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This paper focuses on the error analysis of a first-order, time-discrete scheme for solving the nonlinear Allen-Cahn equation. The discretization of the nonlinear potential is achieved through the EIEQ method, which employs an auxiliary variable to linearize the nonlinear double-well potential effectively. The energy stability of the scheme is demonstrated, along with its decoupled type implementation. Under a set of reasonable assumptions related to boundedness and continuity, an extensive error analysis is performed. This analysis results in the establishment of L-2 and H-1 error bounds for the numerical solution. Furthermore, a variety of numerical examples are conducted to illustrate the accuracy of the EIEQ scheme, highlighting its effectiveness in addressing complex dynamical systems governed by the Allen-Cahn equation.
Keyword ：
Allen-Cahn equation Allen-Cahn equation EIEQ EIEQ Error estimate Error estimate Unconditionally energy stable Unconditionally energy stable
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| GB/T 7714 | Zhang, Jun , Song, Fangying , Yang, Xiaofeng et al. Error analysis of the explicit-invariant energy quadratization (EIEQ) numerical scheme for solving the Allen-Cahn equation [J]. | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2024 , 457 . |
| MLA | Zhang, Jun et al. "Error analysis of the explicit-invariant energy quadratization (EIEQ) numerical scheme for solving the Allen-Cahn equation" . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 457 (2024) . |
| APA | Zhang, Jun , Song, Fangying , Yang, Xiaofeng , Zhang, Yu . Error analysis of the explicit-invariant energy quadratization (EIEQ) numerical scheme for solving the Allen-Cahn equation . | JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS , 2024 , 457 . |
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The design of interconnection networks is a fundamental aspect of high-performance computing (HPC) systems. Among the available topologies, the Galaxyfly network stands out as a low-diameter and flexible-radix network for HPC applications. Given the paramount importance of collective communication in HPC performance, in this paper, we present two different all-to-all broadcast algorithms for the Galaxyfly network, which adhere to the supernode-first rule and the router-first rule, respectively. Our performance evaluation validates their effectiveness and shows that the first algorithm has a higher degree of utilization of network channels, and that the second algorithm can significantly reduce the average time for routers to collect packets from the supernode.
Keyword ：
algorithm algorithm all-to-all broadcast all-to-all broadcast Galaxyfly network Galaxyfly network interconnection network interconnection network
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| GB/T 7714 | Zhuang, Hongbin , Chang, Jou-Ming , Li, Xiao-Yan et al. All-to-All Broadcast Algorithm in Galaxyfly Networks dagger [J]. | MATHEMATICS , 2023 , 11 (11) . |
| MLA | Zhuang, Hongbin et al. "All-to-All Broadcast Algorithm in Galaxyfly Networks dagger" . | MATHEMATICS 11 . 11 (2023) . |
| APA | Zhuang, Hongbin , Chang, Jou-Ming , Li, Xiao-Yan , Song, Fangying , Lin, Qinying . All-to-All Broadcast Algorithm in Galaxyfly Networks dagger . | MATHEMATICS , 2023 , 11 (11) . |
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The pancake graph is an interconnection network that plays a vital role in designing parallel and distributed systems. Due to the unavoidable occurrence of edge faults in large-scale networks and the wide application of path and cycle structures, it is essential and practical to explore the embedding of Hamiltonian paths and cycles in faulty networks. However, existing fault models ignore practical distributions of faulty edges so that only linear edge faults can be tolerated. This paper introduces a powerful fault model named the partitioned fault model. Based on this model, we study the existence of Hamiltonian paths and cycles on pancake graphs with large-scale faulty edges for the first time. We show that the n-dimensional pancake graph Pn admits a Hamiltonian path between any two vertices, avoiding ∑i=4n((i-4)((i-2)!-2)-1)+1 partition-edge faults for 4 ≤ n≤ 7, and avoiding ∑i=8n((i-1)!2-1)+399 faulty partition-edges for n≥ 8. Moreover, we prove that Pn admits a Hamiltonian cycle, avoiding ∑i=4n((i-4)((i-2)!-2)-1)+2 faulty partition-edges for 4 ≤ n≤ 7, and avoiding ∑i=8n((i-1)!2-1)+400 partition-edge faults for n≥ 8. The comparison results show that our results are a large-scale enhancement of existing results. © 2022, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
Keyword ：
