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Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems 一类随机李-泊松系统的李-泊松数值方法
IF 1.1 4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2024-01-01 DOI: 10.4208/ijnam2024-1004
Qianqian Liu, Lijin Wang
{"title":"Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems","authors":"Qianqian Liu, Lijin Wang","doi":"10.4208/ijnam2024-1004","DOIUrl":"https://doi.org/10.4208/ijnam2024-1004","url":null,"abstract":"We propose a numerical method based on the Lie-Poisson reduction for a class of\u0000stochastic Lie-Poisson systems. Such system is transformed to SDE on the dual $mathfrak{g}^∗$ of the Lie\u0000algebra related to the Lie group manifold where the system is located, which is also the reduced\u0000form of a stochastic Hamiltonian system on the cotangent bundle of the Lie group by momentum\u0000mapping. Stochastic Poisson integrators are obtained by discretely reducing stochastic symplectic\u0000methods on the cotangent bundle to integrators on $mathfrak{g}^∗.$ Stochastic generating functions creating\u0000stochastic symplectic methods are used to construct the schemes. An application to the stochastic\u0000rigid body system illustrates the theory and provides numerical validation of the method.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof By Truncation on the Sample-Time Space 随机双曲标量守恒定律有限体积法的收敛性：采样时间空间截断证明
IF 1.1 4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2024-01-01 DOI: 10.4208/ijnam2024-1005
Sylvain Dotti
{"title":"Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof By Truncation on the Sample-Time Space","authors":"Sylvain Dotti","doi":"10.4208/ijnam2024-1005","DOIUrl":"https://doi.org/10.4208/ijnam2024-1005","url":null,"abstract":"We prove the almost sure convergence of the explicit-in-time Finite Volume Method\u0000with monotone fluxes towards the unique solution of the scalar hyperbolic balance law with locally\u0000Lipschitz continuous flux and additive noise driven by a cylindrical Wiener process. We use the\u0000standard CFL condition and a martingale exponential inequality on sets whose probabilities are\u0000converging towards one. Then, with the help of stopping times on those sets, we apply theorems\u0000of convergence for approximate kinetic solutions of balance laws with stochastic forcing.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"70 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Posteriori Error Estimates for Darcy-Forchheimer’s Problem Coupled with the Convection-Diffusion-Reaction Equation 达西-福克海默问题与对流-扩散-反作用方程的后验误差估计
IF 1.1 4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2024-01-01 DOI: 10.4208/ijnam2024-1003
Faouzi Triki,Toni Sayah, Georges Semaan
{"title":"A Posteriori Error Estimates for Darcy-Forchheimer’s Problem Coupled with the Convection-Diffusion-Reaction Equation","authors":"Faouzi Triki,Toni Sayah, Georges Semaan","doi":"10.4208/ijnam2024-1003","DOIUrl":"https://doi.org/10.4208/ijnam2024-1003","url":null,"abstract":"In this work we derive a posteriori error estimates for the convection-diffusion-reaction\u0000equation coupled with the Darcy-Forchheimer problem by a nonlinear external source depending\u0000on the concentration of the fluid. We introduce the variational formulation associated to the\u0000problem, and discretize it by using the finite element method. We prove optimal a posteriori\u0000errors with two types of calculable error indicators. The first one is linked to the linearization and\u0000the second one to the discretization. Then we find upper and lower error bounds under additional\u0000regularity assumptions on the exact solutions. Finally, numerical computations are performed to\u0000show the effectiveness of the obtained error indicators.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"44 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unconditionally Energy Stable and First-Order Accurate Numerical Schemes for the Heat Equation with Uncertain Temperature-Dependent Conductivity 具有不确定温度相关电导率的热方程的无条件能量稳定和一阶精确数值格式
4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1035
Fiordilino, J. A., Winger, M.
