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Second-order a priori and a posteriori error estimations for integral boundary value problems of nonlinear singularly perturbed parameterized form 非线性奇异扰动参数化形式积分边界值问题的二阶先验和后验误差估计
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-21 DOI: 10.1007/s11075-024-01918-5
Shashikant Kumar, Sunil Kumar, Pratibhamoy Das
{"title":"Second-order a priori and a posteriori error estimations for integral boundary value problems of nonlinear singularly perturbed parameterized form","authors":"Shashikant Kumar, Sunil Kumar, Pratibhamoy Das","doi":"10.1007/s11075-024-01918-5","DOIUrl":"https://doi.org/10.1007/s11075-024-01918-5","url":null,"abstract":"<p>In this work, we present the <i>a priori</i> and <i>a posteriori</i> error analysis of a hybrid difference scheme for integral boundary value problems of nonlinear singularly perturbed parameterized form. The discretization for the nonlinear parameterized equation constitutes a hybrid difference scheme which is based on a suitable combination of the trapezoidal scheme and the backward difference scheme. Further, we employ the composite trapezoidal scheme for the discretization of the nonlocal boundary condition. <i>A priori</i> error estimation is provided for the proposed hybrid scheme, which leads to second-order uniform convergence on various <i>a priori</i> defined meshes. Moreover, a detailed <i>a posteriori</i> error analysis is carried out for the present hybrid scheme which provides a proper discretization of the error equidistribution at each partition. Numerical results strongly validate the theoretical findings for nonlinear problems with integral boundary conditions.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"8 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
An efficient hybrid numerical approach for solving two-dimensional fractional cable model involving time-fractional operator of distributed order with error analysis 一种高效的混合数值方法，用于求解涉及分布阶时间分数算子的二维分数电缆模型并进行误差分析
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-17 DOI: 10.1007/s11075-024-01913-w
Saeed Kosari, Peng Xu, Jana Shafi, MohammadHossein Derakhshan
{"title":"An efficient hybrid numerical approach for solving two-dimensional fractional cable model involving time-fractional operator of distributed order with error analysis","authors":"Saeed Kosari, Peng Xu, Jana Shafi, MohammadHossein Derakhshan","doi":"10.1007/s11075-024-01913-w","DOIUrl":"https://doi.org/10.1007/s11075-024-01913-w","url":null,"abstract":"<p>In this article, we study and examine an efficient numerical approach to obtain approximate solutions of the two-dimensional fractional cable model involving the time-fractional operator of distributed order. A hybrid numerical approach is used to approximate the proposed fractional model. For approximating the integral part of the distributed order including Caputo fractional derivative, the combination of Gauss quadrature rule and finite difference are used. As well as, for the integral part of the distributed order including Riemann Liouville fractional derivatives, from the mid-point quadrature rule and shifted Grünwald estimation are applied. Also, to approximate the proposed model in the space direction, the Legendre spectral numerical approach is used in order to calculate the full-discrete numerical approach. In this work, error analysis and convergence are studied. In the end, to show the effectiveness of the proposed approach, two numerical examples are stated and checked.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"151 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Solving semi-discrete optimal transport problems: star shapedeness and Newton’s method 解决半离散最优传输问题：星形整形和牛顿法
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-15 DOI: 10.1007/s11075-024-01903-y
Luca Dieci, Daniyar Omarov
{"title":"Solving semi-discrete optimal transport problems: star shapedeness and Newton’s method","authors":"Luca Dieci, Daniyar Omarov","doi":"10.1007/s11075-024-01903-y","DOIUrl":"https://doi.org/10.1007/s11075-024-01903-y","url":null,"abstract":"<p>In this work, we propose a novel implementation of Newton’s method for solving semi-discrete optimal transport (OT) problems for cost functions which are a positive combination of <i>p</i>-norms, <span>(1&lt;p&lt;infty )</span>. It is well understood that the solution of a semi-discrete OT problem is equivalent to finding a partition of a bounded region in Laguerre cells, and we prove that the Laguerre cells are star-shaped with respect to the target points. By exploiting the geometry of the Laguerre cells, we obtain an efficient and reliable implementation of Newton’s method to find the sought network structure. We provide implementation details and extensive results in support of our technique in 2-d problems, as well as comparison with other approaches used in the literature.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"39 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
