a new mass-preserving splitting domain decomposition method over multiple sub-domains for solving parabolic equations. The domain is divided into non-overlapping multi-bock sub-domains. On the interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit) flux scheme. The solutions and fluxes in the interiors of sub-domains are computed by the splitting one-dimensional implicit solution-flux coupled scheme. The important feature is that the proposed scheme is mass conservative over multiple non-overlapping sub-domains. Analyzing the mass-preserving S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique and the domain decomposition at each time step. We prove theoretically that our scheme satisfies conservation of mass over multi-block non-overlapping sub-domains and it is unconditionally stable. We further prove the convergence and obtain the error estimate in L2-norm. Numerical experiments confirm theoretical results.

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29

Chaoui, Slim, Omar Alruwaili, Chafaa Hamrouni, Aarif Alutaybi, and Afif Masmoudi. "On the Performance of Coded Cooperative Communication with Multiple Energy-Harvesting Relays and Error-Prone Forwarding." Applied Sciences 13, no. 5 (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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30

PAN, GUI-XIA, YI-MIN LIU, XIAO-FENG YIN, WEN ZHANG, and ZHAN-JUN ZHANG. "A QUANTUM SPLITTING SCHEME OF ARBITRARY TWO-QUBIT STATE USING FOUR-QUBIT CLUSTER STATE." International Journal of Quantum Information 06, no. 05 (2008): 1033–40. http://dx.doi.org/10.1142/s021974990800433x.

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In this paper, we explore the channel capacity of the 4-qubit 4-term cluster state (i.e. (|0000〉 + |0011〉 + |1100〉 - |1111〉)/2) for splitting arbitrary two-qubit quantum information. After our extensive investigations, we found that 4 out of 12 possible distributions of the 4 qubits can be utilized to realize bipartite splitting. In terms of each distribution, the corresponding splitting scheme is presented and LOCCs (local operation and classical communication) are explicitly given.

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31

Podoll, K., and D. Robinson. "Splitting of the Body Image as Somesthetic Aura Symptom in Migraine." Cephalalgia 22, no. 1 (2002): 62–65. http://dx.doi.org/10.1046/j.1468-2982.2002.00316.x.

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Splitting of the body image was illustrated as a somesthetic aura symptom in three of the 562 entries to the national Migraine Art competitions, confirming previous descriptions of this rare phenomenon in the migraine literature. In this type of self-experienced paroxysmal body schema disturbance, the own body is perceived as being split, usually in the mid-line, into two halves that may be displaced or separated from each other. Splitting of the body image most frequently applies to the migraine sufferer's head. The said phenomenon, the pathomechanisms of which are obscure, must not be confused with the visual illusion of illusory splitting.

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32

ZUO, XUE-QIN, YI-MIN LIU, WEN ZHANG, XIAO-FENG YIN, and ZHAN-JUN ZHANG. "MINIMAL CLASSICAL COMMUNICATION COST AND MEASUREMENT COMPLEXITY IN SPLITTING TWO-QUBIT QUANTUM INFORMATION VIA ASYMMETRIC W STATES." International Journal of Quantum Information 06, no. 06 (2008): 1245–53. http://dx.doi.org/10.1142/s0219749908004419.

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We propose a scheme for splitting a two-qubit quantum information by using two asymmetric W states as quantum channel. In this scheme the split state is assumed to be completely known by the sender. Because of this, during the splitting process, the sender only needs to perform a two-qubit projective measurement. Once the sender announces the measurement result in terms of the prior agreement, then using this message the two receivers can recover the quantum information via their mutual assistance. We calculate the success probability and classical communication cost of the scheme. In general, the splitting success probability (SSP) is 1/4 and the average classical communication cost is 0.25 bit. However, we find that for some states the SSP can reach 0.5 or even unity after consuming a little additional classical resource.

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33

Do, Dinh-Thuan. "Optimal Energy Harvesting Scheme for Power Beacon-Assisted Wireless-Powered Networks." Indonesian Journal of Electrical Engineering and Computer Science 7, no. 3 (2017): 802. http://dx.doi.org/10.11591/ijeecs.v7.i3.pp802-808.

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In this paper, we consider one-way relay with energy harvesting system based on power beacon (PB), in which the relay node harvests transmitted power from the PB station to forward signals to destination. We also analyse the relay network model with amplify-and-forward (AF) protocol for information cooperation and Power Splitting-based Relaying (PSR) protocol for power transfer. In particular, the outage probability and optimal energy harvesting (EH) power splitting fraction of novel scheme in are presented. We obtain analytical closed-form expression of optimal energy harvesting (EH) power splitting fraction to minimize the outage probability of system. Using numerical and analytical simulations, the performances of different cases are presented and discussed.

