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THE DECAY ESTIMATES FOR MAGNETOHYDRODYNAMIC EQUATIONS WITH COULOMB FORCE

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Acta Mathematica Scientia 2020年第6期40卷 1928-1940页

作者： Wenxuan ZHENG Zhong TAN School of Mathematical Sciences Xiamen UniversityXiamen 361005China School of Mechanical and Electronic Engineering Tarim UniversityAlar 843300China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing Xiamen UniversityXiamen 361005China

In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb *** spectral analysis and energy methods,we obtain the optimal time decay estimate of the *** show that the global classi... 详细信息

关键词： lower convergence rates upper decay rates spectral analysis energy method

Deepstsnet: Reconstructing the Quantum State-Resolved Thermochemical Nonequilibrium Flowfield Using Deep Neural Operator Learning With Scarce Data

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SSRN 2023年

作者： Lv, Jiaqi Hong, Qizhen Wang, Xiaoyong Mao, Zhiping Sun, Quanhua State Key Laboratory of High Temperature Gas Dynamics Institute of Mechanics Chinese Academy of Sciences Beijing100190 China School of Engineering Science University of Chinese Academy of Sciences Beijing100049 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen361005 China

The hypersonic flow is in a thermochemical nonequilibrium state due to the high-temperature caused by the strong shock compression. In a thermochemical nonequilibrium flow, the distribution of molecular internal energy levels strongly deviates from the equilibrium distribution (i.e., the Boltzmann distribution). It is intractable to directly obtain the microscopic nonequilibrium distribution from existed experimental measurements usually described by macroscopic field variables such as temperature or velocity. Motivated by the idea of deep multi-scale multi-physics neural network (DeepMMNet) proposed in [Mao et al, J. Comput. Phys., 2021], we develop in this paper a data assimilation framework called DeepStSNet to accurately reconstruct the quantum state-resolved thermochemical nonequilibrium flowfield by using sparse experimental measurements of vibrational temperature and pre-trained deep neural operator networks (DeepONets). In particular, we first construct several DeepONets to express the coupled dynamics between field variables in the thermochemical nonequilibrium flow and to approximate the state-to-state (StS) approach, which traces the variation of each vibrational level of molecule accurately. These proposed DeepONets are then trained by using the numerical simulation data, and would later be served as building blocks for the DeepStSNet. We demonstrate the effectiveness and accuracy of DeepONets with different test cases showing that the density and energy of vibrational groups as well as the temperature and velocity fields are predicted with high accuracy. We then extend the architectures of DeepMMNet by considering a simplified thermochemical nonequilibrium model, i.e., the 2T model, showing that the entire thermochemical nonequilibrium flowfield is well predicted by using scattered measurements of full or even partial field variables. We next consider a more accurate and complex thermochemical nonequilibrium model, i.e., the StS-CGM model, and develop a

关键词： Multiphysics

High-Order Discontinuous Galerkin Schemes for Radiation Hydrodynamics Equations in the Equilibrium-Diffusion Limit

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SSRN 2023年

作者： Peng, Gang Luo, Dongmi Qiu, Jianxian Chen, Yibing Command and Control Engineering College Army Engineering University of PLA Nanjing210007 China Institute of Applied Physics and Computational Mathematics National Key Laboratory of Computational Physics Beijing100088 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, the high-order discontinuous Galerkin (DG) schemes are presented for radiation hydrodynamics equations in the equilibrium-diffusion limit. The governor equations contain two parts, one part is a hyperbolic system of conservation laws and another part is a radiative heat transfer term. The discrete schemes are constructed in three parts and possess characteristics of the high-order, oscillation-free and efficient. Firstly, the DG method and local DG (LDG) method are respectively used to discretize the hyperbolic system and radiative heat transfer term to achieve high-order spatial accuracy. Secondly, the combination of the KXRCF shock detector, Hermite WENO limiter, and positivity-preserving limiter is utilized to enhance the robustness of the discretization schemes. Finally, the implicit-explicit schemes are employed for time discretization to enable the schemes suitable for larger time steps. Numerical experiments have verified that the new schemes are high-order accuracy, oscillation-free and efficiency. © 2023, The Authors. All rights reserved.

