In this work, we present a novel approach to type design by using Fourier-type series to generate letterforms. We construct a Fourier-type series for functions in $L^2(S^1,mathbb C)$ based on triangles of constant width instead of circles to model the curves and shapes that define individual characters. In order to compute the coefficients of the series, we construct an isomorphism $mathcal R:L^2(S^1,mathbb C)to L^2(S^1,mathbb C)$ and study its application to letterforms, thus presenting an alternative to the common use of B'ezier curves. The proposed method demonstrates potential for creative experimentation in modern type design.
{"title":"Application of a Fourier-Type Series Approach based on Triangles of Constant Width to Letterforms","authors":"Micha Wasem, Florence Yerly","doi":"arxiv-2409.11958","DOIUrl":"https://doi.org/arxiv-2409.11958","url":null,"abstract":"In this work, we present a novel approach to type design by using\u0000Fourier-type series to generate letterforms. We construct a Fourier-type series\u0000for functions in $L^2(S^1,mathbb C)$ based on triangles of constant width\u0000instead of circles to model the curves and shapes that define individual\u0000characters. In order to compute the coefficients of the series, we construct an\u0000isomorphism $mathcal R:L^2(S^1,mathbb C)to L^2(S^1,mathbb C)$ and study its\u0000application to letterforms, thus presenting an alternative to the common use of\u0000B'ezier curves. The proposed method demonstrates potential for creative\u0000experimentation in modern type design.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259036","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}
Andrea BordignonCERMICS, Geneviève DussonLMB, Éric CancèsCERMICS, MATHERIALS, Gaspard KemlinLAMFA, Rafael Antonio Lainez ReyesIANS, Benjamin StammIANS
In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation functionals of practical interest.
{"title":"Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals","authors":"Andrea BordignonCERMICS, Geneviève DussonLMB, Éric CancèsCERMICS, MATHERIALS, Gaspard KemlinLAMFA, Rafael Antonio Lainez ReyesIANS, Benjamin StammIANS","doi":"arxiv-2409.11769","DOIUrl":"https://doi.org/arxiv-2409.11769","url":null,"abstract":"In this article, we derive fully guaranteed error bounds for the energy of\u0000convex nonlinear mean-field models. These results apply in particular to\u0000Kohn-Sham equations with convex density functionals, which includes the reduced\u0000Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact\u0000exchange-density functional (which is unfortunately not explicit and therefore\u0000not usable in practice). We then decompose the obtained bounds into two parts,\u0000one depending on the chosen discretization and one depending on the number of\u0000iterations performed in the self-consistent algorithm used to solve the\u0000nonlinear eigenvalue problem, paving the way for adaptive refinement\u0000strategies. The accuracy of the bounds is demonstrated on a series of test\u0000cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated\u0000with the rHF model and discretized with planewaves. We also show that, although\u0000not anymore guaranteed, the error bounds remain very accurate for a Silicon\u0000crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation\u0000functionals of practical interest.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259032","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}
This work introduces a new solution-transfer process for slab-based space-time finite element methods. The new transfer process is based on Hsieh-Clough-Tocher (HCT) splines and satisfies the following requirements: (i) it maintains high-order accuracy up to 4th order, (ii) it preserves a discrete maximum principle, (iii) it enforces mass conservation, and (iv) it constructs a smooth, continuous surrogate solution in between space-time slabs. While many existing transfer methods meet the first three requirements, the fourth requirement is crucial for enabling visualization and boundary condition enforcement for space-time applications. In this paper, we derive an error bound for our HCT spline-based transfer process. Additionally, we conduct numerical experiments quantifying the conservative nature and order of accuracy of the transfer process. Lastly, we present a qualitative evaluation of the visualization properties of the smooth surrogate solution.
