{"title":"Stabilized time-series expansions for high-order finite element solutions of partial differential equations","authors":"Ahmad Deeb, Denys Dutykh","doi":"10.1111/sapm.12708","DOIUrl":null,"url":null,"abstract":"<p>Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time-Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high-order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier–Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time-series integration even when resummation techniques like the Borel–Padé–Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher-order FEs. Our approach involves the incorporation of artificial diffusion terms on the left-hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"153 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.12708","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time-Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high-order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier–Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time-series integration even when resummation techniques like the Borel–Padé–Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher-order FEs. Our approach involves the incorporation of artificial diffusion terms on the left-hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.
期刊介绍:
Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.