Pub Date : 2024-01-01DOI: 10.4208/cicp.oa-2023-0184
Xuyang Na, Xuejun Xu
In this paper, we revisit some nonoverlapping domain decomposition methods for solving diffusion problems with discontinuous coefficients. We discover some interesting phenomena, that is, the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients in some special cases. Detailedly, in the case of two subdomains, we find that their convergence rates are $mathcal{O}(ν_1/ν_2)$ if $ν_1 < ν_2,$ where $ν_1, ν_2$ are coefficients of two subdomains. Moreover, in the case of many subdomains with red-black partition, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are $1+epsilon(1+{rm log}(H/h))^2$ and $C+epsilon(1+ {rm log}(H/h))^2,$ respectively, where $epsilon$ equals ${rm min}{ν_R/ν_B,ν_B/ν_R}$ and $ν_R,ν_B$ are the coefficients of red and black domains. By contrast, Neumann-Neumann algorithm and Dirichlet-Dirichlet algorithm could not obtain such good convergence results in these cases. Finally, numerical experiments are preformed to confirm our findings.
{"title":"Domain Decomposition Methods for Diffusion Problems with Discontinuous Coefficients Revisited","authors":"Xuyang Na, Xuejun Xu","doi":"10.4208/cicp.oa-2023-0184","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0184","url":null,"abstract":"In this paper, we revisit some nonoverlapping domain decomposition methods for solving diffusion problems with discontinuous coefficients. We discover some\u0000interesting phenomena, that is, the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients in some special cases. Detailedly,\u0000in the case of two subdomains, we find that their convergence rates are $mathcal{O}(ν_1/ν_2)$ if $ν_1 < ν_2,$ where $ν_1, ν_2$ are coefficients of two subdomains. Moreover, in the case of\u0000many subdomains with red-black partition, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are $1+epsilon(1+{rm log}(H/h))^2$ and $C+epsilon(1+ {rm log}(H/h))^2,$ respectively, where $epsilon$ equals ${rm min}{ν_R/ν_B,ν_B/ν_R}$ and $ν_R,ν_B$ are the coefficients of red and black domains. By contrast, Neumann-Neumann algorithm and\u0000Dirichlet-Dirichlet algorithm could not obtain such good convergence results in these\u0000cases. Finally, numerical experiments are preformed to confirm our findings.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"17 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139665098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01DOI: 10.4208/cicp.oa-2023-0189
Ang Li,Hongtao Yang,Yulong Gao, Yonghai Li
This paper is devoted to constructing and analyzing a new upwind finite volume element method for anisotropic convection-diffusion-reaction problems on general quadrilateral meshes. We prove the coercivity, and establish the optimal error estimates in $H^1$ and $L^2$ norm respectively. The novelty is the discretization of convection term, which takes the two terms Taylor expansion. This scheme has not only optimal first-order accuracy in $H^1$ norm, but also optimal second-order accuracy in $L^2$ norm, both for dominant diffusion and dominant convection. Numerical experiments confirm the theoretical results.
{"title":"A New Upwind Finite Volume Element Method for Convection-Diffusion-Reaction Problems on Quadrilateral Meshes","authors":"Ang Li,Hongtao Yang,Yulong Gao, Yonghai Li","doi":"10.4208/cicp.oa-2023-0189","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0189","url":null,"abstract":"This paper is devoted to constructing and analyzing a new upwind finite\u0000volume element method for anisotropic convection-diffusion-reaction problems on\u0000general quadrilateral meshes. We prove the coercivity, and establish the optimal error estimates in $H^1$ and $L^2$ norm respectively. The novelty is the discretization of convection term, which takes the two terms Taylor expansion. This scheme has not only\u0000optimal first-order accuracy in $H^1$ norm, but also optimal second-order accuracy in $L^2$ norm, both for dominant diffusion and dominant convection. Numerical experiments\u0000confirm the theoretical results.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"22 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is proposed for simulating compressible multi-material flows on the adaptive quadrilateral meshes. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of the fluid mixture. The coupled system is discretized in the reference element and we use a kind of Taylor expansion basis functions to construct the interpolation polynomials of the variables. We show the property that the material derivatives of the basis functions in the DG discretization are equal to zero, with which the scheme is simplified. In addition, the mesh velocity in the ALE framework is obtained by using the adaptive mesh method from [H.Z. Tang and T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. NUMER. ANAL]. This adaptive mesh method can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduces the numerical dissipation near the material interfaces in the simulations. With the help of this adaptive mesh method, the resolution of the solution near the target regions can be greatly improved and the computational efficiency of the simulation is increased. Our scheme can be applied in the simulations for the gas and water media efficiently, and it is more concise compared to some other methods such as the indirect ALE methods. Several examples including the gas-water flow problem are presented to demonstrate the efficiency of our scheme, and the results show that our scheme can capture the wave structures sharply with high robustness.
