{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}