An Edge-based cascadic multigrid method for $$H(textbf{curl})$$ problems 针对 $$H(textbf{curl})$$问题的基于边缘的级联多网格方法
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-15 DOI: 10.1007/s11075-024-01917-6
Jinxuan Wang, Kejia Pan, Xiaoxin Wu
{"title":"An Edge-based cascadic multigrid method for $$H(textbf{curl})$$ problems","authors":"Jinxuan Wang, Kejia Pan, Xiaoxin Wu","doi":"10.1007/s11075-024-01917-6","DOIUrl":"https://doi.org/10.1007/s11075-024-01917-6","url":null,"abstract":"<p>An efficient extrapolation cascadic multigird (EXCMG) method is developed to solve large linear systems resulting from edge element discretizations of 3D <span>(H(textbf{curl}))</span> problems on rectangular meshes. By treating edge unknowns as defined on the midpoints of edges, following the similar idea of the nodal EXCMG method, we design a new prolongation operator for 3D edge-based discretizations, which is used to construct a high-order approximation to the finite element solution on the refined grid. This good initial guess greatly reduces the number of iterations required by the multigrid smoother. Furthermore, the divergence correction technique is employed to further speed up the convergence of the multigrid method. Numerical examples including problems with high-contrast discontinuous coefficients are presented to validate the effectiveness of the proposed EXCMG method. The edge-based EXCMG method is more efficient than the auxiliary-space Maxwell solver (AMS) for definite problems in the considered geometrical configuration, and it can also efficiently solve large-scale indefinite problems encountered in engineering and scientific fields.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 IEQ-FEMs for the Cahn-Hilliard equation and Allen-Cahn equation 卡恩-希利亚德方程和艾伦-卡恩方程的无条件能量稳定 IEQ-FEMs
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-13 DOI: 10.1007/s11075-024-01910-z
Yaoyao Chen, Hailiang Liu, Nianyu Yi, Peimeng Yin
{"title":"Unconditionally energy stable IEQ-FEMs for the Cahn-Hilliard equation and Allen-Cahn equation","authors":"Yaoyao Chen, Hailiang Liu, Nianyu Yi, Peimeng Yin","doi":"10.1007/s11075-024-01910-z","DOIUrl":"https://doi.org/10.1007/s11075-024-01910-z","url":null,"abstract":"<p>In this paper, we present several unconditionally energy-stable invariant energy quadratization (IEQ) finite element methods (FEMs) with linear, first- and second-order accuracy for solving both the Cahn-Hilliard equation and the Allen-Cahn equation. For time discretization, we compare three distinct IEQ-FEM schemes that position the intermediate function introduced by the IEQ approach in different function spaces: finite element space, continuous function space, or a combination of these spaces. Rigorous proofs establishing the existence and uniqueness of the numerical solution, along with analyses of energy dissipation for both equations and mass conservation for the Cahn-Hilliard equation, are provided. The proposed schemes’ accuracy, efficiency, and solution properties are demonstrated through numerical experiments.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"399 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 fast numerical algorithm for finding all real solutions to a system of N nonlinear equations in a finite domain 在有限域中寻找 N 个非线性方程组所有实解的快速数值算法
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-12 DOI: 10.1007/s11075-024-01908-7
Fernando Chueca-Díez, Alfonso M. Gañán-Calvo
{"title":"A fast numerical algorithm for finding all real solutions to a system of N nonlinear equations in a finite domain","authors":"Fernando Chueca-Díez, Alfonso M. Gañán-Calvo","doi":"10.1007/s11075-024-01908-7","DOIUrl":"https://doi.org/10.1007/s11075-024-01908-7","url":null,"abstract":"<p>A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica, Matlab, and Python as examples) do not offer a solution, is to find all the real solutions of a system of <i>n</i> nonlinear equations in a certain finite domain of the <i>n</i>-dimensional space of variables. We present two similar algorithms of minimum length and computational weight to solve this problem, in which one resembles a graphical tool of edge detection in an image extended to <i>n</i> dimensions. To do this, we discretize the <i>n</i>-dimensional space sector in which the solutions are sought. Once the discretized hypersurfaces (edges) defined by each nonlinear equation of the <i>n</i>-dimensional system have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in each <i>n</i>-dimensional tile or cell containing at least one solution marks the approximate locations of all the hyperpoints that constitute the solutions. This makes the final Newton-Raphson step rapidly convergent to all the existent solutions in the predefined space sector with the desired degree of accuracy.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"23 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Edge importance in complex networks 复杂网络中边缘的重要性
