Pub Date : 2024-05-01DOI: 10.4208/cicp.oa-2023-0256
Menghuo Chen,Yuanqing Wu,Xiaoyu Feng, Shuyu Sun
In this study, we apply first-order exponential time differencing (ETD) methods to solve benchmark problems for the diffuse-interface model using the Peng-Robinson equation of state. We demonstrate the unconditional stability of the proposed algorithm within the ETD framework. Additionally, we analyzed the complexity of the algorithm, revealing that computations like matrix multiplications and inversions in each time step exhibit complexity strictly less than $mathcal{O}(n^2),$ where $n$ represents the number of variables or grid points. The main objective was to develop an algorithm with enhanced performance and robustness. To achieve this, we avoid iterative solutions (such as matrix inversion) in each time step, as they are sensitive to matrix properties. Instead, we adopted a hierarchical matrix ($mathcal{H}$-matrix) approximation for the matrix inverse and matrix exponential used in each time step. By leveraging hierarchical matrices with a rank $k ≪ n,$ we achieve a complexity of $O(kn{rm log}(n))$ for their product with an $n$-vector, which outperforms the traditional $mathcal{O}(n^2)$ complexity. Overall, our focus is on creating an unconditionally stable algorithm with improved computational efficiency and reliability.
{"title":"Unconstrained ETD Methods on the Diffuse-Interface Model with the Peng-Robinson Equation of State","authors":"Menghuo Chen,Yuanqing Wu,Xiaoyu Feng, Shuyu Sun","doi":"10.4208/cicp.oa-2023-0256","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0256","url":null,"abstract":"In this study, we apply first-order exponential time differencing (ETD) methods to solve benchmark problems for the diffuse-interface model using the Peng-Robinson equation of state. We demonstrate the unconditional stability of the proposed algorithm within the ETD framework. Additionally, we analyzed the complexity of the algorithm, revealing that computations like matrix multiplications and inversions in each time step exhibit complexity strictly less than $mathcal{O}(n^2),$ where $n$ represents\u0000the number of variables or grid points. The main objective was to develop an algorithm with enhanced performance and robustness. To achieve this, we avoid iterative\u0000solutions (such as matrix inversion) in each time step, as they are sensitive to matrix\u0000properties. Instead, we adopted a hierarchical matrix ($mathcal{H}$-matrix) approximation for\u0000the matrix inverse and matrix exponential used in each time step. By leveraging hierarchical matrices with a rank $k ≪ n,$ we achieve a complexity of $O(kn{rm log}(n))$ for\u0000their product with an $n$-vector, which outperforms the traditional $mathcal{O}(n^2)$ complexity.\u0000Overall, our focus is on creating an unconditionally stable algorithm with improved\u0000computational efficiency and reliability.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"18 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940517","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 : 2024-05-01DOI: 10.4208/cicp.oa-2023-0172
Yukun Li,Corey Prachniak, Yi Zhang
This paper proposes and analyzes a novel fully discrete finite element scheme with an interpolation operator for stochastic Cahn-Hilliard equations with functional-type noise. The nonlinear term satisfies a one-sided Lipschitz condition and the diffusion term is globally Lipschitz continuous. The novelties of this paper are threefold. Firstly, the $L^2$-stability ($L^∞$ in time) and $H^2$-stability ($L^2$ in time) are proved for the proposed scheme. The idea is to utilize the special structure of the matrix assembled by the nonlinear term. None of these stability results has been proved for the fully implicit scheme in existing literature due to the difficulty arising from the interaction of the nonlinearity and the multiplicative noise. Secondly, higher moment stability in $L^2$-norm of the discrete solution is established based on the previous stability results. Thirdly, the Hölder continuity in time for the strong solution is established under the minimum assumption of the strong solution. Based on these findings, the strong convergence in $H^{−1}$-norm of the discrete solution is discussed. Several numerical experiments including stability and convergence are also presented to validate our theoretical results.
