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Effective highly accurate time integrators for linear Klein–Gordon equations across the scales 跨尺度线性克莱因-戈登方程的有效高精度时间积分器
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2024-09-11 DOI: 10.1515/jnma-2023-0070
Karolina Kropielnicka, Karolina Lademann, Katharina Schratz
{"title":"Effective highly accurate time integrators for linear Klein–Gordon equations across the scales","authors":"Karolina Kropielnicka, Karolina Lademann, Katharina Schratz","doi":"10.1515/jnma-2023-0070","DOIUrl":"https://doi.org/10.1515/jnma-2023-0070","url":null,"abstract":"We propose an efficient approach for time integration of Klein–Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192664","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
Obituary for Professor Yuri Kuznetsov 尤里-库兹涅佐夫教授讣告
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2024-06-01 DOI: 10.1515/jnma-2024-1078
{"title":"Obituary for Professor Yuri Kuznetsov","authors":"","doi":"10.1515/jnma-2024-1078","DOIUrl":"https://doi.org/10.1515/jnma-2024-1078","url":null,"abstract":"","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141396858","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
Analysis and Computation of a Weak Galerkin Scheme for Solving the 2D/3D Stationary Stokes Interface Problems with High-Order Elements 用高阶元素解决二维/三维静态斯托克斯界面问题的弱 Galerkin 方案的分析与计算
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2024-04-04 DOI: 10.1515/jnma-2023-0112
Raman Kumar, Bhupen Deka
{"title":"Analysis and Computation of a Weak Galerkin Scheme for Solving the 2D/3D Stationary Stokes Interface Problems with High-Order Elements","authors":"Raman Kumar, Bhupen Deka","doi":"10.1515/jnma-2023-0112","DOIUrl":"https://doi.org/10.1515/jnma-2023-0112","url":null,"abstract":"In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ<jats:sup> <jats:italic>d</jats:italic> </jats:sup> (<jats:italic>d</jats:italic> = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree <jats:italic>k</jats:italic> ≥ 1 for the velocity and polynomials of degree <jats:italic>k</jats:italic> – 1 for the pressure, both are discontinuous. Optimal convergence rates of order <jats:italic>k</jats:italic> + 1 for the velocity and order <jats:italic>k</jats:italic> for the pressure are established in <jats:italic>L</jats:italic> <jats:sup>2</jats:sup>-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575326","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 evaluation of symmetry-adapted n-correlations via recursive contraction of sparse symmetric tensors 通过稀疏对称张量的递归收缩优化对称适配 n 相关性评估
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2024-03-30 DOI: 10.1515/jnma-2024-0025
Illia Kaliuzhnyi, Christoph Ortner
{"title":"Optimal evaluation of symmetry-adapted n-correlations via recursive contraction of sparse symmetric tensors","authors":"Illia Kaliuzhnyi, Christoph Ortner","doi":"10.1515/jnma-2024-0025","DOIUrl":"https://doi.org/10.1515/jnma-2024-0025","url":null,"abstract":"We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant (or equi-variant) under permutations and rotations. This task arises in the evaluation linear models as well as equivariant neural network models of many-particle systems. The theoretical bottleneck is the contraction of a high-dimensional symmetric and sparse tensor with a specific sparsity pattern that is directly related to the symmetries imposed on the polynomial. The sparsity of this tensor makes it challenging to construct a highly efficient evaluation scheme. The references [10, 11] introduced a recursive evaluation strategy that relied on a number of heuristics, but performed well in tests. In the present work, we propose an explicit construction of such a recursive evaluation strategy and show that it is in fact optimal in the limit of infinite polynomial degree.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575164","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
A fully well-balanced hydrodynamic reconstruction 完全平衡的水动力重建
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2024-03-25 DOI: 10.1515/jnma-2023-0065
Christophe Berthon, Victor Michel-Dansac
{"title":"A fully well-balanced hydrodynamic reconstruction","authors":"Christophe Berthon, Victor Michel-Dansac","doi":"10.1515/jnma-2023-0065","DOIUrl":"https://doi.org/10.1515/jnma-2023-0065","url":null,"abstract":"The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297416","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
Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations 探索Keller-Segel-Navier-Stokes方程的数值爆破现象
2区 数学
Journal of Numerical Mathematics Pub Date : 2023-10-16 DOI: 10.1515/jnma-2023-0016
Jesús Bonilla, Juan Vicente Gutiérrez-Santacreu
{"title":"Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations","authors":"Jesús Bonilla, Juan Vicente Gutiérrez-Santacreu","doi":"10.1515/jnma-2023-0016","DOIUrl":"https://doi.org/10.1515/jnma-2023-0016","url":null,"abstract":"Abstract The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2 π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L 1 (Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2 π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4 π may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136114648","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
Local parameter selection in the C0 interior penalty method for the biharmonic equation 双调和方程C0内罚法的局部参数选择
2区 数学
Journal of Numerical Mathematics Pub Date : 2023-09-11 DOI: 10.1515/jnma-2023-0028
Philipp Bringmann, Carsten Carstensen, Julian Streitberger
{"title":"Local parameter selection in the C<sup>0</sup> interior penalty method for the biharmonic equation","authors":"Philipp Bringmann, Carsten Carstensen, Julian Streitberger","doi":"10.1515/jnma-2023-0028","DOIUrl":"https://doi.org/10.1515/jnma-2023-0028","url":null,"abstract":"Abstract The symmetric 0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh-refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. a detailed documentation of C 0 interior penalty method in.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135979905","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
On the discrete Sobolev inequalities 关于离散Sobolev不等式
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2023-09-05 DOI: 10.1515/jnma-2023-0086
Sédrick Kameni Ngwamou, Michael Ndjinga
{"title":"On the discrete Sobolev inequalities","authors":"Sédrick Kameni Ngwamou, Michael Ndjinga","doi":"10.1515/jnma-2023-0086","DOIUrl":"https://doi.org/10.1515/jnma-2023-0086","url":null,"abstract":"Abstract We prove a discrete version of the famous Sobolev inequalities [1] in R d for d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $mathbb{R}^{d} text { for } d in mathbb{N}^{*}, p in[1,+infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d into L d d − 1 $B Vleft(mathbb{R}^{d}right) text { into } mathrm{L}^{frac{d}{d-1}}$ [1, theorem 9.9],[12, theorem 1.1] by introducing discrete analogs of the directional total variations. In the case p > d (Gagliardo-Nirenberg inequality), we adapt the proof of the continuous case ( [1, theorem 9.9], [9, theorem 4.8]) and use techniques from [3, 5]. In the case p > d (Morrey’s inequality), we simplify and extend the proof of [12, theorem 1.1] to more general meshes.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80063195","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
Frontmatter 头版头条
2区 数学
Journal of Numerical Mathematics Pub Date : 2023-09-01 DOI: 10.1515/jnma-2023-frontmatter3
{"title":"Frontmatter","authors":"","doi":"10.1515/jnma-2023-frontmatter3","DOIUrl":"https://doi.org/10.1515/jnma-2023-frontmatter3","url":null,"abstract":"","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135200505","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
Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn–Hilliard systems with bounded mass source 有界质量源Cahn-Hilliard系统松弛标量辅助变量格式的稳定性和收敛性
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2023-08-30 DOI: 10.1515/jnma-2023-0021
K. F. Lam, Ru Wang
{"title":"Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn–Hilliard systems with bounded mass source","authors":"K. F. Lam, Ru Wang","doi":"10.1515/jnma-2023-0021","DOIUrl":"https://doi.org/10.1515/jnma-2023-0021","url":null,"abstract":"Abstract The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn–Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg–Landau energy. Hence, compared to previous works, the stability of SAV-discrete solutions for such systems is not immediate. We establish, with a bounded mass source, stability and convergence of time discrete solutions for a first-order relaxed SAV scheme in the sense of Jiang et al. (2022), and apply our ideas to Cahn–Hilliard systems with mass source appearing in diblock co-polymer phase separation, tumor growth, image inpainting and segmentation.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83313876","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
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