本文提出了一种直接任意拉格朗日-欧拉（ALE）非连续加勒金（DG）方案，用于模拟自适应四边形网格上的可压缩多材料流动。我们的方案将与体积分数模型相关的保守方程与描述流体混合物动力学的欧拉方程耦合在一起。耦合系统在参考元素中离散化，我们使用一种泰勒扩展基函数来构建变量的插值多项式。我们证明了 DG 离散中基函数的材料导数等于零的特性，从而简化了方案。此外，ALE 框架中的网格速度是通过使用自适应网格方法获得的[H.Z. Tang and T. Tang, Adaptive mesh methodsfor one and two-dimensional hyperbolic conservation laws, SIAM J. NUMER.这种自适应网格方法可以自动将网格节点集中在梯度值较大的区域附近，大大减少了模拟中材料界面附近的数值耗散。在这种自适应网格方法的帮助下，目标区域附近的解的分辨率可以大大提高，模拟的计算效率也随之提高。我们的方案可以有效地应用于气体和水介质的模拟，与其他方法（如间接 ALE 方法）相比更加简洁。为了证明我们的方案的高效性，我们给出了包括气体-水流问题在内的几个例子，结果表明我们的方案可以清晰地捕捉波浪结构，并具有很高的鲁棒性。
{"title":"An Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Scheme for Compressible Multi-Material Flows on Adaptive Quadrilateral Meshes","authors":"Xiaolong Zhao,Shicang Song,Xijun Yu,Shijun Zou, Fang Qing","doi":"10.4208/cicp.oa-2023-0015","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0015","url":null,"abstract":"In this paper, a direct arbitrary Lagrangian-Eulerian (ALE) discontinuous\u0000Galerkin (DG) scheme is proposed for simulating compressible multi-material flows\u0000on the adaptive quadrilateral meshes. Our scheme couples a conservative equation\u0000related to the volume-fraction model with the Euler equations for describing the dynamics of the fluid mixture. The coupled system is discretized in the reference element\u0000and we use a kind of Taylor expansion basis functions to construct the interpolation\u0000polynomials of the variables. We show the property that the material derivatives of\u0000the basis functions in the DG discretization are equal to zero, with which the scheme\u0000is simplified. In addition, the mesh velocity in the ALE framework is obtained by using the adaptive mesh method from [H.Z. Tang and T. Tang, Adaptive mesh methods\u0000for one-and two-dimensional hyperbolic conservation laws, SIAM J. NUMER. ANAL].\u0000This adaptive mesh method can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduces the numerical dissipation near\u0000the material interfaces in the simulations. With the help of this adaptive mesh method,\u0000the resolution of the solution near the target regions can be greatly improved and the\u0000computational efficiency of the simulation is increased. Our scheme can be applied in\u0000the simulations for the gas and water media efficiently, and it is more concise compared\u0000to some other methods such as the indirect ALE methods. Several examples including\u0000the gas-water flow problem are presented to demonstrate the efficiency of our scheme, and the results show that our scheme can capture the wave structures sharply with\u0000high robustness.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"37 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01DOI: 10.4208/cicp.oa-2022-0197
Qianting Ma,Yang Wang, Tieyong Zeng
Images captured under insufficient light conditions often suffer from noticeable degradation of visibility, brightness and contrast. Existing methods pose limitations on enhancing low-visibility images, especially for diverse low-light conditions. In this paper, we first propose a new variational model for estimating the illumination map based on fractional-order differential. Once the illumination map is obtained, we directly inject the well-constructed illumination map into a general image restoration model, whose regularization terms can be viewed as an adaptive mapping. Since the regularization term in the restoration part can be arbitrary, one can model the regularization term by using different off-the-shelf denoisers and do not need to explicitly design various priors on the reflectance component. Because of flexibility of the model, the desired enhanced results can be solved efficiently by techniques like the plug-and-play inspired algorithm. Numerical experiments based on three public datasets demonstrate that our proposed method outperforms other competing methods, including deep learning approaches, under three commonly used metrics in terms of visual quality and image quality assessment.