{"title":"Analysis of a Mixed Finite Element Method for Stochastic Cahn-Hilliard Equation with Multiplicative Noise","authors":"Yukun Li,Corey Prachniak, Yi Zhang","doi":"10.4208/cicp.oa-2023-0172","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0172","url":null,"abstract":"This paper proposes and analyzes a novel fully discrete finite element scheme\u0000with an interpolation operator for stochastic Cahn-Hilliard equations with functional-type noise. The nonlinear term satisfies a one-sided Lipschitz condition and the diffusion term is globally Lipschitz continuous. The novelties of this paper are threefold.\u0000Firstly, the $L^2$-stability ($L^∞$ in time) and $H^2$-stability ($L^2$ in time) are proved for the\u0000proposed scheme. The idea is to utilize the special structure of the matrix assembled by the nonlinear term. None of these stability results has been proved for the\u0000fully implicit scheme in existing literature due to the difficulty arising from the interaction of the nonlinearity and the multiplicative noise. Secondly, higher moment\u0000stability in $L^2$-norm of the discrete solution is established based on the previous stability results. Thirdly, the Hölder continuity in time for the strong solution is established under the minimum assumption of the strong solution. Based on these findings, the\u0000strong convergence in $H^{−1}$-norm of the discrete solution is discussed. Several numerical experiments including stability and convergence are also presented to validate our\u0000theoretical results.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"6 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940616","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 : 2024-04-01DOI: 10.4208/cicp.oa-2023-0232
Mingzhu Zhang, Lijun Yi
The aim of this paper is to propose and analyze two postprocessing techniques for improving the accuracy of the $C^1$- and $C^0$-continuous Galerkin (CG) time stepping methods for nonlinear second-order initial value problems, respectively. We first derive several optimal a priori error estimates and nodal superconvergent estimates for the $C^1$- and $C^0$-$CG$ methods. Then we propose two simple but efficient local postprocessing techniques for the $C^1$- and $C^0$-$CG$ methods, respectively. The key idea of the postprocessing techniques is to add a certain higher order generalized Jacobi polynomial of degree $k+1$ to the $C^1$- or $C^0$-$CG$ approximation of degree $k$ on each local time step. We prove that, for problems with regular solutions, such postprocessing techniques improve the global convergence rates for the $L^2$-, $H^1$- and $L^∞$-error estimates of the $C^1$- and $C^0$-$CG$ methods with quasi-uniform meshes by one order. As applications, we apply the superconvergent postprocessing techniques to the $C^1$- and $C^0$-$CG$ time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.
{"title":"Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations","authors":"Mingzhu Zhang, Lijun Yi","doi":"10.4208/cicp.oa-2023-0232","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0232","url":null,"abstract":"The aim of this paper is to propose and analyze two postprocessing techniques for improving the accuracy of the $C^1$- and $C^0$-continuous Galerkin (CG) time\u0000stepping methods for nonlinear second-order initial value problems, respectively. We\u0000first derive several optimal a priori error estimates and nodal superconvergent estimates for the $C^1$- and $C^0$-$CG$ methods. Then we propose two simple but efficient local\u0000postprocessing techniques for the $C^1$- and $C^0$-$CG$ methods, respectively. The key idea\u0000of the postprocessing techniques is to add a certain higher order generalized Jacobi\u0000polynomial of degree $k+1$ to the $C^1$- or $C^0$-$CG$ approximation of degree $k$ on each local\u0000time step. We prove that, for problems with regular solutions, such postprocessing\u0000techniques improve the global convergence rates for the $L^2$-, $H^1$- and $L^∞$-error estimates of the $C^1$- and $C^0$-$CG$ methods with quasi-uniform meshes by one order. As\u0000applications, we apply the superconvergent postprocessing techniques to the $C^1$- and $C^0$-$CG$ time discretization of nonlinear wave equations. Several numerical examples\u0000are presented to verify the theoretical results.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"31 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583184","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 : 2024-04-01DOI: 10.4208/cicp.oa-2023-0196
Jingfeng Wang, Guanghui Hu
The dual consistency is an important issue in developing stable DWR error estimation towards the goal-oriented mesh adaptivity. In this paper, such an issue is studied in depth based on a Newton-GMG framework for the steady Euler equations. Theoretically, the numerical framework is redescribed using the Petrov-Galerkin scheme, based on which the dual consistency is depicted. It is found that for a problem with general configuration, a boundary modification technique is an effective approach to preserve the dual consistency in our numerical framework. Numerically, a geometrical multigrid is proposed for solving the dual problem, and a regularization term is designed to guarantee the convergence of the iteration. The following features of our method can be observed from numerical experiments, i). a stable numerical convergence of the quantity of interest can be obtained smoothly for problems with different configurations, and ii). towards accurate calculation of quantity of interest, mesh grids can be saved significantly using the proposed dual-consistent DWR method, compared with the dual-inconsistent one.
