Applied Numerical Mathematics最新文献

{"title":"Stroboscopic averaging methods to study autoresonance and other problems with slowly varying forcing frequencies","authors":"M.P. Calvo ,&nbsp;J.M. Sanz-Serna ,&nbsp;Beibei Zhu","doi":"10.1016/j.apnum.2025.04.004","DOIUrl":"10.1016/j.apnum.2025.04.004","url":null,"abstract":"<div><div>Autoresonance is a phenomenon of physical interest that may take place when a nonlinear oscillator is forced at a frequency that varies slowly. The stroboscopic averaging method (SAM), which provides an efficient numerical technique for the integration of highly oscillatory systems, cannot be used directly to study autoresonance due to the slow changes of the forcing frequency. We study how to modify SAM to cater for such slow variations. Numerical experiments show the computational advantages of using SAM.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 15-24"},"PeriodicalIF":2.2,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143838831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong convergence rates of Galerkin finite element methods for SWEs with cubic polynomial nonlinearity","authors":"Ruisheng Qi ,&nbsp;Xiaojie Wang","doi":"10.1016/j.apnum.2025.04.001","DOIUrl":"10.1016/j.apnum.2025.04.001","url":null,"abstract":"<div><div>In the present work, strong approximation errors are analyzed for both the spatial semi-discretization and the spatio-temporal fully discretization of stochastic wave equations (SWEs) with cubic polynomial nonlinearities and additive noises. The fully discretization is achieved by the standard Galerkin finite element method in space and a novel exponential time integrator combined with the averaged vector field approach. The newly proposed scheme is proved to exactly satisfy a trace formula based on an energy functional. Recovering the convergence rates of the scheme, however, meets essential difficulties, due to the lack of the global monotonicity condition. To overcome this issue, we derive the exponential integrability property of the considered numerical approximations, by the energy functional. Armed with these properties, we obtain the strong convergence rates of the approximations in both spatial and temporal direction. Finally, numerical results are presented to verify the previously theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 112-137"},"PeriodicalIF":2.2,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143860247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Wachspress-Habetler extension to the HSS iteration method in Rn×n","authors":"Thomas Smotzer,&nbsp;John Buoni","doi":"10.1016/j.apnum.2025.04.002","DOIUrl":"10.1016/j.apnum.2025.04.002","url":null,"abstract":"<div><div>In the study of implicit iterations for the two dimensional heat and Helmhotz equations, one constructs a splitting of the form <span><math><mi>A</mi><mo>=</mo><mi>U</mi><mo>+</mo><mi>V</mi></math></span> where <em>U</em> and <em>V</em> are the difference approximations parallel to the <em>x</em> and <em>y</em> axis, respectively. In the commutative case for <em>U</em> and <em>V</em>, several investigations have taken place. For the noncommutative case, a symmetric positive definite matrix <em>F</em> is found, such that <span><math><mi>U</mi><mi>F</mi><mi>V</mi><mo>=</mo><mi>V</mi><mi>F</mi><mi>U</mi></math></span> and then, the investigations use this to address the non-commutative nature of <em>U</em> with <em>V</em>. The purpose of this paper is to study the same problem type for the commutativity of <em>H</em> and <em>S</em> in the <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> splitting of <span><math><mi>A</mi><mo>=</mo><mi>H</mi><mo>+</mo><mi>S</mi></math></span>, where <em>H</em> and <em>S</em> are the symmetric and skew-symmetric parts of <em>A</em>, respectively. Although throughout the literature in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msup></math></span>, the <em>H</em> of <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> stands for hermitian, we use it in the symmetric matrix case. One then applies this result to the <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> iteration method. Since it is well known that the commutativity of <em>U</em> and <em>V</em> plays an important role in the analysis of <em>ADI</em> methods; especially for the solution of the Helmhotz Equation, one hopes that this commutativity will improve performance of the <span><math><mi>H</mi><mi>S</mi><mi>S</mi></math></span> method. This extension is similar to that of Wachspress and Habetler's variation of the Peaceman-Rachford method. Along the way, suitable conditions are found for <em>A</em>, which yield a symmetric non-zero matrix <span><math><mi>P</mi><mo>=</mo><msqrt><mrow><mi>F</mi></mrow></msqrt></math></span> for which <span><math><mi>N</mi><mspace></mspace><mo>=</mo><mspace></mspace><mi>P</mi><mi>A</mi><mi>P</mi></math></span> is a normal matrix.