{"title":"Spline-based solution transfer for space-time methods in 2D+t","authors":"Logan Larose, Jude T. Anderson, David M. Williams","doi":"arxiv-2409.11639","DOIUrl":"https://doi.org/arxiv-2409.11639","url":null,"abstract":"This work introduces a new solution-transfer process for slab-based\u0000space-time finite element methods. The new transfer process is based on\u0000Hsieh-Clough-Tocher (HCT) splines and satisfies the following requirements: (i)\u0000it maintains high-order accuracy up to 4th order, (ii) it preserves a discrete\u0000maximum principle, (iii) it enforces mass conservation, and (iv) it constructs\u0000a smooth, continuous surrogate solution in between space-time slabs. While many\u0000existing transfer methods meet the first three requirements, the fourth\u0000requirement is crucial for enabling visualization and boundary condition\u0000enforcement for space-time applications. In this paper, we derive an error\u0000bound for our HCT spline-based transfer process. Additionally, we conduct\u0000numerical experiments quantifying the conservative nature and order of accuracy\u0000of the transfer process. Lastly, we present a qualitative evaluation of the\u0000visualization properties of the smooth surrogate solution.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259037","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}
Daniele Moretto, Andrea Franceschini, Massimiliano Ferronato
Recent advancements in computational capabilities have significantly enhanced the numerical simulation of complex multiphysics and multidomain problems. However, mesh generation remains a primary bottleneck in these simulations. To address this challenge, non-conforming grids are often utilized, which necessitates the development of robust and efficient intergrid interpolator operators. This paper presents a novel approach for transferring variable fields across non-conforming meshes within a mortar framework, where weak continuity conditions are imposed. The key contribution of our work is the introduction of an innovative algorithm that utilizes Radial Basis Function (RBF) interpolations to compute the mortar integral, offering a compelling alternative to traditional projection-based algorithms. Pairing RBF methods with numerical integration techniques, we propose an efficient algorithm tailored for complex three-dimensional scenarios. This paper details the formulation, analysis, and validation of the proposed RBF algorithm through a series of numerical examples, demonstrating its effectiveness. Furthermore, the details of the implementation are discussed and a test case involving a complex geometry is presented, to illustrate the applicability and advantages of our approach in addressing real-world problems.
{"title":"A novel Mortar Method Integration using Radial Basis Functions","authors":"Daniele Moretto, Andrea Franceschini, Massimiliano Ferronato","doi":"arxiv-2409.11735","DOIUrl":"https://doi.org/arxiv-2409.11735","url":null,"abstract":"Recent advancements in computational capabilities have significantly enhanced\u0000the numerical simulation of complex multiphysics and multidomain problems.\u0000However, mesh generation remains a primary bottleneck in these simulations. To\u0000address this challenge, non-conforming grids are often utilized, which\u0000necessitates the development of robust and efficient intergrid interpolator\u0000operators. This paper presents a novel approach for transferring variable\u0000fields across non-conforming meshes within a mortar framework, where weak\u0000continuity conditions are imposed. The key contribution of our work is the\u0000introduction of an innovative algorithm that utilizes Radial Basis Function\u0000(RBF) interpolations to compute the mortar integral, offering a compelling\u0000alternative to traditional projection-based algorithms. Pairing RBF methods\u0000with numerical integration techniques, we propose an efficient algorithm\u0000tailored for complex three-dimensional scenarios. This paper details the\u0000formulation, analysis, and validation of the proposed RBF algorithm through a\u0000series of numerical examples, demonstrating its effectiveness. Furthermore, the\u0000details of the implementation are discussed and a test case involving a complex\u0000geometry is presented, to illustrate the applicability and advantages of our\u0000approach in addressing real-world problems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259033","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}
Fredrik Fryklund, Leslie Greengard, Shidong Jiang, Samuel Potter
Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial differential equation itself, one first evaluates a volume integral to account for the source distribution within the domain, followed by solving a boundary integral equation to impose the specified boundary conditions. Here, we present a new fast algorithm which is easy to implement and compatible with virtually any discretization technique, including unstructured domain triangulations, such as those used in standard finite element or finite volume methods. Our approach combines earlier work on potential theory for the heat equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT), and the dual-space multilevel kernel-splitting (DMK) framework. It is insensitive to flaws in the triangulation, permitting not just nonconforming elements, but arbitrary aspect ratio triangles, gaps and various other degeneracies. On a single CPU core, the scheme computes the solution at a rate comparable to that of the fast Fourier transform (FFT) in work per gridpoint.