{"title":"Variational Low-Light Image Enhancement Based on Fractional-Order Differential","authors":"Qianting Ma,Yang Wang, Tieyong Zeng","doi":"10.4208/cicp.oa-2022-0197","DOIUrl":"https://doi.org/10.4208/cicp.oa-2022-0197","url":null,"abstract":"Images captured under insufficient light conditions often suffer from noticeable degradation of visibility, brightness and contrast. Existing methods pose limitations on enhancing low-visibility images, especially for diverse low-light conditions.\u0000In this paper, we first propose a new variational model for estimating the illumination\u0000map based on fractional-order differential. Once the illumination map is obtained,\u0000we directly inject the well-constructed illumination map into a general image restoration model, whose regularization terms can be viewed as an adaptive mapping. Since\u0000the regularization term in the restoration part can be arbitrary, one can model the\u0000regularization term by using different off-the-shelf denoisers and do not need to explicitly design various priors on the reflectance component. Because of flexibility of\u0000the model, the desired enhanced results can be solved efficiently by techniques like\u0000the plug-and-play inspired algorithm. Numerical experiments based on three public\u0000datasets demonstrate that our proposed method outperforms other competing methods, including deep learning approaches, under three commonly used metrics in terms\u0000of visual quality and image quality assessment.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"172 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01DOI: 10.4208/cicp.oa-2023-0121
Simin Shekarpaz,Fanhai Zeng, George Karniadakis
We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the challenge of applying PINNs to forward dynamical systems and demonstrates improved accuracy through its application to neuron models. Specifically, we apply operator splitting to decompose the original neuron model into sub-problems that are then solved using PINNs. Moreover, we develop an $L^1$ scheme for discretizing fractional derivatives in fractional neuron models, leading to improved accuracy and efficiency. The results of this study highlight the potential of splitting PINNs in solving both integer- and fractional-order neuron models, as well as other similar systems in computational science and engineering.
{"title":"Splitting Physics-Informed Neural Networks for Inferring the Dynamics of Integer- and Fractional-Order Neuron Models","authors":"Simin Shekarpaz,Fanhai Zeng, George Karniadakis","doi":"10.4208/cicp.oa-2023-0121","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0121","url":null,"abstract":"We introduce a new approach for solving forward systems of differential\u0000equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the challenge of applying PINNs to forward dynamical systems and demonstrates improved\u0000accuracy through its application to neuron models. Specifically, we apply operator\u0000splitting to decompose the original neuron model into sub-problems that are then\u0000solved using PINNs. Moreover, we develop an $L^1$ scheme for discretizing fractional\u0000derivatives in fractional neuron models, leading to improved accuracy and efficiency.\u0000The results of this study highlight the potential of splitting PINNs in solving both\u0000integer- and fractional-order neuron models, as well as other similar systems in computational science and engineering.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"255 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we extend the unified gas kinetic particle (UGKP) method to the frequency-dependent radiative transfer equation with both absorption-emission and scattering processes. The extended UGKP method could capture the diffusion and free transport limit and provide a smooth transition in the physical and frequency space in the regime between the above two limits. The proposed scheme has the properties of asymptotic-preserving and regime-adaptive, which make it an accurate and efficient scheme in the simulation of multiscale photon transport problems. In the UGKP formulation of flux construction and distribution closure, the coefficients of the non-equilibrium free stream distribution and near-equilibrium Planck expansion are independent of the time step. Therefore, even with a large CFL number, the UGKP can preserve a physically consistent ratio of the non-equilibrium and the near-equilibrium proportion. The methodology of scheme construction is a coupled evolution of the macroscopic energy equation and the microscopic radiant intensity equation, where the numerical flux in the macroscopic energy equation and the closure in the microscopic radiant intensity equation are constructed based on the integral solution. Both numerical dissipation and computational complexity are well controlled, especially in the optically thick regime. 2D multi-thread code on a general unstructured mesh has been developed. Several numerical tests have been simulated to verify the numerical scheme and code, covering a wide range of flow regimes. The numerical scheme and code we developed are highly demanded and widely applicable in high-energy engineering applications.