{"title":"Towards the Efficient Calculation of Quantity of Interest from Steady Euler Equations I: A Dual-Consistent DWR-Based $h$-Adaptive Newton-GMG Solver","authors":"Jingfeng Wang, Guanghui Hu","doi":"10.4208/cicp.oa-2023-0196","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0196","url":null,"abstract":"The dual consistency is an important issue in developing stable DWR error estimation towards the goal-oriented mesh adaptivity. In this paper, such an issue\u0000is studied in depth based on a Newton-GMG framework for the steady Euler equations. Theoretically, the numerical framework is redescribed using the Petrov-Galerkin\u0000scheme, based on which the dual consistency is depicted. It is found that for a problem with general configuration, a boundary modification technique is an effective approach to preserve the dual consistency in our numerical framework. Numerically,\u0000a geometrical multigrid is proposed for solving the dual problem, and a regularization term is designed to guarantee the convergence of the iteration. The following\u0000features of our method can be observed from numerical experiments, i). a stable numerical convergence of the quantity of interest can be obtained smoothly for problems\u0000with different configurations, and ii). towards accurate calculation of quantity of interest, mesh grids can be saved significantly using the proposed dual-consistent DWR\u0000method, compared with the dual-inconsistent one.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"47 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583187","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 : 2024-04-01DOI: 10.4208/cicp.oa-2023-0133
Chuwen Ma, Shi Jin
An implicit, asymptotic-preserving and energy-charge-conserving (APECC) Particle-In-Cell (PIC) method is proposed to solve the Vlasov-Maxwell (VM) equations in the quasi-neutral regime. Charge conservation is enforced by particle orbital averaging and fixed sub-time steps. The truncation error depending on the number of sub-time steps is further analyzed. The temporal discretization is chosen by the Crank-Nicolson method to conserve the discrete energy exactly. The key step in the asymptotic-preserving iteration for the nonlinear system is based on a decomposition of the current density deduced from the Vlasov equation in the source of the Maxwell model. Moreover, we show that the convergence is independent of the quasineutral parameter. Extensive numerical experiments show that the proposed method can achieve asymptotic preservation and energy-charge conservation.
{"title":"An Implicit, Asymptotic-Preserving and Energy-Charge-Conserving Method for the Vlasov-Maxwell System Near Quasi-Neutrality","authors":"Chuwen Ma, Shi Jin","doi":"10.4208/cicp.oa-2023-0133","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0133","url":null,"abstract":"An implicit, asymptotic-preserving and energy-charge-conserving (APECC)\u0000Particle-In-Cell (PIC) method is proposed to solve the Vlasov-Maxwell (VM) equations in the quasi-neutral regime. Charge conservation is enforced by particle orbital\u0000averaging and fixed sub-time steps. The truncation error depending on the number of sub-time steps is further analyzed. The temporal discretization is chosen by\u0000the Crank-Nicolson method to conserve the discrete energy exactly. The key step in\u0000the asymptotic-preserving iteration for the nonlinear system is based on a decomposition of the current density deduced from the Vlasov equation in the source of the\u0000Maxwell model. Moreover, we show that the convergence is independent of the quasineutral parameter. Extensive numerical experiments show that the proposed method\u0000can achieve asymptotic preservation and energy-charge conservation.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"93 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583012","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 : 2024-04-01DOI: 10.4208/cicp.oa-2023-0253
Janne Siipola
Deep Ritz method is a deep learning paradigm to solve partial differential equations. In this article we study the generalization error of the Deep Ritz method. We focus on the quintessential problem which is the Poisson’s equation. We show that generalization error of the Deep Ritz method converges to zero with rate $frac{C}{sqrt{n}},$ and we discuss about the constant $C.$ Results are obtained for shallow and residual neural networks with smooth activation functions.