Fault tolerance Fault tolerance Fault tolerant computer systems Fault tolerant computer systems Graph theory Graph theory Hamiltonians Hamiltonians Interconnection networks (circuit switching) Interconnection networks (circuit switching)
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| GB/T 7714 | Zhao, Kun , Zhuang, Hongbin , Li, Xiao-Yan et al. Partition-Edge Fault-Tolerant Hamiltonicity of Pancake Graphs [C] . 2022 : 192-204 . |
| MLA | Zhao, Kun et al. "Partition-Edge Fault-Tolerant Hamiltonicity of Pancake Graphs" . (2022) : 192-204 . |
| APA | Zhao, Kun , Zhuang, Hongbin , Li, Xiao-Yan , Song, Fangying , Su, Lichao . Partition-Edge Fault-Tolerant Hamiltonicity of Pancake Graphs . (2022) : 192-204 . |
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We propose a novel time-stepping scheme for solving the Allen-Cahn equation. We first rewrite the free energy into an equivalent form and then obtain a new Allen-Cahn equation by energy variational formula of L2-gradient flow. Using leapfrog formula, a new linear scheme is obtained, and we prove that the numerical scheme is unconditionally energy stable and uniquely solvable, and the discrete energy is in agreement with the original free energy. In addition, we also discuss the uniform boundedness and error estimate of numerical solution, the results show that the numerical solution is uniformly bounded in H2-norm, and error estimate shows that the time-stepping scheme can achieve second-order accuracy in time direction. At last, several numerical tests are illustrated to verify the theoretical results. The numerical strategy developed in this paper can be easily applied to other gradient flow models.
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| GB/T 7714 | Lin, Shimin , Song, Fangying , Sun, Tao et al. A Novel Second-Order and Unconditionally Energy Stable Numerical Scheme for Allen-Cahn Equation [J]. | MATHEMATICAL PROBLEMS IN ENGINEERING , 2022 , 2022 . |
| MLA | Lin, Shimin et al. "A Novel Second-Order and Unconditionally Energy Stable Numerical Scheme for Allen-Cahn Equation" . | MATHEMATICAL PROBLEMS IN ENGINEERING 2022 (2022) . |
| APA | Lin, Shimin , Song, Fangying , Sun, Tao , Zhang, Jun . A Novel Second-Order and Unconditionally Energy Stable Numerical Scheme for Allen-Cahn Equation . | MATHEMATICAL PROBLEMS IN ENGINEERING , 2022 , 2022 . |
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| GB/T 7714 | Chen, Abel C. H. , Jia, Wen-Kang , Hwang, Feng-Jang et al. Machine learning and deep learning methods for wireless network applications [J]. | EURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING , 2022 , 2022 (1) . |
| MLA | Chen, Abel C. H. et al. "Machine learning and deep learning methods for wireless network applications" . | EURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING 2022 . 1 (2022) . |
| APA | Chen, Abel C. H. , Jia, Wen-Kang , Hwang, Feng-Jang , Liu, Genggeng , Song, Fangying , Pu, Lianrong . Machine learning and deep learning methods for wireless network applications . | EURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING , 2022 , 2022 (1) . |
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Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30-50 y+ to 0.1-0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.
Keyword ：
fractional conservations laws fractional conservations laws fractional PINN fractional PINN physics-informed learning physics-informed learning turbulent flows turbulent flows variable fractional model variable fractional model
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| GB/T 7714 | Song, Fangying , Karniadakis, George Em . Variable-Order Fractional Models for Wall-Bounded Turbulent Flows [J]. | ENTROPY , 2021 , 23 (6) . |
| MLA | Song, Fangying et al. "Variable-Order Fractional Models for Wall-Bounded Turbulent Flows" . | ENTROPY 23 . 6 (2021) . |
| APA | Song, Fangying , Karniadakis, George Em . Variable-Order Fractional Models for Wall-Bounded Turbulent Flows . | ENTROPY , 2021 , 23 (6) . |
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In this paper, we propose a second-order implicit-explicit (IMEX) time-stepping scheme for the Gray-Scott (GS) model. In order to achieve a fast, stable and efficient scheme in time, we develop a linear second-order time-stepping scheme for solving GS model. We prove the linear stability and convergence of the proposed scheme. The advantage of our numerical scheme is stable, that is, there is no need for other stability restrictions to the nonlinear terms, only a relatively small time step is needed. Moreover, we prove that the numerical solutions achieve second-order accuracy in the time. Finally, several numerical experiments are conducted to verify the accuracy both in space and time of the numerical scheme. Numerical examples illustrate the accuracy and efficiency of the IMEX scheme is effective for the GS model.
