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学者姓名：宋方应
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Abstract ：
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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采用分数阶物理信息网络(fPINN)求解时间分数阶Nernst-Plank方程，并对其解决时间分数阶N-P的正问题与反问题的准确性和有效性进行说明.在这基础上，分析离散化时间分数阶算子所导致的离散误差、采样误差、神经网络优化误差对最终求解的影响.同时，分析离散误差与取样误差的关系，并发现当固定离散误差后存在最好的训练点集大小使得求解误差最低.最后，展示神经网络求解反问题的准确性与效率.
Keyword ：
分数阶物理信息神经网络 分数阶物理信息神经网络 时间分数阶Nernst-Plank方程 时间分数阶Nernst-Plank方程 深度学习 深度学习 误差分析 误差分析
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| GB/T 7714 | 徐国泰 , 李娴娟 , 宋方应 . 基于深度学习的分数阶Nernst-Plank方程求解 [J]. | 福州大学学报(自然科学版) , 2024 , 52 (04) : 379-386 . |
| MLA | 徐国泰 et al. "基于深度学习的分数阶Nernst-Plank方程求解" . | 福州大学学报(自然科学版) 52 . 04 (2024) : 379-386 . |
| APA | 徐国泰 , 李娴娟 , 宋方应 . 基于深度学习的分数阶Nernst-Plank方程求解 . | 福州大学学报(自然科学版) , 2024 , 52 (04) , 379-386 . |
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In this paper, we used Fourier Neural Operator (FNO) networks to solve reaction-diffusion equations. The FNO is a novel framework designed to solve partial differential equations by learning mappings between infinite-dimensional functional spaces. We applied the FNO to the Surface Quasi-Geostrophic (SQG) equation, and we tested the model with two significantly different initial conditions: Vortex Initial Conditions and Sinusoidal Initial Conditions. Furthermore, we explored the generalization ability of the model by evaluating its performance when trained on Vortex Initial Conditions and applied to Sinusoidal Initial Conditions. Additionally, we investigated the modes (frequency parameters) used during training, analyzing their impact on the experimental results, and we determined the most suitable modes for this study. Next, we conducted experiments on the number of convolutional layers. The results showed that the performance of the models did not differ significantly when using two, three, or four layers, with the performance of two or three layers even slightly surpassing that of four layers. However, as the number of layers increased to five, the performance improved significantly. Beyond 10 layers, overfitting became evident. Based on these observations, we selected the optimal number of layers to ensure the best model performance. Given the autoregressive nature of the FNO, we also applied it to solve the Gray-Scott (GS) model, analyzing the impact of different input time steps on the performance of the model during recursive solving. The results indicated that the FNO requires sufficient information to capture the long-term evolution of the equations. However, compared to traditional methods, the FNO offers a significant advantage by requiring almost no additional computation time when predicting with new initial conditions.
Keyword ：
Fourier Neural Operator (FNO) Fourier Neural Operator (FNO) Gray-Scott equation Gray-Scott equation multi-layer convolution multi-layer convolution Surface Quasi-Geostrophic (SQG) equation Surface Quasi-Geostrophic (SQG) equation
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| GB/T 7714 | Hao, Yaobin , Song, Fangying . Fourier Neural Operator Networks for Solving Reaction-Diffusion Equations [J]. | FLUIDS , 2024 , 9 (11) . |
| MLA | Hao, Yaobin et al. "Fourier Neural Operator Networks for Solving Reaction-Diffusion Equations" . | FLUIDS 9 . 11 (2024) . |
| APA | Hao, Yaobin , Song, Fangying . Fourier Neural Operator Networks for Solving Reaction-Diffusion Equations . | FLUIDS , 2024 , 9 (11) . |
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We present a numerical method for approximating the two-dimensional surface quasi-geostrophic equation. We first reformulate the equation into an equivalent system by using the scalar auxiliary variable approach. Then we propose first-order and second-order discretization schemes for solving the new surface quasi-geostrophic system in time. Furthermore, we prove that both of these schemes based on the scalar auxiliary variable approach satisfy an unconditionally energy stability property. Since the method gives two linear equations with constant coefficients in each time step. We have an efficient approach and accurate for solving these spacial fractional diffusion equations. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for inviscid problems and the problems with small viscosity.