关键词： Hydrodynamics

High Order Implicit Finite Difference Schemes with a Semi-Implicit Weno Reconstruction for Nonlinear Degenerate Parabolic Equations

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SSRN 2022年

作者： Zhang, Peng Xiong, Tao School of Mathematical Sciences Xiamen University Fujian Xiamen361005 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, we develop implicit finite difference schemes with a semi-implicit weighted essentially non-oscillatory (WENO) reconstruction for nonlinear degenerate parabolic equations. Such degenerate equations would exist finite speed front propagation which is similar to discontinuous solutions for hyperbolic equations. We propose to use the finite difference WENO scheme with an implicit time discretization, to avoid the parabolic time step condition from an explicit time discretization, namely ∆t = O(∆x2) where ∆x is the mesh size. We further simplify the resulting nonlinear system by employing a semi-implicit WENO reconstruction, that is, the nonlinear weights are treated explicitly while keeping the linear reconstruction of the derivatives implicitly. The resulting scheme maintains the original high order accuracy, while saves much more computational cost than solving a fully implicit nonlinear system. Numerical experiments are performed to show the high order accuracy, large time step condition with better efficiency as compared to an explicit scheme, and good performance for capturing front propagation. © 2022, The Authors. All rights reserved.

关键词： Finite difference method

A prediction-correction based iterative convolution-thresholding method for topology optimization of heat transfer problems

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arXiv

arXiv 2024年

作者： Chen, Huangxin Dong, Piaopiao Wang, Dong Wang, Xiao-Ping School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian361005 China School of Science and Engineering The Chinese University of Hong Kong Guangdong Shenzhen518172 China Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big Data Guangdong518172 China

In this paper, we propose an iterative convolution-thresholding method (ICTM) based on prediction-correction for solving the topology optimization problem in steady-state heat transfer equations. The problem is formulated as a constrained minimization problem of the complementary energy, incorporating a perimeter/surface-area regularization term, while satisfying a steady-state heat transfer equation. The decision variables of the optimization problem represent the domains of different materials and are represented by indicator functions. The perimeter/surface-area term of the domain is approximated using Gaussian kernel convolution with indicator functions. In each iteration, the indicator function is updated using a prediction-correction approach. The prediction step is based on the variation of the objective functional by imposing the constraints, while the correction step ensures the monotonically decreasing behavior of the objective functional. Numerical results demonstrate the efficiency and robustness of our proposed method, particularly when compared to classical approaches based on the ICTM. Copyright © 2024, The Authors. All rights reserved.

关键词： Convolution

NON-SPLITTING EULERIAN-LAGRANGIAN WENO SCHEMES FOR TWO-DIMENSIONAL NONLINEAR CONVECTION-DIFFUSION EQUATIONS

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arXiv

arXiv 2024年

作者： Zheng, Nanyi Cai, Xiaofeng Qiu, Jing-Mei Qiu, Jianxian Department of Mathematical Sciences University of Delaware NewarkDE19716 United States Research Center for Mathematics Advanced Institute of Natural Sciences Beijing Normal University Zhuhai519087 China Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science BNU-HKBU United International College Zhuhai519087 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme. © 2024, CC BY.

关键词： Mesh generation

A Prediction-Correction Based Iterative Convolution-Thresholding Method for Topology Optimization of Heat Transfer Problems

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SSRN 2024年

关键词： Convolution

A Compact Simple Hweno Scheme with Ader Time Discretization for Hyperbolic Conservation Laws I: Structured Meshes

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SSRN 2023年

作者： Luo, Dongmi Li, Shiyi Qiu, Jianxian Zhu, Jun Chen, Yibing Institute of Applied Physics and Computational Mathematics Beijing100088 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China MIIT Nanjing University of Aeronautics and Astronautics Jiangsu Nanjing210016 China

In this paper, a compact and high order ADER (Arbitrary high order using DERivatives) scheme using the simple HWENO method (ADER-SHWENO) is proposed for hyperbolic conservation laws. The newly-developed method employs the Lax-Wendroff procedure to convert time derivatives to spatial derivatives, which provides the time evolution of the variables at the cell interfaces. This information is required for the simple HWENO reconstructions, which take advantages of the simple WENO and the classic HWENO. Compared with the original Runge-Kutta HWENO method (RK-HWENO), the new method has two advantages. Firstly, RK-HWENO method must solve the additional equations for reconstructions, which is avoided for the new method. Secondly, the SHWENO reconstruction is performed once with one stencil and is different from the classic HWENO methods, in which both the function and its derivative values are reconstructed with two different stencils, respectively. Thus the new method is more efficient than the RK-HWENO method. Moreover, the new method is more compact than the existing ADER-WENO method. Besides, the new method makes the best use of the information in the ADER method. Thus, the time evolution of the cell averages of the derivatives is simpler than that developed in the work [Li et. al., 447 (2021), 110661.]. Numerical tests indicate that the new method can achieve high order for smooth solutions both in space and time, keep non-oscillatory at discontinuities. © 2023, The Authors. All rights reserved.

关键词： Runge Kutta methods

A compact simple HWENO scheme with ADER time discretization for hyperbolic conservation laws I: structured meshes

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arXiv

arXiv 2023年

关键词： Runge Kutta methods