Keyword ：
consistency analysis consistency analysis error estimation error estimation Gray-scott model Gray-scott model IMEX scheme IMEX scheme stability stability
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| GB/T 7714 | Song, Fangying , Zhang, Jun , Wang, Jinrong . Convergence analysis and error estimate of second-order implicit-explicit scheme for Gray-Scott model [J]. | INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS , 2021 , 98 (11) : 2330-2340 . |
| MLA | Song, Fangying et al. "Convergence analysis and error estimate of second-order implicit-explicit scheme for Gray-Scott model" . | INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 98 . 11 (2021) : 2330-2340 . |
| APA | Song, Fangying , Zhang, Jun , Wang, Jinrong . Convergence analysis and error estimate of second-order implicit-explicit scheme for Gray-Scott model . | INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS , 2021 , 98 (11) , 2330-2340 . |
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In this paper, we propose a second-order operator splitting spectral element method for solving fractional reaction-diffusion equations. In order to achieve a fast second-order scheme in time, we decompose the original equation into linear and nonlinear sub-equations, and combine a quarter-time nonlinear solver and a half-time linear solver followed by final quarter-time nonlinear solver. The spatial discretization is eigen-decomposition based on spectral element method. Since this method gives a full diagonal representation of the fractional operator and gets an exponential convergence in space. We have an accurate and efficient approach for solving spacial fractional reaction-diffusion equations. Some numerical experiments are carried out to demonstrate the accuracy and efficiency of this method. Finally, we apply the proposed method to investigate the effect of the fractional order in the fractional reaction-diffusion equations. © The Author(s) 2020.
Keyword ：
eigen-decomposition; exponential convergence; Fractional reaction-diffusion equations; operator splitting; spectral element method eigen-decomposition; exponential convergence; Fractional reaction-diffusion equations; operator splitting; spectral element method
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| GB/T 7714 | Li, Q. , Song, F. . Splitting spectral element method for fractional reaction-diffusion equations [J]. | Journal of Algorithms and Computational Technology , 2020 , 14 . |
| MLA | Li, Q. et al. "Splitting spectral element method for fractional reaction-diffusion equations" . | Journal of Algorithms and Computational Technology 14 (2020) . |
| APA | Li, Q. , Song, F. . Splitting spectral element method for fractional reaction-diffusion equations . | Journal of Algorithms and Computational Technology , 2020 , 14 . |
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In order to generate a probability density function (PDF) for fitting the probability distributions of practical data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be utilised as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of the trained deep learning model can be used to estimate the PDF. Numerical experiments with single and mixed distributions are conducted to evaluate the performance of the proposed method. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method. (C) 2019 Elsevier B.V. All rights reserved.
Keyword ：
Cumulative distribution function Cumulative distribution function Neural networks Neural networks Probability density function Probability density function
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| GB/T 7714 | Chen, Chi-Hua , Song, Fangying , Hwang, Feng-Jang et al. A probability density function generator based on neural networks [J]. | PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS , 2020 , 541 . |
| MLA | Chen, Chi-Hua et al. "A probability density function generator based on neural networks" . | PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 541 (2020) . |
| APA | Chen, Chi-Hua , Song, Fangying , Hwang, Feng-Jang , Wu, Ling . A probability density function generator based on neural networks . | PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS , 2020 , 541 . |
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We mathematically investigate the Parker instability in the three-dimensional compressible isentropic viscous magnetohydrodynamic (MHD) fluid with zero resistivity in the presence of a modified gravitational force in a vertical strip domain in which the velocity of the fluid is non-slip on the boundary. We prove the existence of Parker instability solutions in the sense of L-2-norm in Lagrangian coordinates based on a classical bootstrap instability method with finer skills of energy estimates. By an inverse transformation of Lagrangian coordinates, we also get Parker instability solutions in L-2-norm in Eulerian coordinates. (C) 2019 Elsevier Ltd. All rights reserved.
Keyword ：
Bootstrap instability method Bootstrap instability method Magnetohydrodynamic fluid Magnetohydrodynamic fluid Parker instability Parker instability
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| GB/T 7714 | Liu, Mengmeng , Song, Fangying , Wang, Weiwei . On Parker instability under L-2-norm [J]. | NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS , 2020 , 192 . |
| MLA | Liu, Mengmeng et al. "On Parker instability under L-2-norm" . | NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 192 (2020) . |
| APA | Liu, Mengmeng , Song, Fangying , Wang, Weiwei . On Parker instability under L-2-norm . | NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS , 2020 , 192 . |
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