Deep Ritz 方法是一种解决偏微分方程的深度学习范式。本文研究了 Deep Ritz 方法的泛化误差。我们证明了深度里兹方法的泛化误差以$frac{C}{sqrt{n}}的速率收敛为零，并讨论了常数$C。
{"title":"Generalization Error in the Deep Ritz Method with Smooth Activation Functions","authors":"Janne Siipola","doi":"10.4208/cicp.oa-2023-0253","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0253","url":null,"abstract":"Deep Ritz method is a deep learning paradigm to solve partial differential\u0000equations. In this article we study the generalization error of the Deep Ritz method.\u0000We focus on the quintessential problem which is the Poisson’s equation. We show that\u0000generalization error of the Deep Ritz method converges to zero with rate $frac{C}{sqrt{n}},$ and we\u0000discuss about the constant $C.$ Results are obtained for shallow and residual neural\u0000networks with smooth activation functions.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"117 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583009","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 : 2024-04-01DOI: 10.4208/cicp.oa-2023-0038
Wenbin Chen,Jianyu Jing,Qianqian Liu,Cheng Wang, Xiaoming Wang
A second order accurate in time, finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Navier-Stokes system, with logarithmic Flory-Huggins energy potential. In the numerical approximation to the chemical potential, a modified Crank-Nicolson approximation is applied to the singular logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. Moreover, a nonlinear artificial regularization term is included in the chemical potential approximation, which ensures the positivity-preserving property for the logarithmic arguments, i.e., the numerical value of the phase variable is always between −1 and 1 at a point-wise level. Meanwhile, the convective term in the phase field evolutionary equation is updated in a semi-implicit way, with second order accurate temporal approximation. The fluid momentum equation is also computed by a semi-implicit algorithm. The unique solvability and the positivity-preserving property of the second order scheme is proved, accomplished by an iteration process. A modified total energy stability of the second order scheme is also derived. Some numerical results are presented to demonstrate the accuracy and the robust performance of the proposed second order scheme.
{"title":"A Second Order Numerical Scheme of the Cahn-Hilliard-Navier-Stokes System with Flory-Huggins Potential","authors":"Wenbin Chen,Jianyu Jing,Qianqian Liu,Cheng Wang, Xiaoming Wang","doi":"10.4208/cicp.oa-2023-0038","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0038","url":null,"abstract":"A second order accurate in time, finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Navier-Stokes system, with logarithmic\u0000Flory-Huggins energy potential. In the numerical approximation to the chemical potential, a modified Crank-Nicolson approximation is applied to the singular logarithmic nonlinear term, while the expansive term is updated by an explicit second order\u0000Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. Moreover, a nonlinear artificial regularization term is included in\u0000the chemical potential approximation, which ensures the positivity-preserving property for the logarithmic arguments, i.e., the numerical value of the phase variable is\u0000always between −1 and 1 at a point-wise level. Meanwhile, the convective term in the\u0000phase field evolutionary equation is updated in a semi-implicit way, with second order\u0000accurate temporal approximation. The fluid momentum equation is also computed by\u0000a semi-implicit algorithm. The unique solvability and the positivity-preserving property of the second order scheme is proved, accomplished by an iteration process. A\u0000modified total energy stability of the second order scheme is also derived. Some numerical results are presented to demonstrate the accuracy and the robust performance\u0000of the proposed second order scheme.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"158 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583186","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 : 2024-04-01DOI: 10.4208/cicp.oa-2023-0002
Zhenming Wang,Jun Zhu,Linlin Tian, Ning Zhao
In this paper, a fifth-order hybrid multi-resolution weighted essentially non-oscillatory (WENO) scheme in the finite difference framework is proposed for solving one- and two-dimensional Hamilton-Jacobi equations. Firstly, a new discontinuity sensor is designed based on the extreme values of the highest degree polynomial in the multi-resolution WENO procedures. This hybrid strategy does not contain any human parameters related to specific problems and can identify the troubled grid points accurately and automatically. Secondly, a hybrid multi-resolution WENO scheme for Hamilton-Jacobi equations is developed based on the above discontinuity sensor and a simplified multi-resolution WENO scheme, which yields uniform high-order accuracy in smooth regions and sharply resolves discontinuities. Compared with the existing multi-resolution WENO scheme, the method developed in this paper can inherit its many advantages and is more efficient. Finally, some benchmark numerical experiments are performed to demonstrate the performance of the presented fifth-order hybrid multi-resolution WENO scheme for one- and two-dimensional Hamilton-Jacobi equations.