{"title":"A Unified Gas-Kinetic Particle Method for Frequency-Dependent Radiative Transfer Equations with Isotropic Scattering Process on Unstructured Mesh","authors":"Yuan Hu,Chang Liu,Huayun Shen,Shiyang Zou, Baolin Tian","doi":"10.4208/cicp.oa-2023-0161","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0161","url":null,"abstract":"In this paper, we extend the unified gas kinetic particle (UGKP) method to\u0000the frequency-dependent radiative transfer equation with both absorption-emission\u0000and scattering processes. The extended UGKP method could capture the diffusion\u0000and free transport limit and provide a smooth transition in the physical and frequency\u0000space in the regime between the above two limits. The proposed scheme has the properties of asymptotic-preserving and regime-adaptive, which make it an accurate and\u0000efficient scheme in the simulation of multiscale photon transport problems. In the\u0000UGKP formulation of flux construction and distribution closure, the coefficients of the\u0000non-equilibrium free stream distribution and near-equilibrium Planck expansion are\u0000independent of the time step. Therefore, even with a large CFL number, the UGKP can\u0000preserve a physically consistent ratio of the non-equilibrium and the near-equilibrium\u0000proportion. The methodology of scheme construction is a coupled evolution of the\u0000macroscopic energy equation and the microscopic radiant intensity equation, where\u0000the numerical flux in the macroscopic energy equation and the closure in the microscopic radiant intensity equation are constructed based on the integral solution. Both\u0000numerical dissipation and computational complexity are well controlled, especially in\u0000the optically thick regime. 2D multi-thread code on a general unstructured mesh has\u0000been developed. Several numerical tests have been simulated to verify the numerical scheme and code, covering a wide range of flow regimes. The numerical scheme\u0000and code we developed are highly demanded and widely applicable in high-energy\u0000engineering applications.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"172 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01DOI: 10.4208/cicp.oa-2023-0163
Qiuqi Li,Chang Liu,Mengnan Li, Pingwen Zhang
Parametric dynamical systems are widely used to model physical systems, but their numerical simulation can be computationally demanding due to nonlinearity, long-time simulation, and multi-query requirements. Model reduction methods aim to reduce computation complexity and improve simulation efficiency. However, traditional model reduction methods are inefficient for parametric dynamical systems with nonlinear structures. To address this challenge, we propose an adaptive method based on local dynamic mode decomposition (DMD) to construct an efficient and reliable surrogate model. We propose an improved greedy algorithm to generate the atoms set $Theta$ based on a sequence of relatively small training sets, which could reduce the effect of large training set. At each enrichment step, we construct a local sub-surrogate model using the Taylor expansion and DMD, resulting in the ability to predict the state at any time without solving the original dynamical system. Moreover, our method provides the best approximation almost everywhere over the parameter domain with certain smoothness assumptions, thanks to the gradient information. At last, three concrete examples are presented to illustrate the effectiveness of the proposed method.
