International Journal of Numerical Analysis and Modeling最新文献

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.

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

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

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

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

引用次数: 1

A Sharp $alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation 二维时间调质分数阶Fokker-Planck方程梯度网格上的Sharp $alpha$-Robust $L1$格式

4区数学

International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1033

Can Wang, Weihua Deng null, Xiangong Tang

引用次数: 0

A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids 直角网格上半线性二阶椭圆问题局部不连续Galerkin逼近的后验误差估计

4区数学

International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1034

Mahboub Baccouch

{"title":"A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids","authors":"Mahboub Baccouch","doi":"10.4208/ijnam2023-1034","DOIUrl":"https://doi.org/10.4208/ijnam2023-1034","url":null,"abstract":". In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type (cid:0) ∆ u = f ( x ;u ). We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order p +2 towards the p -degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most p are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The ﬁrst error can be expressed as a linear combination of two ( p +1)- degree Radau polynomials in the x - and y (cid:0) directions. The second error converges to zero with order p + 2 in the L 2 -norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the L 2 -norm under mesh reﬁnement at the optimal rate. The order of convergence is proved to be p + 2. We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.","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":"135195303","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 Finite Volume Element Solution Based on Postprocessing Technique Over Arbitrary Convex Polygonal Meshes 基于后处理技术的任意凸多边形网格有限体积元解

4区数学

International Journal of Numerical Analysis and Modeling Pub Date : 2023-06-01 DOI: 10.4208/ijnam2023-1026

Yanlong Zhang null, Yanhui Zhou

{"title":"A Finite Volume Element Solution Based on Postprocessing Technique Over Arbitrary Convex Polygonal Meshes","authors":"Yanlong Zhang null, Yanhui Zhou","doi":"10.4208/ijnam2023-1026","DOIUrl":"https://doi.org/10.4208/ijnam2023-1026","url":null,"abstract":". A special (cid:12)nite volume element method based on postprocessing technique is proposed to solve the anisotropic di(cid:11)usion problem on arbitrary convex polygonal meshes. The shape function of polygonal (cid:12)nite element method is constructed by Wachspress generalized barycentric coordinate, and by adding some element-wise bubble functions to the (cid:12)nite element solution, we get a new (cid:12)nite volume element solution that satis(cid:12)es the local conservation law on a certain dual mesh. The postprocessing algorithm only needs to solve a local linear algebraic system on each primary cell, so that it is easy to implement. More interesting is that, a general construction of the bubble functions is introduced on each polygonal cell, which enables us to prove the existence and uniqueness of the post-processed solution on arbitrary convex polygonal meshes with full anisotropic di(cid:11)usion tensor. The optimal H 1 and L 2 error estimates of the post-processed solution are also obtained. Finally, the local conservation property and convergence of the new polygonal (cid:12)nite volume element solution are veri(cid:12)ed by numerical experiments.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"1 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":"135143757","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