arXiv - MATH - Numerical Analysis最新文献_第2页

Numerical approximation of bi-harmonic wave maps into spheres 进入球面的双谐波图的数值逼近

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11366

Ľubomír Baňas, Sebastian Herr

{"title":"Numerical approximation of bi-harmonic wave maps into spheres","authors":"Ľubomír Baňas, Sebastian Herr","doi":"arxiv-2409.11366","DOIUrl":"https://doi.org/arxiv-2409.11366","url":null,"abstract":"We construct a structure preserving non-conforming finite element\u0000approximation scheme for the bi-harmonic wave maps into spheres equation. It\u0000satisfies a discrete energy law and preserves the non-convex sphere constraint\u0000of the continuous problem. The discrete sphere constraint is enforced at the\u0000mesh-points via a discrete Lagrange multiplier. This approach restricts the\u0000spatial approximation to the (non-conforming) linear finite elements. We show\u0000that the numerical approximation converges to the weak solution of the\u0000continuous problem in spatial dimension $d=1$. The convergence analysis in\u0000dimensions $d>1$ is complicated by the lack of a discrete product rule as well\u0000as the low regularity of the numerical approximation in the non-conforming\u0000setting. Hence, we show convergence of the numerical approximation in\u0000higher-dimensions by introducing additional stabilization terms in the\u0000numerical approximation. We present numerical experiments to demonstrate the\u0000performance of the proposed numerical approximation and to illustrate the\u0000regularizing effect of the bi-Laplacian which prevents the formation of\u0000singularities.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Spectral Volume from a DG perspective: Oscillation Elimination, Stability, and Optimal Error Estimates 从 DG 角度看频谱量：振荡消除、稳定性和最佳误差估计

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.10871

Zhuoyun Li, Kailiang Wu

{"title":"Spectral Volume from a DG perspective: Oscillation Elimination, Stability, and Optimal Error Estimates","authors":"Zhuoyun Li, Kailiang Wu","doi":"arxiv-2409.10871","DOIUrl":"https://doi.org/arxiv-2409.10871","url":null,"abstract":"The discontinuous Galerkin (DG) method and the spectral volume (SV) method\u0000are two widely-used numerical methodologies for solving hyperbolic conservation\u0000laws. In this paper, we demonstrate that under specific subdivision\u0000assumptions, the SV method can be represented in a DG form with a different\u0000inner product. Building on this insight, we extend the oscillation-eliminating\u0000(OE) technique, recently proposed in [M. Peng, Z. Sun, and K. Wu, {it\u0000Mathematics of Computation}, https://doi.org/10.1090/mcom/3998], to develop a\u0000new fully-discrete OESV method. The OE technique is non-intrusive, efficient,\u0000and straightforward to implement, acting as a simple post-processing filter to\u0000effectively suppress spurious oscillations. From a DG perspective, we present a\u0000comprehensive framework to theoretically analyze the stability and accuracy of\u0000both general Runge-Kutta SV (RKSV) schemes and the novel OESV method. For the\u0000linear advection equation, we conduct an energy analysis of the fully-discrete\u0000RKSV method, identifying an upwind condition crucial for stability.\u0000Furthermore, we establish optimal error estimates for the OESV schemes,\u0000overcoming nonlinear challenges through error decomposition and treating the OE\u0000procedure as additional source terms in the RKSV schemes. Extensive numerical\u0000experiments validate our theoretical findings and demonstrate the effectiveness\u0000and robustness of the proposed OESV method. This work enhances the theoretical\u0000understanding and practical application of SV schemes for hyperbolic\u0000conservation laws, making the OESV method a promising approach for\u0000high-resolution simulations.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Local discontinuous Galerkin method for nonlinear BSPDEs of Neumann boundary conditions with deep backward dynamic programming time-marching 采用深度后向动态程序设计时间行进的 Neumann 边界条件非线性 BSPDE 的局部非连续 Galerkin 方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11004

