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Energy stable semi-implicit schemes for the 2D Allen–Cahn and fractional Cahn–Hilliard equations 二维Allen-Cahn和分数Cahn-Hilliard方程的能量稳定半隐式格式
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-31 DOI: 10.1093/imanum/draf010
Xinyu Cheng
{"title":"Energy stable semi-implicit schemes for the 2D Allen–Cahn and fractional Cahn–Hilliard equations","authors":"Xinyu Cheng","doi":"10.1093/imanum/draf010","DOIUrl":"https://doi.org/10.1093/imanum/draf010","url":null,"abstract":"In this work, we are interested in a class of numerical schemes for certain phase field models. It is well known that unconditional energy stability (energy decays in time regardless of the size of the time step) provides a fidelity check in practical numerical simulations. In recent work (Li, D. (2022b, Why large time-stepping methods for the Cahn–Hilliard equation is stable. Math. Comp., 91, 2501–2515)), a type of semi-implicit scheme for the Cahn–Hilliard (CH) equation with regular potential was developed satisfying the energy-decay property. In this paper, we extend such semi-implicit schemes to the Allen–Cahn equation and the fractional CH equation with a rigorous proof of similar energy stability. Models in two spatial dimensions are discussed.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143744936","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 convergence of the arbitrary Lagrangian–Eulerian interface tracking method for two-phase Navier–Stokes flow without surface tension 无表面张力两相Navier-Stokes流的任意拉格朗日-欧拉界面跟踪方法的最优收敛性
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-29 DOI: 10.1093/imanum/draf003
Buyang Li, Shu Ma, Weifeng Qiu
{"title":"Optimal convergence of the arbitrary Lagrangian–Eulerian interface tracking method for two-phase Navier–Stokes flow without surface tension","authors":"Buyang Li, Shu Ma, Weifeng Qiu","doi":"10.1093/imanum/draf003","DOIUrl":"https://doi.org/10.1093/imanum/draf003","url":null,"abstract":"Optimal-order convergence in the $H^{1}$ norm is proved for an arbitrary Lagrangian–Eulerian (ALE) interface tracking finite element method (FEM) for the sharp interface model of two-phase Navier–Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid’s velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete ALE interface tracking FEM is shown to be $O(h^{k})$ in the $L^infty (0, T; H^{1}(varOmega ))$ norm for the Taylor–Hood finite elements of degree $k geqslant 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143736499","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
Finite element approximation of the Einstein tensor 爱因斯坦张量的有限元近似
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-29 DOI: 10.1093/imanum/draf004
Evan S Gawlik, Michael Neunteufel
{"title":"Finite element approximation of the Einstein tensor","authors":"Evan S Gawlik, Michael Neunteufel","doi":"10.1093/imanum/draf004","DOIUrl":"https://doi.org/10.1093/imanum/draf004","url":null,"abstract":"We construct and analyse finite element approximations of the Einstein tensor in dimension $N ge 3$. We focus on the setting where a smooth Riemannian metric tensor $g$ on a polyhedral domain $varOmega subset mathbb{R}^{N}$ has been approximated by a piecewise polynomial metric $g_{h}$ on a simplicial triangulation $mathcal{T}$ of $varOmega $ having maximum element diameter $h$. We assume that $g_{h}$ possesses single-valued tangential–tangential components on every codimension-$1$ simplex in $mathcal{T}$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $g_{h}$ to the Einstein curvature of $g$ under refinement of the triangulation. We show that in the $H^{-2}(varOmega )$-norm this convergence takes place at a rate of $O(h^{r+1})$ when $g_{h}$ is an optimal-order interpolant of $g$ that is piecewise polynomial of degree $r ge 1$. We provide numerical evidence to support this claim. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143736497","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
Complex generalized Gauss–Radau quadrature rules for Hankel transforms of integer order 整数阶Hankel变换的复广义Gauss-Radau正交规则
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-23 DOI: 10.1093/imanum/draf007
Haiyong Wang, Menghan Wu
{"title":"Complex generalized Gauss–Radau quadrature rules for Hankel transforms of integer order","authors":"Haiyong Wang, Menghan Wu","doi":"10.1093/imanum/draf007","DOIUrl":"https://doi.org/10.1093/imanum/draf007","url":null,"abstract":"Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to $[0,1/2]$. In this paper we introduce a new family of Gaussian quadrature rules for Hankel transforms of integer order. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss–Radau quadrature rules that guarantee existence can be constructed and their nodes and weights can be calculated from a half-size Gaussian quadrature rule with respect to the generalized Prudnikov weight function. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to show the performance of the proposed rules.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"61 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143677760","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
Numerical schemes for radial Dunkl processes 径向Dunkl过程的数值格式
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-12 DOI: 10.1093/imanum/draf005
Hoang-Long Ngo, Dai Taguchi