{"title":"An Adaptive Method Based on Local Dynamic Mode Decomposition for Parametric Dynamical Systems","authors":"Qiuqi Li,Chang Liu,Mengnan Li, Pingwen Zhang","doi":"10.4208/cicp.oa-2023-0163","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0163","url":null,"abstract":"Parametric dynamical systems are widely used to model physical systems,\u0000but their numerical simulation can be computationally demanding due to nonlinearity,\u0000long-time simulation, and multi-query requirements. Model reduction methods aim\u0000to reduce computation complexity and improve simulation efficiency. However, traditional model reduction methods are inefficient for parametric dynamical systems with\u0000nonlinear structures. To address this challenge, we propose an adaptive method based\u0000on local dynamic mode decomposition (DMD) to construct an efficient and reliable\u0000surrogate model. We propose an improved greedy algorithm to generate the atoms set $Theta$ based on a sequence of relatively small training sets, which could reduce the effect of\u0000large training set. At each enrichment step, we construct a local sub-surrogate model\u0000using the Taylor expansion and DMD, resulting in the ability to predict the state at any\u0000time without solving the original dynamical system. Moreover, our method provides\u0000the best approximation almost everywhere over the parameter domain with certain\u0000smoothness assumptions, thanks to the gradient information. At last, three concrete\u0000examples are presented to illustrate the effectiveness of the proposed method.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"38 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139664728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01DOI: 10.4208/cicp.oa-2023-0141
Pei Zhang,Yanli Wang, Zhennan Zhou
In this work, we consider the Fokker-Planck equation of the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model for neuron networks. Due to the firing events of neurons at the microscopic level, this Fokker-Planck equation contains dynamic boundary conditions involving specific internal points. To efficiently solve this problem and explore the properties of the unknown, we construct a flexible numerical scheme for the Fokker-Planck equation in the framework of spectral methods that can accurately handle the dynamic boundary condition. This numerical scheme is stable with suitable choices of test function spaces, and asymptotic preserving, and it is easily extendable to variant models with multiple time scales. We also present extensive numerical examples to verify the scheme properties, including order of convergence and time efficiency, and explore unique properties of the model, including blow-up phenomena for the NNLIF model and learning and discriminative properties for the NNLIF model with learning rules.
{"title":"A Spectral Method for a Fokker-Planck Equation in Neuroscience with Applications in Neuron Networks with Learning Rules","authors":"Pei Zhang,Yanli Wang, Zhennan Zhou","doi":"10.4208/cicp.oa-2023-0141","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0141","url":null,"abstract":"In this work, we consider the Fokker-Planck equation of the Nonlinear Noisy\u0000Leaky Integrate-and-Fire (NNLIF) model for neuron networks. Due to the firing events\u0000of neurons at the microscopic level, this Fokker-Planck equation contains dynamic\u0000boundary conditions involving specific internal points. To efficiently solve this problem and explore the properties of the unknown, we construct a flexible numerical\u0000scheme for the Fokker-Planck equation in the framework of spectral methods that can\u0000accurately handle the dynamic boundary condition. This numerical scheme is stable\u0000with suitable choices of test function spaces, and asymptotic preserving, and it is easily extendable to variant models with multiple time scales. We also present extensive\u0000numerical examples to verify the scheme properties, including order of convergence\u0000and time efficiency, and explore unique properties of the model, including blow-up\u0000phenomena for the NNLIF model and learning and discriminative properties for the\u0000NNLIF model with learning rules.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"297 2 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139664987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0136
Shuai Miao,Jiming Wu, Yanzhong Yao
Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is distorted or the problem is discontinuous, so interpolation algorithms of auxiliary unknowns are required. Interpolation algorithms are not only difficult to construct, but also bring extra computation. In this paper, an interpolation-free cell-centered finite volume scheme is proposed for the heterogeneous and anisotropic convection-diffusion problems on arbitrary polyhedral meshes. We propose a new interpolation-free discretization method for diffusion term, and two new second-order upwind algorithms for convection term. Most interestingly, the scheme can be adapted to any mesh topology and can handle any discontinuity strictly. Numerical experiments show that this new scheme is robust, possesses a small stencil, and has approximately second-order accuracy for both diffusion-dominated and convection-dominated problems.