{"title":"Hybrid Finite Difference Fifth-Order Multi-Resolution WENO Scheme for Hamilton-Jacobi Equations","authors":"Zhenming Wang,Jun Zhu,Linlin Tian, Ning Zhao","doi":"10.4208/cicp.oa-2023-0002","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0002","url":null,"abstract":"In this paper, a fifth-order hybrid multi-resolution weighted essentially non-oscillatory (WENO) scheme in the finite difference framework is proposed for solving\u0000one- and two-dimensional Hamilton-Jacobi equations. Firstly, a new discontinuity sensor is designed based on the extreme values of the highest degree polynomial in the\u0000multi-resolution WENO procedures. This hybrid strategy does not contain any human parameters related to specific problems and can identify the troubled grid points\u0000accurately and automatically. Secondly, a hybrid multi-resolution WENO scheme for\u0000Hamilton-Jacobi equations is developed based on the above discontinuity sensor and a\u0000simplified multi-resolution WENO scheme, which yields uniform high-order accuracy\u0000in smooth regions and sharply resolves discontinuities. Compared with the existing\u0000multi-resolution WENO scheme, the method developed in this paper can inherit its\u0000many advantages and is more efficient. Finally, some benchmark numerical experiments are performed to demonstrate the performance of the presented fifth-order hybrid multi-resolution WENO scheme for one- and two-dimensional Hamilton-Jacobi\u0000equations.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"6 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583292","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 : 2024-04-01DOI: 10.4208/cicp.oa-2023-0211
Shi Jin,Zheng Ma, Keke Wu
In this paper, we present two novel Asymptotic-Preserving Neural Networks (APNNs) for tackling multiscale time-dependent kinetic problems, encompassing the linear transport equation and Bhatnagar-Gross-Krook (BGK) equation in all ranges of Knudsen number. Our primary objective is to devise accurate APNN approaches for resolving multiscale kinetic equations, which is also efficient in the small Knudsen number regime. The first APNN for linear transport equation is based on even-odd decomposition, which relaxes the stringent conservation prerequisites while concurrently introducing an auxiliary deep neural network. We conclude that enforcing the initial condition for the linear transport equation with inflow boundary conditions is crucial for this network. For the Boltzmann-BGK equation, the APNN incorporates the conservation of mass, momentum, and total energy into the APNN framework as well as exact boundary conditions. A notable finding of this study is that approximating the zeroth, first, and second moments—which govern the conservation of density, momentum, and energy for the Boltzmann-BGK equation, is simpler than the distribution itself. Another interesting phenomenon observed in the training process is that the convergence of density is swifter than that of momentum and energy. Finally, we investigate several benchmark problems to demonstrate the efficacy of our proposed APNN methods.