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"215 ","pages":"Pages 49-58"},"PeriodicalIF":2.2,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"General enrichments of stable GFEM for interface problems: Theory and extreme learning machine construction","authors":"Dongmei Wang ,&nbsp;Hengguang Li ,&nbsp;Qinghui Zhang","doi":"10.1016/j.apnum.2025.03.009","DOIUrl":"10.1016/j.apnum.2025.03.009","url":null,"abstract":"<div><div>Generalized finite element methods (GFEMs), when applied to interface problems (IPs), need to be enriched with special functions to enhance approximation accuracy. These functions include distance functions, one-side distance functions, level set functions, and exponential forms of level set function. For the IP with geometrically complex interface curves, computation of the distance function or level set function could be challenging, and algorithms of computational geometry are usually involved. Moreover, theoretical analysis on optimal convergence of the GFEM enriched by these functions has not been fully investigated. In this study we propose a general enrichment scheme, based on which all the aforementioned enrichments can be viewed as special examples. We prove that a stable GFEM (SGFEM) with such a new enrichment scheme reaches the optimal convergence rate. Most importantly, the new scheme provides an instruction to construct machine learning (ML) based enrichments, which advances the ability of GFEM to handle geometrically complex interfaces. Two ML methods, deep neural network (DNN) and extreme learning machine (ELM), are studied. Among them, the ELM is highly suggested because it exhibits high accuracy for the interface curve with complex geometries. The learning dimension for the ML is one dimension less than that of the domain so that the proposed ML algorithm can be implemented efficiently. The numerical experiments demonstrate that the SGFEM with the ELM enrichment achieves the optimal convergence rates for the IP, as predicted theoretically.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 143-159"},"PeriodicalIF":2.2,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143824352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A unified space-time finite element scheme for a quasilinear parabolic problem","authors":"I. Toulopoulos","doi":"10.1016/j.apnum.2025.03.006","DOIUrl":"10.1016/j.apnum.2025.03.006","url":null,"abstract":"<div><div>A new approach is presented to obtain stabilized space - time finite element schemes for solving in a unified space-time way a quasilinear parabolic model problem. The procedure consists in introducing first upwind diffusion terms with an appropriate scaling factor in the initial space-time finite element scheme. Then additional interface jump terms are introduced for ensuring the consistency of the final finite element discetzation. A discretization error analysis is presented and a priori error estimates in an appropriate discrete norm are shown. The corresponding convergence rates are optimal with respect to the regularity of the solution and are confirmed through a series of numerical tests.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 127-142"},"PeriodicalIF":2.2,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A revised MRMIL Riemannian conjugate gradient method with simplified global convergence properties","authors":"Nasiru Salihu ,&nbsp;Poom Kumam ,&nbsp;Sani Salisu ,&nbsp;Lin Wang ,&nbsp;Kanokwan Sitthithakerngkiet","doi":"10.1016/j.apnum.2025.03.007","DOIUrl":"10.1016/j.apnum.2025.03.007","url":null,"abstract":"<div><div>In this work, we propose an effective coefficient for the conjugate gradient (CG) method. First, we present the coefficient for Euclidean optimization, explaining its motivation, and then extend it to Riemannian optimization. We analyze the convergence of the CG method generated by this coefficient in the context of Riemannian optimization, ensuring that the generated search direction satisfies the sufficient descent property. This property ensures that the sequence generated converges to a minimizer of the underlying function. We test the effectiveness of the proposed coefficient numerically on various Riemannian optimization problems, demonstrating favorable performance compared to existing Riemannian CG methods and other coefficients of similar class. These results also extend to Euclidean optimization, where such findings have not yet been established.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the unconditional long-time L2-stability of the BDF2 time stepping scheme for the two-dimensional Navier-Stokes equations","authors":"Ming-Cheng Shiue","doi":"10.1016/j.apnum.2025.03.008","DOIUrl":"10.1016/j.apnum.2025.03.008","url":null,"abstract":"<div><div>In this note, we study the long-time <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> stability of the BDF2 time stepping scheme for the two-dimensional Navier-Stokes equations with homogenous Dirichlet boundary condition. More precisely, the numerical scheme obtained from using the backward differentiation formula (BDF2) in time is proven to be long time stable in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for any size of the time step <span><math><mi>Δ</mi><mi>t</mi><mo>&gt;</mo><mn>0</mn></math></span>. In addition, less regularity of the solution is required to derive the unconditional long-time <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> stability of the BDF2 time stepping scheme. This improves the results in <span><span>[1]</span></span> and <span><span>[9]</span></span>.