{"title":"An Interpolation-Free Cell-Centered Finite Volume Scheme for 3D Anisotropic Convection-Diffusion Equations on Arbitrary Polyhedral Meshes","authors":"Shuai Miao,Jiming Wu, Yanzhong Yao","doi":"10.4208/cicp.oa-2023-0136","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0136","url":null,"abstract":"Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is\u0000distorted or the problem is discontinuous, so interpolation algorithms of auxiliary\u0000unknowns are required. Interpolation algorithms are not only difficult to construct,\u0000but also bring extra computation. In this paper, an interpolation-free cell-centered finite volume scheme is proposed for the heterogeneous and anisotropic convection-diffusion problems on arbitrary polyhedral meshes. We propose a new interpolation-free discretization method for diffusion term, and two new second-order upwind algorithms for convection term. Most interestingly, the scheme can be adapted to any mesh\u0000topology and can handle any discontinuity strictly. Numerical experiments show that\u0000this new scheme is robust, possesses a small stencil, and has approximately second-order accuracy for both diffusion-dominated and convection-dominated problems.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"2 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138580166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0083
Xiaolu Gu,Juan Cheng, Chiwang Shu
The arbitrary Lagrangian-Eulerian (ALE) method is widely used in the field of compressible multi-material and multi-phase flow problems. In order to implement the indirect ALE approach for the simulation of compressible flow in the context of high order discontinuous Galerkin (DG) discretizations, we present a high order positivity-preserving DG remapping method based on a moving mesh solver in this paper. This remapping method is based on the ALE-DG method developed by Klingenberg et al. [17, 18] to solve the trivial equation $frac{∂u}{∂t} = 0$ on a moving mesh, which is the old mesh before remapping at $t = 0$ and is the new mesh after remapping at $t = T.$ An appropriate selection of the final pseudo-time $T$ can always satisfy the relatively mild smoothness requirement (Lipschitz continuity) on the mesh movement velocity, which guarantees the high order accuracy of the remapping procedure. We use a multi-resolution weighted essentially non-oscillatory (WENO) limiter which can keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth regions. We further employ an effective linear scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order accuracy. Numerical experiments are provided to illustrate the high order accuracy, essentially non-oscillatory performance and positivity-preserving of our remapping algorithm. In addition, the performance of the ALE simulation based on the DG framework with our remapping algorithm is examined in one- and two-dimensional Euler equations.
任意拉格朗日-欧勒（ALE）方法被广泛应用于可压缩多材料和多相流问题领域。为了在高阶非连续伽勒金（DG）离散化背景下实现可压缩流动模拟的间接 ALE 方法，我们在本文中提出了一种基于移动网格求解器的高阶正向保留 DG 重映射方法。这种重映射方法基于 Klingenberg 等人开发的 ALE-DG 方法[17, 18]，在移动网格上求解三元方程 $frac{∂u}{∂t} = 0$，移动网格是在 $t = 0$ 时重映射前的旧网格和在 $t = T 时重映射后的新网格。最终伪时间 $T$ 的适当选择总能满足对网格移动速度相对温和的平滑性要求（Lipschitz 连续性），从而保证重映射过程的高阶精度。我们使用了多分辨率加权本质非振荡（WENO）限制器，它可以在强不连续性附近保持本质非振荡特性，同时在平滑区域保持高阶精度。我们进一步采用了有效的线性缩放限制器，在不牺牲守恒性和原有高阶精度的情况下，保持相关物理变量的正向性。我们提供了数值实验，以说明我们的重映射算法具有高阶精度、基本无振荡性能和保留正性的特点。此外，还在一维和二维欧拉方程中检验了基于 DG 框架的 ALE 仿真与我们的重映射算法的性能。
{"title":"A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow","authors":"Xiaolu Gu,Juan Cheng, Chiwang Shu","doi":"10.4208/cicp.oa-2023-0083","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0083","url":null,"abstract":"The arbitrary Lagrangian-Eulerian (ALE) method is widely used in the field\u0000of compressible multi-material and multi-phase flow problems. In order to implement the indirect ALE approach for the simulation of compressible flow in the context\u0000of high order discontinuous Galerkin (DG) discretizations, we present a high order\u0000positivity-preserving DG remapping method based on a moving mesh solver in this\u0000paper. This remapping method is based on the ALE-DG method developed by Klingenberg et al. [17, 18] to solve the trivial equation $frac{∂u}{∂t} = 0$ on a moving mesh, which\u0000is the old mesh before remapping at $t = 0$ and is the new mesh after remapping at $t = T.$ An appropriate selection of the final pseudo-time $T$ can always satisfy the relatively mild smoothness requirement (Lipschitz continuity) on the mesh movement\u0000velocity, which guarantees the high order accuracy of the remapping procedure. We\u0000use a multi-resolution weighted essentially non-oscillatory (WENO) limiter which can\u0000keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth regions. We further employ an effective linear\u0000scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order accuracy. Numerical experiments are\u0000provided to illustrate the high order accuracy, essentially non-oscillatory performance\u0000and positivity-preserving of our remapping algorithm. In addition, the performance\u0000of the ALE simulation based on the DG framework with our remapping algorithm is\u0000examined in one- and two-dimensional Euler equations.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"25 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138567980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：