{"title":"Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations","authors":"Shi Jin,Zheng Ma, Keke Wu","doi":"10.4208/cicp.oa-2023-0211","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0211","url":null,"abstract":"In this paper, we present two novel Asymptotic-Preserving Neural Networks (APNNs) for tackling multiscale time-dependent kinetic problems, encompassing the linear transport equation and Bhatnagar-Gross-Krook (BGK) equation in all\u0000ranges of Knudsen number. Our primary objective is to devise accurate APNN approaches for resolving multiscale kinetic equations, which is also efficient in the small\u0000Knudsen number regime. The first APNN for linear transport equation is based on\u0000even-odd decomposition, which relaxes the stringent conservation prerequisites while\u0000concurrently introducing an auxiliary deep neural network. We conclude that enforcing the initial condition for the linear transport equation with inflow boundary\u0000conditions is crucial for this network. For the Boltzmann-BGK equation, the APNN\u0000incorporates the conservation of mass, momentum, and total energy into the APNN\u0000framework as well as exact boundary conditions. A notable finding of this study is that\u0000approximating the zeroth, first, and second moments—which govern the conservation\u0000of density, momentum, and energy for the Boltzmann-BGK equation, is simpler than\u0000the distribution itself. Another interesting phenomenon observed in the training process is that the convergence of density is swifter than that of momentum and energy.\u0000Finally, we investigate several benchmark problems to demonstrate the efficacy of our\u0000proposed APNN methods.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"15 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583190","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}
The high-order gas-kinetic scheme (HGKS) features good robustness, high efficiency and satisfactory accuracy,the performance of which can be further improved combined with WENO-AO (WENO with adaptive order) scheme for reconstruction. To reduce computational costs in the reconstruction procedure, this paper proposes to combine HGKS with a hybrid WENO-AO scheme. The hybrid WENO-AO scheme reconstructs target variables using upwind linear approximation directly if all extreme points of the reconstruction polynomials for these variables are outside the large stencil. Otherwise, the WENO-AO scheme is used. Unlike combining the hybrid WENO scheme with traditional Riemann solvers, the troubled cell indicator of the hybrid WENO-AO method is fully utilized in the spatial reconstruction process of HGKS. During normal and tangential reconstruction, the gas-kinetic scheme flux not only needs to reconstruct the conservative variables on the left and right interfaces but also to reconstruct the derivative terms of the conservative variables. By reducing the number of times that the WENO-AO scheme is used, the calculation cost is reduced. The high-order gas-kinetic scheme with the hybrid WENO-AO method retains original robustness and accuracy of the WENO5-AO GKS, while exhibits higher computational efficiency.
{"title":"An Efficient High-Order Gas-Kinetic Scheme with Hybrid WENO-AO Method for the Euler and Navier-Stokes Solutions","authors":"Junlei Mu,Congshan Zhuo,Qingdian Zhang,Sha Liu, Chengwen Zhong","doi":"10.4208/cicp.oa-2023-0108","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0108","url":null,"abstract":"The high-order gas-kinetic scheme (HGKS) features good robustness, high\u0000efficiency and satisfactory accuracy,the performance of which can be further improved\u0000combined with WENO-AO (WENO with adaptive order) scheme for reconstruction.\u0000To reduce computational costs in the reconstruction procedure, this paper proposes\u0000to combine HGKS with a hybrid WENO-AO scheme. The hybrid WENO-AO scheme\u0000reconstructs target variables using upwind linear approximation directly if all extreme\u0000points of the reconstruction polynomials for these variables are outside the large stencil. Otherwise, the WENO-AO scheme is used. Unlike combining the hybrid WENO\u0000scheme with traditional Riemann solvers, the troubled cell indicator of the hybrid\u0000WENO-AO method is fully utilized in the spatial reconstruction process of HGKS.\u0000During normal and tangential reconstruction, the gas-kinetic scheme flux not only\u0000needs to reconstruct the conservative variables on the left and right interfaces but also\u0000to reconstruct the derivative terms of the conservative variables. By reducing the number of times that the WENO-AO scheme is used, the calculation cost is reduced. The\u0000high-order gas-kinetic scheme with the hybrid WENO-AO method retains original robustness and accuracy of the WENO5-AO GKS, while exhibits higher computational\u0000efficiency.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"117 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583290","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}
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