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34

Mingalev, I., Z. Suvorova, O. Ahmetov, and O. Mingalev. "The explicit splitting scheme for Maxwell's equations." Matematicheskoe modelirovanie 30, no. 12 (2018): 17–38. http://dx.doi.org/10.31857/s023408790001934-1.

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35

Vabishchevich, P. N., M. V. Vasil’eva, and A. E. Kolesov. "Splitting scheme for poroelasticity and thermoelasticity problems." Computational Mathematics and Mathematical Physics 54, no. 8 (2014): 1305–15. http://dx.doi.org/10.1134/s0965542514080132.

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36

Lian, Yong-Sheng, and Ruquan Wang. "An implicit kinetic flux vector splitting scheme." Communications in Nonlinear Science and Numerical Simulation 2, no. 3 (1997): 186–90. http://dx.doi.org/10.1016/s1007-5704(97)90025-3.

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37

Čiegis, Raimondas, and Violeta Pakenienė. "Šredingerio uždavinio su stiprinimo procesu sprendimas simetrizuota išskaidymo schema." Lietuvos matematikos rinkinys 41 (December 17, 2001): 504–10. http://dx.doi.org/10.15388/lmr.2001.34637.

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In this paper we consider one-dimensional nonlinear Shrödinger equation. The equation includes an absorption term and the solution is periodically amplified in order to compensate the lose of the energy. We present a finite difference approximation by a symetrical splitting scheme. The convergence of the discrete solution is proved.

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38

Grigor'yev, F. V., and A. S. Nuzhnyy. "Radionuclide Transport in Geological Environment Modeled in the GeRa Code with an Account Taken of Radioactive Decay Chains: Numerical Scheme Verification and Calculation Specifics." Radioactive Waste 27, no. 2 (2024): 95–101. http://dx.doi.org/10.25283/2587-9707-2024-2-95-101.

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The paper describes a numerical scheme implemented in the GeRa code to solve the problem of radionuclide transport in porous medium taking into account radioactive decay chains. It is based on a scheme of splitting by physical processes and the use of separate modules for solving transport and radioactive decay chain problems. Verification of the numerical scheme was performed using an analytical solution for a specific case of the problem formulation in a homogeneous porous medium. The study explores the error introduced into the numerical solution by the use of the splitting scheme. The study provides recommendations regarding the selection of appropriate parameters for the numerical scheme to calculate the transport of nuclide chains with different sorption properties.

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39

Wang, Bo, Dong Liang, and Tongjun Sun. "The Conservative Splitting High-Order Compact Finite Difference Scheme for Two-Dimensional Schrödinger Equations." International Journal of Computational Methods 15, no. 01 (2017): 1750079. http://dx.doi.org/10.1142/s0219876217500797.

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In this paper, a new conservative and splitting fourth-order compact difference scheme is proposed and analyzed for solving two-dimensional linear Schrödinger equations. The proposed splitting high-order compact scheme in two dimensions has the excellent property that it preserves the conservations of charge and energy. We strictly prove that the scheme satisfies the charge and energy conservations and it is unconditionally stable. We also prove the optimal error estimate of fourth-order accuracy in spatial step and second-order accuracy in time step. The scheme can be easily implemented and extended to higher dimensional problems. Numerical examples are presented to confirm our theoretical results.

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40

Chen, Jing-Bo. "High-order time discretizations in seismic modeling." GEOPHYSICS 72, no. 5 (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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41

Boda, Lívia, Istvan Faragó, and Tamás Kalmár-Nagy. "The average method is much better than average." Journal of Computational and Applied Mechanics 16, no. 1 (2021): 37–56. http://dx.doi.org/10.32973/jcam.2021.003.

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Operator splitting is a powerful method for the numerical investigation of complex time-dependent models, where the stationary (elliptic) part consists of a sum of several structurally simpler sub-operators. As an alternative to the classical splitting methods, a new splitting scheme is proposed here, the Average Method with sequential splitting. In this method, a decomposition of the original problem is sought in terms of commuting matrices. Wedemonstrate that third-order accuracy can be achieved with the Average Method. The computational performance of the method is investigated, yielding run times 1-2 orders of magnitude faster than traditional methods.

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42

Yang, Zhaohua, Dan Li, and Yuzhe Sun. "Analysis of Kerr Noise in Angular-Rate Sensing Based on Mode Splitting in a Whispering-Gallery-Mode Microresonator." Micromachines 10, no. 2 (2019): 150. http://dx.doi.org/10.3390/mi10020150.