{"title":"Unconditionally Energy Stable and First-Order Accurate Numerical Schemes for the Heat Equation with Uncertain Temperature-Dependent Conductivity","authors":"Fiordilino, J. A., Winger, M.","doi":"10.4208/ijnam2023-1035","DOIUrl":"https://doi.org/10.4208/ijnam2023-1035","url":null,"abstract":"In this paper, we present first-order accurate numerical methods for solution of the heat equation with uncertain temperature-dependent thermal conductivity. Each algorithm yields a shared coefficient matrix for the ensemble set improving computational efficiency. Both mixed and Robin-type boundary conditions are treated. In contrast with alternative, related methodologies, stability and convergence are unconditional. In particular, we prove unconditional, energy stability and optimal-order error estimates. A battery of numerical tests are presented to illustrate both the theory and application of these algorithms.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135219537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A Novel Deep Neural Network Algorithm for the Helmholtz Scattering Problem In the Unbounded Domain 无界域Helmholtz散射问题的一种新的深度神经网络算法
4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1032
Andy L Yang
{"title":"A Novel Deep Neural Network Algorithm for the Helmholtz Scattering Problem In the Unbounded Domain","authors":"Andy L Yang","doi":"10.4208/ijnam2023-1032","DOIUrl":"https://doi.org/10.4208/ijnam2023-1032","url":null,"abstract":". In this paper, we develop a novel meshless, ray-based deep neural network algorithm for solving the high-frequency Helmholtz scattering problem in the unbounded domain. While our recent work [44] designed a deep neural network method for solving the Helmholtz equation over (cid:12)nite bounded domains, this paper deals with the more general and di(cid:14)cult case of unbounded regions. By using the perfectly matched layer method, the original mathematical model in the unbounded domain is transformed into a new format of second-order system in a (cid:12)nite bounded domain with simple homogeneous Dirichlet boundary conditions. Compared with the Helmholtz equation in the bounded domain, the new system is equipped with variable coe(cid:14)cients. Then, a deep neural network algorithm is designed for the new system, where the rays in various random directions are used as the basis of the numerical solution. Various numerical examples have been carried out to demonstrate the accuracy and e(cid:14)ciency of the proposed numerical method. The proposed method has the advantage of easy implementation and meshless while maintaining high accuracy. To the best of the author’s knowledge, this is the (cid:12)rst deep neural network method to solve the Helmholtz equation in the unbounded domain.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Conforming Dg Method for the Biharmonic Equation on Polytopal Meshes 多边形网格双调和方程的调和Dg法
4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1037
Xiu Ye null, Shangyou Zhang
{"title":"A Conforming Dg Method for the Biharmonic Equation on Polytopal Meshes","authors":"Xiu Ye null, Shangyou Zhang","doi":"10.4208/ijnam2023-1037","DOIUrl":"https://doi.org/10.4208/ijnam2023-1037","url":null,"abstract":"A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki Equation with Modified Physics-Informed Neural Networks (Pinns) 修正物理信息神经网络(Pinns)的Allen-Cahn-Ohta-Kawasaki方程数值逼近
4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1030
Jingjing Xu, Jia Zhao null, Yanxiang Zhao
{"title":"Numerical Approximations of the Allen-Cahn-Ohta-Kawasaki Equation with Modified Physics-Informed Neural Networks (Pinns)","authors":"Jingjing Xu, Jia Zhao null, Yanxiang Zhao","doi":"10.4208/ijnam2023-1030","DOIUrl":"https://doi.org/10.4208/ijnam2023-1030","url":null,"abstract":"","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
High-Order Enriched Finite Element Methods for Elliptic Interface Problems with Discontinuous Solutions 具有不连续解的椭圆界面问题的高阶丰富有限元方法
4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1038
Champike Attanayake, So-Hsiang Chou null, Quanling Deng
{"title":"High-Order Enriched Finite Element Methods for Elliptic Interface Problems with Discontinuous Solutions","authors":"Champike Attanayake, So-Hsiang Chou null, Quanling Deng","doi":"10.4208/ijnam2023-1038","DOIUrl":"https://doi.org/10.4208/ijnam2023-1038","url":null,"abstract":"Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Fully Implicit Method Using Nodal Radial Basis Functions to Solve the Linear Advection Equation 利用节点径向基函数求解线性平流方程的全隐式方法
4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1018
P.-A. Gourdain, M. Evans, H. R. Hasson, J. R. Young, I. West-Abdallah null, M. B. Adams
{"title":"A Fully Implicit Method Using Nodal Radial Basis Functions to Solve the Linear Advection Equation","authors":"P.-A. Gourdain, M. Evans, H. R. Hasson, J. R. Young, I. West-Abdallah null, M. B. Adams","doi":"10.4208/ijnam2023-1018","DOIUrl":"https://doi.org/10.4208/ijnam2023-1018","url":null,"abstract":"","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135421665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improved Long Time Accuracy for Projection Methods for Navier-Stokes Equations Using Emac Formulation 利用Emac公式提高Navier-Stokes方程投影方法的长时间精度
4区 数学
International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1008
Sean Ingimarson, Monika Neda, Leo G. Rebholz, Jorge Reyes null, An Vu
{"title":"Improved Long Time Accuracy for Projection Methods for Navier-Stokes Equations Using Emac Formulation","authors":"Sean Ingimarson, Monika Neda, Leo G. Rebholz, Jorge Reyes null, An Vu","doi":"10.4208/ijnam2023-1008","DOIUrl":"https://doi.org/10.4208/ijnam2023-1008","url":null,"abstract":"","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136370904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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