Keyword ：
fractional reaction-diffusion equation fractional reaction-diffusion equation Scalar auxiliary variable Scalar auxiliary variable unconditionally stable unconditionally stable
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| GB/T 7714 | Shi, Shengtao , Song, Fangying . Scalar auxiliary variable approache for the surface quasi-geostrophic equation [J]. | JOURNAL OF ALGORITHMS & COMPUTATIONAL TECHNOLOGY , 2023 , 17 . |
| MLA | Shi, Shengtao et al. "Scalar auxiliary variable approache for the surface quasi-geostrophic equation" . | JOURNAL OF ALGORITHMS & COMPUTATIONAL TECHNOLOGY 17 (2023) . |
| APA | Shi, Shengtao , Song, Fangying . Scalar auxiliary variable approache for the surface quasi-geostrophic equation . | JOURNAL OF ALGORITHMS & COMPUTATIONAL TECHNOLOGY , 2023 , 17 . |
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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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研究科氏力下带有内表面张力的不可压缩黏弹性流体的瑞利—泰勒问题的稳定性.建立稳定性准则，并在该准则下证明了瑞利—泰勒问题的解关于时间的指数稳定性.由于所得到的稳定性准则不依赖于旋转速度大小，故稳定性结果表明在瑞利—泰勒问题中科氏力不会产生失稳效果.
Keyword ：
Stokes估计 Stokes估计 瑞利-泰勒问题 瑞利-泰勒问题 科氏力 科氏力 稳定性 稳定性 稳定条件 稳定条件 黏弹性流体 黏弹性流体
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| GB/T 7714 | 吴兴奇 , 刘蒙蒙 , 宋方应 . 科氏力下带有内表面张力的黏弹性流体的瑞利-泰勒问题的稳定性 [J]. | 福州大学学报(自然科学版) , 2022 , 50 (02) : 155-160 . |
| MLA | 吴兴奇 et al. "科氏力下带有内表面张力的黏弹性流体的瑞利-泰勒问题的稳定性" . | 福州大学学报(自然科学版) 50 . 02 (2022) : 155-160 . |
| APA | 吴兴奇 , 刘蒙蒙 , 宋方应 . 科氏力下带有内表面张力的黏弹性流体的瑞利-泰勒问题的稳定性 . | 福州大学学报(自然科学版) , 2022 , 50 (02) , 155-160 . |
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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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研究旋转作用下无磁耗散的磁Rayleigh-Bénard问题的稳定性.建立了旋转作用下的磁Rayleigh-Bénard问题的稳定性条件，并在该条件下证明磁Rayleigh-Bénard问题存在全局的稳定性解，该解关于时间代数衰减.
Keyword ：
Rayleigh-Bénard问题 Rayleigh-Bénard问题 热不稳定性 热不稳定性 磁流体 磁流体 科氏力 科氏力
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| GB/T 7714 | 邵慧敏 , 刘蒙蒙 , 宋方应 . 旋转作用下无磁耗散的磁Rayleigh-Bénard问题的稳定性 [J]. | 福州大学学报(自然科学版) , 2022 , 50 (04) : 447-451 . |
| MLA | 邵慧敏 et al. "旋转作用下无磁耗散的磁Rayleigh-Bénard问题的稳定性" . | 福州大学学报(自然科学版) 50 . 04 (2022) : 447-451 . |
| APA | 邵慧敏 , 刘蒙蒙 , 宋方应 . 旋转作用下无磁耗散的磁Rayleigh-Bénard问题的稳定性 . | 福州大学学报(自然科学版) , 2022 , 50 (04) , 447-451 . |
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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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| 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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