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Whispering-gallery-mode (WGM) microresonators have shown their potential in high-precision gyroscopes because of their small volume and high-quality factors. However, Kerr noise can always be the limit of accuracy. Angular-rate sensing based on mode splitting treats backscattering as a measured signal, which can induce mode splitting, while it is considered as a main source of noise in conventional resonator optical gyroscopes. Meanwhile, mode splitting also provides superior noise suppression owing to its self-reference scheme. Kerr noise in this scheme has not been defined and solved yet. Here, the mechanism of the Kerr noise in the measurement is analyzed and the mathematical expressions are derived, indicating the relationship between the Kerr noise and the output of the system. The influence caused by Kerr noise on the output is simulated and discussed. Simulations show that the deviation of the splitting caused by Kerr noise is 1.913 × 10−5 Hz at an angular rate of 5 × 106 °/s and the corresponding deviation of the angular rate is 9.26 × 10−9 °/s. It has been proven that angular-rate sensing based on mode splitting offers good suppression of Kerr noise.

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43

Liu, Chein-Shan, Chih-Wen Chang, and Chia-Cheng Tsai. "Numerical Simulations of Complex Helmholtz Equations Using Two-Block Splitting Iterative Schemes with Optimal Values of Parameters." AppliedMath 4, no. 4 (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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44

Wilkins, Jesse L. M., and Anderson Norton. "The Splitting Loope." Journal for Research in Mathematics Education 42, no. 4 (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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45

Volosova, N. K., K. A. Volosov, A. K. Volosova, M. I. Karlov, D. F. Pastuhov, and Yu F. Pastuhov. "Explicit Difference Scheme N-fold Splitting For the Vortex Equation in a Viscous Incompressible Fluid." Вестник Пермского университета. Математика. Механика. Информатика, no. 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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46

Tiam Kapen, Pascalin, and 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 (December 2, 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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47

Geiser, Jürgen. "Higher-Order Splitting Method for Elastic Wave Propagation." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–31. http://dx.doi.org/10.1155/2008/291968.

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Motivated by seismological problems, we have studied a fourth-order split scheme for the elastic wave equation. We split in the spatial directions and obtain locally one-dimensional systems to be solved. We have analyzed the new scheme and obtained results showing consistency and stability. We have used the split scheme to solve problems in two and three dimensions. We have also looked at the influence of singular forcing terms on the convergence properties of the scheme.

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48

Vovchanskyi, M. B. "Splitting for some classes of homeomorphic and coalescing stochastic flows." ESAIM: Probability and Statistics 28 (2024): 75–109. http://dx.doi.org/10.1051/ps/2024004.

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The splitting scheme (the Kato-Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E. Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield estimates of the speed of the convergence under additional regularity assumptions.

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49

Maeda, Kazuhiko. "(Invited) Z-Scheme Water Splitting Using Dye-Sensitized Metal Oxide Nanosheets." ECS Meeting Abstracts MA2023-01, no. 37 (2023): 2148. http://dx.doi.org/10.1149/ma2023-01372148mtgabs.

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Solar water splitting by a heterogeneous photocatalyst, producing H2 and O2, is a potential solution for realizing large-scale H2 production. While dye-sensitized metal oxides are good candidates as H2 evolution photocatalysts for solar-driven Z-scheme water splitting, their solar-to-hydrogen (STH) energy conversion efficiencies remains low because of uncontrolled charge recombination reactions. Here it is shown that modification of Ru dye-sensitized, Pt- intercalated HCa2Nb3O10 nanosheets (Ru/Pt/HCa2Nb3O10) with both amorphous Al2O3 and poly(styrenesulfonate) (PSS) improves the STH efficiency of Z-scheme overall water splitting by a factor of ~100, when the nanosheets are used in combination with a WO3-based O2 evolution photocatalyst and an I3 –/I– redox mediator, relative to an analogous system that uses unmodified Ru/Pt/HCa2Nb3O10. The Al2O3 and PSS modifiers, which have previously been shown to suppress back electron transfer reactions in a dye-sensitized H2 evolution photocatalyst, enabled operation of the Z-scheme system even at low intensity of simulated sunlight without a decrease in the STH values. By using the optimized photocatalyst, PSS/Ru/Al2O3/Pt/HCa2Nb3O10, a maximum STH of 0.12% and an apparent quantum yield of 4.1% at 420 nm were obtained, by far the highest among dye-sensitized water splitting systems and also comparable to conventional semiconductor-based suspended particulate photocatalyst systems.

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50

Seydaoğlu, Muaz. "An integral formulation for the global error of Lie Trotter splitting scheme." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 9, no. 1 (2019): 36–40. http://dx.doi.org/10.11121/ijocta.01.2019.00625.

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An ordinary differential equation (ODE) can be split into simpler sub equations and each of the sub equations is solved subsequently by a numerical method. Such a procedure involves splitting error and numerical error caused by the time stepping methods applied to sub equations. The aim of the paper is to present an integral formula for the global error expansion of a splitting procedure combined with any numerical ODE solver.

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