Yixiang Dai, Yunzhang Li, Jing Zhang

引用次数: 0

A Fractional spectral method for weakly singular Volterra integro-differential equations with delays of the third-kind 带第三类延迟的弱奇异 Volterra 积分微分方程的分数谱方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.10861

Borui Zhao

引用次数: 0

A lattice Boltzmann method for Biot's consolidation model of linear poroelasticity 线性孔弹性毕奥固结模型的格点玻尔兹曼法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11382

Stephan B. Lunowa, Barbara Wohlmuth

{"title":"A lattice Boltzmann method for Biot's consolidation model of linear poroelasticity","authors":"Stephan B. Lunowa, Barbara Wohlmuth","doi":"arxiv-2409.11382","DOIUrl":"https://doi.org/arxiv-2409.11382","url":null,"abstract":"Biot's consolidation model is a classical model for the evolution of\u0000deformable porous media saturated by a fluid and has various interdisciplinary\u0000applications. While numerical solution methods to solve poroelasticity by\u0000typical schemes such as finite differences, finite volumes or finite elements\u0000have been intensely studied, lattice Boltzmann methods for poroelasticity have\u0000not been developed yet. In this work, we propose a novel semi-implicit coupling\u0000of lattice Boltzmann methods to solve Biot's consolidation model in two\u0000dimensions. To this end, we use a single-relaxation-time lattice Boltzmann\u0000method for reaction-diffusion equations to solve the Darcy flow and combine it\u0000with a recent pseudo-time multi-relaxation-time lattice Boltzmann scheme for\u0000quasi-static linear elasticity by Boolakee, Geier and De Lorenzis (2023, DOI:\u000010.1016/j.cma.2022.115756). The numerical results demonstrate that naive\u0000coupling schemes lead to instabilities when the poroelastic system is strongly\u0000coupled. However, the newly developed centered coupling scheme using fully\u0000explicit and semi-implicit contributions is stable and accurate in all\u0000considered cases, even for the Biot--Willis coefficient being one. Furthermore,\u0000the numerical results for Terzaghi's consolidation problem and a\u0000two-dimensional extension thereof highlight that the scheme is even able to\u0000capture discontinuous solutions arising from instantaneous loading.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Nonconvex models for recovering images corrupted by salt-and-pepper noise on surfaces 用于恢复被表面椒盐噪声破坏的图像的非凸模型

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11139

Yuan Liu, Peiqi Yu, Chao Zeng

引用次数: 0

A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems 鲍尔-费克定理的非线性概括和解决非线性特征值问题的新颖迭代法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11098

Ronald Katende

{"title":"A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems","authors":"Ronald Katende","doi":"arxiv-2409.11098","DOIUrl":"https://doi.org/arxiv-2409.11098","url":null,"abstract":"Nonlinear eigenvalue problems (NEPs) present significant challenges due to\u0000their inherent complexity and the limitations of traditional linear eigenvalue\u0000theory. This paper addresses these challenges by introducing a nonlinear\u0000generalization of the Bauer-Fike theorem, which serves as a foundational result\u0000in classical eigenvalue theory. This generalization provides a robust\u0000theoretical framework for understanding the sensitivity of eigenvalues in NEPs,\u0000extending the applicability of the Bauer-Fike theorem beyond linear cases.\u0000Building on this theoretical foundation, we propose novel iterative methods\u0000designed to efficiently solve NEPs. These methods leverage the generalized\u0000theorem to improve convergence rates and accuracy, making them particularly\u0000effective for complex NEPs with dense spectra. The adaptive contour integral\u0000method, in particular, is highlighted for its ability to identify multiple\u0000eigenvalues within a specified region of the complex plane, even in cases where\u0000eigenvalues are closely clustered. The efficacy of the proposed methods is\u0000demonstrated through a series of numerical experiments, which illustrate their\u0000superior performance compared to existing approaches. These results underscore\u0000the practical applicability of our methods in various scientific and\u0000engineering contexts. In conclusion, this paper represents a significant\u0000advancement in the study of NEPs by providing a unified theoretical framework\u0000and effective computational tools, thereby bridging the gap between theory and\u0000practice in the field of nonlinear eigenvalue problems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A node-based uniform strain virtual element method for elastoplastic solids 基于节点的弹塑性固体均匀应变虚拟元素法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.10808