在过去的二十年里，已经开发出几种快速、稳健和高阶精确的方法，用于利用势理论求解复杂几何中的泊松方程。在这种方法中，我们不是将边际微分方程本身离散化，而是首先求体积积分来计算域内的源分布，然后求解边界积分方程来施加指定的边界条件。在这里，我们提出了一种新的快速算法，这种算法易于实现，而且几乎与任何离散化技术兼容，包括非结构化域三角测量，如标准有限元或有限体积方法中使用的算法。我们的方法结合了早先在氦方程势理论、渐近分析、非均匀快速傅立叶变换（NUFFT）和双空间多级内核拆分（DMK）框架方面的工作。它对三角剖分中的缺陷很敏感，不仅允许不规则的元素，还允许任意长宽比的三角形、间隙和其他各种退行性。在单个 CPU 内核上，该方案计算解的速度可与快速傅立叶变换 (FFT) 计算每个网格点的工作量相媲美。
{"title":"A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials","authors":"Fredrik Fryklund, Leslie Greengard, Shidong Jiang, Samuel Potter","doi":"arxiv-2409.11998","DOIUrl":"https://doi.org/arxiv-2409.11998","url":null,"abstract":"Over the last two decades, several fast, robust, and high-order accurate\u0000methods have been developed for solving the Poisson equation in complicated\u0000geometry using potential theory. In this approach, rather than discretizing the\u0000partial differential equation itself, one first evaluates a volume integral to\u0000account for the source distribution within the domain, followed by solving a\u0000boundary integral equation to impose the specified boundary conditions. Here,\u0000we present a new fast algorithm which is easy to implement and compatible with\u0000virtually any discretization technique, including unstructured domain\u0000triangulations, such as those used in standard finite element or finite volume\u0000methods. Our approach combines earlier work on potential theory for the heat\u0000equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT),\u0000and the dual-space multilevel kernel-splitting (DMK) framework. It is\u0000insensitive to flaws in the triangulation, permitting not just nonconforming\u0000elements, but arbitrary aspect ratio triangles, gaps and various other\u0000degeneracies. On a single CPU core, the scheme computes the solution at a rate\u0000comparable to that of the fast Fourier transform (FFT) in work per gridpoint.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259028","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}
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a unified framework, ensuring stability and accuracy. The schemes are derived and analyzed for their consistency and stability properties, showcasing their effectiveness in practical computational scenarios.
{"title":"Adaptive Time-Step Semi-Implicit One-Step Taylor Scheme for Stiff Ordinary Differential Equations","authors":"S. Boscarino, E. Macca","doi":"arxiv-2409.11990","DOIUrl":"https://doi.org/arxiv-2409.11990","url":null,"abstract":"In this study, we propose high-order implicit and semi-implicit schemes for\u0000solving ordinary differential equations (ODEs) based on Taylor series\u0000expansion. These methods are designed to handle stiff and non-stiff components\u0000within a unified framework, ensuring stability and accuracy. The schemes are\u0000derived and analyzed for their consistency and stability properties, showcasing\u0000their effectiveness in practical computational scenarios.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259029","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}
We study various formulations of the boundary conditions for the Euler equations of gas dynamics from a mathematical and numerical point of view. In the case of one space dimension, we recall the classical results, based on an analysis of the linearized problem. Then we present a more recent formulation of the problem, which allows for nonlinear effects at the boundary of the study domain. This formulation fits naturally into a finite volume discretization, and we present a significant one-dimensional test case.
{"title":"Conditions aux limites fortement non lin{é}aires pour les {é}quations d'Euler de la dynamique des gaz","authors":"François DuboisLMO, LMSSC","doi":"arxiv-2409.11774","DOIUrl":"https://doi.org/arxiv-2409.11774","url":null,"abstract":"We study various formulations of the boundary conditions for the Euler\u0000equations of gas dynamics from a mathematical and numerical point of view. In\u0000the case of one space dimension, we recall the classical results, based on an\u0000analysis of the linearized problem. Then we present a more recent formulation\u0000of the problem, which allows for nonlinear effects at the boundary of the study\u0000domain. This formulation fits naturally into a finite volume discretization,\u0000and we present a significant one-dimensional test case.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259031","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}
Christian Döding, Benjamin Dörich, Patrick Henning
In this work, we study the numerical approximation of minimizers of the Ginzburg-Landau free energy, a common model to describe the behavior of superconductors under magnetic fields. The unknowns are the order parameter, which characterizes the density of superconducting charge carriers, and the magnetic vector potential, which allows to deduce the magnetic field that penetrates the superconductor. Physically important and numerically challenging are especially settings which involve lattices of quantized vortices which can be formed in materials with a large Ginzburg-Landau parameter $kappa$. In particular, $kappa$ introduces a severe mesh resolution condition for numerical approximations. In order to reduce these computational restrictions, we investigate a particular discretization which is based on mixed meshes where we apply a Lagrange finite element approach for the vector potential and a localized orthogonal decomposition (LOD) approach for the order parameter. We justify the proposed method by a rigorous a-priori error analysis (in $L^2$ and $H^1$) in which we keep track of the influence of $kappa$ in all error contributions. This allows us to conclude $kappa$-dependent resolution conditions for the various meshes and which only impose moderate practical constraints compared to a conventional finite element discretization. Finally, our theoretical findings are illustrated by numerical experiments.