{"title":"Numerical schemes for radial Dunkl processes","authors":"Hoang-Long Ngo, Dai Taguchi","doi":"10.1093/imanum/draf005","DOIUrl":"https://doi.org/10.1093/imanum/draf005","url":null,"abstract":"We consider the numerical approximation for a class of radial Dunkl processes corresponding to arbitrary (reduced) root systems in $mathbb{R}^{d}$. This class contains well-known processes such as Bessel processes, Dyson’s Brownian motions and square root of Wishart processes. We propose some semi-implicit and truncated Euler–Maruyama schemes for radial Dunkl processes and study their convergence rate with respect to the $L^{p}$-sup norm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143607954","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-posteriori error estimates for systems of hyperbolic conservation laws via computing negative norms of local residuals 计算局部残差负规范的双曲守恒律系统的后验误差估计
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-11 DOI: 10.1093/imanum/drae111
Jan Giesselmann, Aleksey Sikstel
{"title":"A-posteriori error estimates for systems of hyperbolic conservation laws via computing negative norms of local residuals","authors":"Jan Giesselmann, Aleksey Sikstel","doi":"10.1093/imanum/drae111","DOIUrl":"https://doi.org/10.1093/imanum/drae111","url":null,"abstract":"We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599894","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
Well-posedness of first-order acoustic wave equations and space-time finite element approximation 一阶声波方程的良好拟合与时空有限元近似
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-11 DOI: 10.1093/imanum/drae104
Thomas Führer, Roberto González, Michael Karkulik
{"title":"Well-posedness of first-order acoustic wave equations and space-time finite element approximation","authors":"Thomas Führer, Roberto González, Michael Karkulik","doi":"10.1093/imanum/drae104","DOIUrl":"https://doi.org/10.1093/imanum/drae104","url":null,"abstract":"We study a first-order system formulation of the (acoustic) wave equation and prove that the operator of this system is an isomorphism from an appropriately defined graph space to $L^{2}$. The results rely on well-posedness and stability of the weak and ultraweak formulation of the second-order wave equation. As an application, we define and analyze a space-time least-squares finite element method for solving the wave equation. Some numerical examples for one- and two-dimensional spatial domains are presented.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"86 1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599893","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 noncoforming virtual element approximation for the Oseen eigenvalue problem Oseen特征值问题的非共形虚元逼近
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-11 DOI: 10.1093/imanum/drae108
Dibyendu Adak, Felipe Lepe, Gonzalo Rivera
{"title":"A noncoforming virtual element approximation for the Oseen eigenvalue problem","authors":"Dibyendu Adak, Felipe Lepe, Gonzalo Rivera","doi":"10.1093/imanum/drae108","DOIUrl":"https://doi.org/10.1093/imanum/drae108","url":null,"abstract":"In this paper, we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method that is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators, we prove convergence and error estimates for the method. By employing the theory of compact operators, we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599895","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
Geometry error analysis of a parametric mapping for higher order unfitted space–time methods 高阶非拟合时空方法参数映射的几何误差分析
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-10 DOI: 10.1093/imanum/drae098
Fabian Heimann, Christoph Lehrenfeld
{"title":"Geometry error analysis of a parametric mapping for higher order unfitted space–time methods","authors":"Fabian Heimann, Christoph Lehrenfeld","doi":"10.1093/imanum/drae098","DOIUrl":"https://doi.org/10.1093/imanum/drae098","url":null,"abstract":"In Heimann, Lehrenfeld, and Preuß (2023, SIAM J. Sci. Comp., 45(2), B139–B165), new geometrically unfitted space–time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and time have been introduced. For geometrically higher-order accuracy a parametric mapping on a background space–time tensor-product mesh has been used. In this paper, we concentrate on the geometrical accuracy of the approximation and derive rigorous bounds for the distance between the realized and an ideal mapping in different norms and derive results for the space–time regularity of the parametric mapping. These results are important and lay the ground for the error analysis of corresponding unfitted space–time finite element methods.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599897","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 finite element approximation of a diffuse interface approach to the Stokes–Biot coupling Stokes-Biot耦合扩散界面方法的分析与有限元逼近
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2025-03-10 DOI: 10.1093/imanum/draf002
Francis R A Aznaran, Martina Bukač, Boris Muha, Abner J Salgado
{"title":"Analysis and finite element approximation of a diffuse interface approach to the Stokes–Biot coupling","authors":"Francis R A Aznaran, Martina Bukač, Boris Muha, Abner J Salgado","doi":"10.1093/imanum/draf002","DOIUrl":"https://doi.org/10.1093/imanum/draf002","url":null,"abstract":"We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $mathcal{O}(varepsilon)$ around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretization of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We, in turn, prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"54 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599896","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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