SIAM Journal on Numerical Analysis最新文献_第5页

An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity 低正则势能和非线性非线性薛定谔方程的显式对称指数波积分器

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-08-06 DOI: 10.1137/23m1615656

Weizhu Bao, Chushan Wang

{"title":"An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity","authors":"Weizhu Bao, Chushan Wang","doi":"10.1137/23m1615656","DOIUrl":"https://doi.org/10.1137/23m1615656","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1901-1928, August 2024. Abstract. We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For “good” potential and nonlinearity ([math]-potential and [math]), we establish an optimal second-order error bound in the [math]-norm. For low regularity potential and nonlinearity ([math]-potential and [math]), we obtain a first-order [math]-norm error bound accompanied with a uniform [math]-norm bound of the numerical solution. Moreover, adopting a new technique of regularity compensation oscillation to analyze error cancellation, for some nonresonant time steps, the optimal second-order [math]-norm error bound is proved under a weaker assumption on the nonlinearity: [math]. For all the cases, we also present corresponding fractional order error bounds in the [math]-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"100 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141899472","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}

引用次数: 0

Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications 参数椭圆特征值问题的解析和 Gevrey 类正则性及其应用

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-08-05 DOI: 10.1137/23m1596296

Alexey Chernov, Tùng Lê

引用次数: 0

Discontinuous Galerkin Methods for 3D–1D Systems 三维-一维系统的非连续伽勒金方法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-08-02 DOI: 10.1137/23m1627390

Rami Masri, Miroslav Kuchta, Beatrice Riviere

引用次数: 0

Learning Homogenization for Elliptic Operators 椭圆算子的学习均质化

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-08-02 DOI: 10.1137/23m1585015

Kaushik Bhattacharya, Nikola B. Kovachki, Aakila Rajan, Andrew M. Stuart, Margaret Trautner

{"title":"Learning Homogenization for Elliptic Operators","authors":"Kaushik Bhattacharya, Nikola B. Kovachki, Aakila Rajan, Andrew M. Stuart, Margaret Trautner","doi":"10.1137/23m1585015","DOIUrl":"https://doi.org/10.1137/23m1585015","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1844-1873, August 2024. Abstract. Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context, in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"42 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141880249","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}

引用次数: 0

Optimal [math] Error Analysis of a Loosely Coupled Finite Element Scheme for Thin-Structure Interactions 针对薄结构相互作用的松耦合有限元方案的最优[数学]误差分析

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-07-30 DOI: 10.1137/23m1578401

Buyang Li, Weiwei Sun, Yupei Xie, Wenshan Yu

{"title":"Optimal [math] Error Analysis of a Loosely Coupled Finite Element Scheme for Thin-Structure Interactions","authors":"Buyang Li, Weiwei Sun, Yupei Xie, Wenshan Yu","doi":"10.1137/23m1578401","DOIUrl":"https://doi.org/10.1137/23m1578401","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1782-1813, August 2024. Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard [math] norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thin-structure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the [math] norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"74 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141857639","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}

引用次数: 0

Polynomial Interpolation of Function Averages on Interval Segments 区间段上函数平均值的多项式内插法

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-07-25 DOI: 10.1137/23m1598271

Ludovico Bruni Bruno, Wolfgang Erb

引用次数: 0

Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework 无限延迟方程：在抽象框架中实现数值稳定性和分岔的伪谱离散化

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-07-25 DOI: 10.1137/23m1581133

Francesca Scarabel, Rossana Vermiglio

{"title":"Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework","authors":"Francesca Scarabel, Rossana Vermiglio","doi":"10.1137/23m1581133","DOIUrl":"https://doi.org/10.1137/23m1581133","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1736-1758, August 2024. Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141764421","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}

引用次数: 0

Accurately Recover Global Quasiperiodic Systems by Finite Points 用有限点精确恢复全局准周期系统

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-07-24 DOI: 10.1137/23m1620247

Kai Jiang, Qi Zhou, Pingwen Zhang

{"title":"Accurately Recover Global Quasiperiodic Systems by Finite Points","authors":"Kai Jiang, Qi Zhou, Pingwen Zhang","doi":"10.1137/23m1620247","DOIUrl":"https://doi.org/10.1137/23m1620247","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024. Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higher-dimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141764227","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}

引用次数: 0

Duality-Based Error Control for the Signorini Problem 基于对偶性的西格诺里尼问题误差控制

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-07-23 DOI: 10.1137/22m1534791

Ben S. Ashby, Tristan Pryer

引用次数: 0

Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions 具有非均质 Dirichlet 边界条件的稳定广义 Navier-Stokes 方程的有限元离散化

IF 2.9 2区数学

SIAM Journal on Numerical Analysis Pub Date : 2024-07-23 DOI: 10.1137/23m1607398

Julius Jeßberger, Alex Kaltenbach

引用次数: 0