{"title":"A multiscale approach to the stationary Ginzburg-Landau equations of superconductivity","authors":"Christian Döding, Benjamin Dörich, Patrick Henning","doi":"arxiv-2409.12023","DOIUrl":"https://doi.org/arxiv-2409.12023","url":null,"abstract":"In this work, we study the numerical approximation of minimizers of the\u0000Ginzburg-Landau free energy, a common model to describe the behavior of\u0000superconductors under magnetic fields. The unknowns are the order parameter,\u0000which characterizes the density of superconducting charge carriers, and the\u0000magnetic vector potential, which allows to deduce the magnetic field that\u0000penetrates the superconductor. Physically important and numerically challenging\u0000are especially settings which involve lattices of quantized vortices which can\u0000be formed in materials with a large Ginzburg-Landau parameter $kappa$. In\u0000particular, $kappa$ introduces a severe mesh resolution condition for\u0000numerical approximations. In order to reduce these computational restrictions,\u0000we investigate a particular discretization which is based on mixed meshes where\u0000we apply a Lagrange finite element approach for the vector potential and a\u0000localized orthogonal decomposition (LOD) approach for the order parameter. We\u0000justify the proposed method by a rigorous a-priori error analysis (in $L^2$ and\u0000$H^1$) in which we keep track of the influence of $kappa$ in all error\u0000contributions. This allows us to conclude $kappa$-dependent resolution\u0000conditions for the various meshes and which only impose moderate practical\u0000constraints compared to a conventional finite element discretization. Finally,\u0000our theoretical findings are illustrated by numerical experiments.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259038","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}
Inspired by the unconstrained PPE (UPPE) formulation [Liu, Liu, & Pego 2007 Comm. Pure Appl. Math., 60 pp. 1443], we previously proposed the GePUP formulation [Zhang 2016 J. Sci. Comput., 67 pp. 1134] for numerically solving the incompressible Navier-Stokes equations (INSE) on no-slip domains. In this paper, we propose GePUP-E and GePUP-ES, variants of GePUP that feature (a) electric boundary conditions with no explicit enforcement of the no-penetration condition, (b) equivalence to the no-slip INSE, (c) exponential decay of the divergence of an initially non-solenoidal velocity, and (d) monotonic decrease of the kinetic energy. Different from UPPE, the GePUP-E and GePUP-ES formulations are of strong forms and are designed for finite volume/difference methods under the framework of method of lines. Furthermore, we develop semi-discrete algorithms that preserve (c) and (d) and fully discrete algorithms that are fourth-order accurate for velocity both in time and in space. These algorithms employ algebraically stable time integrators in a black-box manner and only consist of solving a sequence of linear equations in each time step. Results of numerical tests confirm our analysis.