Rodrigo Silva-Valenzuela, Alejandro Ortiz-Bernardin, Edoardo Artioli

引用次数: 0

A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations 地球物理标记单元内模拟中严重条件不良线性系统的稀疏求解器比较

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11515

Marcel Ferrari

{"title":"A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations","authors":"Marcel Ferrari","doi":"arxiv-2409.11515","DOIUrl":"https://doi.org/arxiv-2409.11515","url":null,"abstract":"Solving sparse linear systems is a critical challenge in many scientific and\u0000engineering fields, particularly when these systems are severely\u0000ill-conditioned. This work aims to provide a comprehensive comparison of\u0000various solvers designed for such problems, offering valuable insights and\u0000guidance for domain scientists and researchers. We develop the tools required\u0000to accurately evaluate the performance and correctness of 16 solvers from 11\u0000state-of-the-art numerical libraries, focusing on their effectiveness in\u0000handling ill-conditioned matrices. The solvers were tested on linear systems\u0000arising from a coupled hydro-mechanical marker-in-cell geophysical simulation.\u0000To address the challenge of computing accurate error bounds on the solution, we\u0000introduce the Projected Adam method, a novel algorithm to efficiently compute\u0000the condition number of a matrix without relying on eigenvalues or singular\u0000values. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO,\u0000UMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This\u0000work serves as a resource for selecting appropriate solvers, understanding the\u0000impact of condition numbers, and improving the robustness of numerical\u0000solutions in practical applications.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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 Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-type Equation of State 具有一般辛格型状态方程的相对论流体力学的高阶精确熵稳定方案

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.10872

Linfeng Xu, Shengrong Ding, Kailiang Wu

{"title":"High-order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-type Equation of State","authors":"Linfeng Xu, Shengrong Ding, Kailiang Wu","doi":"arxiv-2409.10872","DOIUrl":"https://doi.org/arxiv-2409.10872","url":null,"abstract":"All the existing entropy stable (ES) schemes for relativistic hydrodynamics\u0000(RHD) in the literature were restricted to the ideal equation of state (EOS),\u0000which however is often a poor approximation for most relativistic flows due to\u0000its inconsistency with the relativistic kinetic theory. This paper develops\u0000high-order ES finite difference schemes for RHD with general Synge-type EOS,\u0000which encompasses a range of special EOSs. We first establish an entropy pair\u0000for the RHD equations with general Synge-type EOS in any space dimensions. We\u0000rigorously prove that the found entropy function is strictly convex and derive\u0000the associated entropy variables, laying the foundation for designing entropy\u0000conservative (EC) and ES schemes. Due to relativistic effects, one cannot\u0000explicitly express primitive variables, fluxes, and entropy variables in terms\u0000of conservative variables. Consequently, this highly complicates the analysis\u0000of the entropy structure of the RHD equations, the investigation of entropy\u0000convexity, and the construction of EC numerical fluxes. By using a suitable set\u0000of parameter variables, we construct novel two-point EC fluxes in a unified\u0000form for general Synge-type EOS. We obtain high-order EC schemes through linear\u0000combinations of the two-point EC fluxes. Arbitrarily high-order accurate ES\u0000schemes are achieved by incorporating dissipation terms into the EC schemes,\u0000based on (weighted) essentially non-oscillatory reconstructions. Additionally,\u0000we derive the general dissipation matrix for general Synge-type EOS based on\u0000the scaled eigenvectors of the RHD system. We also define a suitable average of\u0000the dissipation matrix at the cell interfaces to ensure that the resulting ES\u0000schemes can resolve stationary contact discontinuities accurately. Several\u0000numerical examples are provided to validate the accuracy and effectiveness of\u0000our schemes for RHD with four special EOSs.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0