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-09 DOI: 10.1007/s11075-024-01881-1
Silvia Noschese, Lothar Reichel
{"title":"Edge importance in complex networks","authors":"Silvia Noschese, Lothar Reichel","doi":"10.1007/s11075-024-01881-1","DOIUrl":"https://doi.org/10.1007/s11075-024-01881-1","url":null,"abstract":"<p>Complex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the subgraph centrality of the vertices. It is more difficult to assess the importance of the edges. One approach is to consider the line graph associated with the given network and determine the importance of the vertices of the line graph, but this is fairly complicated except for small networks. This paper compares two approaches to estimate the importance of edges of medium-sized to large networks. One approach computes partial derivatives of the total communicability of the weights of the edges, where a partial derivative of large magnitude indicates that the corresponding edge may be important. Our second approach computes the Perron sensitivity of the edges. A high sensitivity signals that the edge may be important. The performance of these methods and some computational aspects are discussed. Applications of interest include to determine whether a network can be replaced by a network with fewer edges with about the same communicability.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"31 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Estimates of discrete time derivatives for the parabolic-parabolic Robin-Robin coupling method 抛物线-抛物线罗宾-罗宾耦合法的离散时间导数估算
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-09 DOI: 10.1007/s11075-024-01902-z
Erik Burman, Rebecca Durst, Miguel A. Fernández, Johnny Guzmán, Sijing Liu
{"title":"Estimates of discrete time derivatives for the parabolic-parabolic Robin-Robin coupling method","authors":"Erik Burman, Rebecca Durst, Miguel A. Fernández, Johnny Guzmán, Sijing Liu","doi":"10.1007/s11075-024-01902-z","DOIUrl":"https://doi.org/10.1007/s11075-024-01902-z","url":null,"abstract":"<p>We consider a loosely coupled, non-iterative Robin-Robin coupling method proposed and analyzed in Burman et al. (J. Numer. Math. <b>31</b>(1):59–77, 2023) for a parabolic-parabolic interface problem and prove estimates for the discrete time derivatives of the scalar field in different norms. When the interface is flat and perpendicular to two of the edges of the domain we prove error estimates in the <span>(H^2)</span>-norm. Such estimates are key ingredients to analyze a defect correction method for the parabolic-parabolic interface problem. Numerical results are shown to support our findings.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"22 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Discrete non-commutative hungry Toda lattice and its application in matrix computation 离散非交换饥饿户田网格及其在矩阵计算中的应用
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-08 DOI: 10.1007/s11075-024-01915-8
Zheng Wang, Shi-Hao Li, Kang-Ya Lu, Jian-Qing Sun
{"title":"Discrete non-commutative hungry Toda lattice and its application in matrix computation","authors":"Zheng Wang, Shi-Hao Li, Kang-Ya Lu, Jian-Qing Sun","doi":"10.1007/s11075-024-01915-8","DOIUrl":"https://doi.org/10.1007/s11075-024-01915-8","url":null,"abstract":"<p>In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued <span>(theta )</span>-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"41 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Approximate moment functions for logistic stochastic differentialequations 逻辑随机微分方程的近似矩函数
IF 2.1 3区 数学
Numerical Algorithms Pub Date : 2024-08-07 DOI: 10.1007/s11075-024-01911-y
Coşkun Çetin, Jasmina Đorđević
{"title":"Approximate moment functions for logistic stochastic differentialequations","authors":"Coşkun Çetin, Jasmina Đorđević","doi":"10.1007/s11075-024-01911-y","DOIUrl":"https://doi.org/10.1007/s11075-024-01911-y","url":null,"abstract":"<p>In this paper, we introduce a method of successive approximations for moment functions of logistic stochastic differential equations. We first reduce the system of the corresponding moment functions to an infinite system of linear ordinary differential equations. Then, we determine certain upper and lower bounds on the moment functions, and utilize these bounds to solve the resulting systems approximately via suitable truncations, iterations and a local improvement step. After obtaining some general theoretical results on the error norms and describing a general algorithm for logistic SDE, we focus on stochastic Verhulst systems in numerical implementations. We compare their moment approximations with numerical solutions via simulation-based methods that include discretizations of the pathwise solutions as well as other convergent numerical procedures like semi-implicit split-step Euler methods.</p>","PeriodicalId":54709,"journal":{"name":"Numerical Algorithms","volume":"59 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
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