受无约束 PPE (UPPE) 公式 [Liu, Liu, & Pego 2007Comm. Pure Appl. Math., 60 pp. 1443] 的启发，我们之前提出了 GePUP 公式 [Zhang 2016 J. Sci.在本文中，我们提出了 GePUP-E 和 GePUP-ES，它们是 GePUP 的变体，具有以下特点：（a）不明确执行无渗透条件的电边界条件；（b）等效于无滑动 INSE；（c）初始非滑动速度的发散呈指数衰减；（d）动能单调递减。与 UPPE 不同的是，GePUP-E 和 GePUP-ES 形式是强形式的，是在线性方法框架下为有限体积/差分方法设计的。此外，我们还开发了保留（c）和（d）的半离散算法和完全离散算法，这些算法在时间和空间上对速度都有四阶精度。这些算法以黑箱方式使用代数稳定的时间积分器，只需求解每个时间步的线性方程序列。数值测试结果证实了我们的分析。
{"title":"GePUP-ES: High-order Energy-stable Projection Methods for the Incompressible Navier-Stokes Equations with No-slip Conditions","authors":"Yang Li, Xu Wu, Jiatu Yan, Jiang Yang, Qinghai Zhang, Shubo Zhao","doi":"arxiv-2409.11255","DOIUrl":"https://doi.org/arxiv-2409.11255","url":null,"abstract":"Inspired by the unconstrained PPE (UPPE) formulation [Liu, Liu, & Pego 2007\u0000Comm. Pure Appl. Math., 60 pp. 1443], we previously proposed the GePUP\u0000formulation [Zhang 2016 J. Sci. Comput., 67 pp. 1134] for numerically solving\u0000the incompressible Navier-Stokes equations (INSE) on no-slip domains. In this\u0000paper, we propose GePUP-E and GePUP-ES, variants of GePUP that feature (a)\u0000electric boundary conditions with no explicit enforcement of the no-penetration\u0000condition, (b) equivalence to the no-slip INSE, (c) exponential decay of the\u0000divergence of an initially non-solenoidal velocity, and (d) monotonic decrease\u0000of the kinetic energy. Different from UPPE, the GePUP-E and GePUP-ES\u0000formulations are of strong forms and are designed for finite volume/difference\u0000methods under the framework of method of lines. Furthermore, we develop\u0000semi-discrete algorithms that preserve (c) and (d) and fully discrete\u0000algorithms that are fourth-order accurate for velocity both in time and in\u0000space. These algorithms employ algebraically stable time integrators in a\u0000black-box manner and only consist of solving a sequence of linear equations in\u0000each time step. Results of numerical tests confirm our analysis.","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":"142259039","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}
Numerical simulation of large-scale multiphase and multicomponent flow in porous media is a significant field of interest in the petroleum industry. The fully implicit approach is favored in reservoir simulation due to its numerical stability and relaxed constraints on time-step sizes. However, this method requires solving a large nonlinear system at each time step, making the development of efficient and convergent numerical methods crucial for accelerating the nonlinear solvers. In this paper, we present an adaptively coupled subdomain framework based on the domain decomposition method. The solution methods developed within this framework effectively handle strong nonlinearities in global problems by addressing subproblems in the coupled regions. Furthermore, we propose several adaptive coupling strategies and develop a method for leveraging initial guesses to accelerate the solution of nonlinear problems, thereby improving the convergence and parallel performance of nonlinear solvers. A series of numerical experiments validate the effectiveness of the proposed framework. Additionally, by utilizing tens of thousands of processors, we demonstrate the scalability of this approach through a large-scale reservoir simulation with over 2 billion degrees of freedom.
{"title":"Adaptively Coupled Domain Decomposition Method for Multiphase and Multicomponent Porous Media Flows","authors":"Shizhe Li, Li Zhao, Chen-Song Zhang","doi":"arxiv-2409.10875","DOIUrl":"https://doi.org/arxiv-2409.10875","url":null,"abstract":"Numerical simulation of large-scale multiphase and multicomponent flow in\u0000porous media is a significant field of interest in the petroleum industry. The\u0000fully implicit approach is favored in reservoir simulation due to its numerical\u0000stability and relaxed constraints on time-step sizes. However, this method\u0000requires solving a large nonlinear system at each time step, making the\u0000development of efficient and convergent numerical methods crucial for\u0000accelerating the nonlinear solvers. In this paper, we present an adaptively\u0000coupled subdomain framework based on the domain decomposition method. The\u0000solution methods developed within this framework effectively handle strong\u0000nonlinearities in global problems by addressing subproblems in the coupled\u0000regions. Furthermore, we propose several adaptive coupling strategies and\u0000develop a method for leveraging initial guesses to accelerate the solution of\u0000nonlinear problems, thereby improving the convergence and parallel performance\u0000of nonlinear solvers. A series of numerical experiments validate the\u0000effectiveness of the proposed framework. Additionally, by utilizing tens of\u0000thousands of processors, we demonstrate the scalability of this approach\u0000through a large-scale reservoir simulation with over 2 billion degrees of\u0000freedom.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259043","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}
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