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 104-109"},"PeriodicalIF":2.2,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A convergent and stable fourth-order iterative procedure based on Kung-Traub conjecture for nonlinear systems","authors":"Asifa Tassaddiq ,&nbsp;Sania Qureshi ,&nbsp;Ioannis K. Argyros ,&nbsp;Francisco I. Chicharro ,&nbsp;Amanullah Soomro ,&nbsp;Paras Nizamani ,&nbsp;Evren Hincal","doi":"10.1016/j.apnum.2025.03.003","DOIUrl":"10.1016/j.apnum.2025.03.003","url":null,"abstract":"<div><div>Iterative algorithms are essential in computer research because they solve nonlinear models. This study presents a new and efficient approach for finding the roots of nonlinear equations and nonlinear systems of equations. The algorithm follows a two-step process and aims to optimize the iterative process. Based on Kung-Traub conjecture, the method shows optimal convergence and only needs three function evaluations per iteration. This creates a fourth-order optimal iterative procedure with an efficiency index of about 1.5874. This method utilizes a combination of two well-established third-order iterative approaches. The analyses of local and semilocal convergence, as well as stability analysis using complex dynamics, provide substantial improvements compared to current methods. We have extensively tested the proposed algorithm on a range of nonlinear models, including chemical reactions, kinetic synthesis, and non-adiabatic stirred tank reactors, consistently demonstrating accurate and efficient results. In terms of both speed and accuracy, it outperforms contemporary algorithms.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 54-79"},"PeriodicalIF":2.2,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hamiltonian boundary value methods applied to KdV-KdV systems","authors":"Qian Luo ,&nbsp;Aiguo Xiao ,&nbsp;Xuqiong Luo ,&nbsp;Xiaoqiang Yan","doi":"10.1016/j.apnum.2025.02.006","DOIUrl":"10.1016/j.apnum.2025.02.006","url":null,"abstract":"<div><div>In this paper, we propose a highly accurate scheme for two KdV systems of the Boussinesq type under periodic boundary conditions. The proposed scheme combines the Fourier-Galerkin method for spatial discretization with Hamiltonian boundary value methods for time integration, ensuring the conservation of discrete mass and energy. By expanding the system in Fourier series, the equations are firstly transformed into Hamiltonian form, preserving the original Hamiltonian structure. Applying the Fourier-Galerkin method for semi-discretization in space, we obtain a large-scale system of Hamiltonian ordinary differential equations, which is then solved using a class of energy-conserving Runge-Kutta methods, known as Hamiltonian boundary value methods. The efficiency of this approach is assessed, and several numerical examples are provided to demonstrate the effectiveness of the method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 1-27"},"PeriodicalIF":2.2,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143725163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Splitting techniques for DAEs with port-Hamiltonian applications","authors":"Andreas Bartel ,&nbsp;Malak Diab ,&nbsp;Andreas Frommer ,&nbsp;Michael Günther ,&nbsp;Nicole Marheineke","doi":"10.1016/j.apnum.2025.03.004","DOIUrl":"10.1016/j.apnum.2025.03.004","url":null,"abstract":"<div><div>In the simulation of differential-algebraic equations (DAEs), it is essential to employ numerical schemes that take into account the inherent structure and maintain explicit or hidden algebraic constraints. This paper focuses on operator splitting techniques for coupled systems and aims at preserving the structure in the port-Hamiltonian framework. The study explores two decomposition strategies: one considering the underlying coupled subsystem structure and the other addressing energy-associated properties such as conservation and dissipation. We show that for coupled index-1 DAEs with and without private index-2 variables, the splitting schemes on top of a dimension-reducing decomposition achieve the same convergence rate as in the case of ordinary differential equations. Additionally, we discuss an energy-associated decomposition for linear time-invariant port-Hamiltonian index-1 DAEs and introduce generalized Cayley transforms to uphold energy conservation. The effectiveness of both strategies is evaluated using port-Hamiltonian benchmark examples from electric circuits.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"214 ","pages":"Pages 28-53"},"PeriodicalIF":2.2,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}