{"title":"The variational multiscale element free Galerkin method for three-dimensional steady magnetohydrodynamics duct flows","authors":"Xiaohua Zhang , Yujie Fan","doi":"10.1016/j.jocs.2025.102683","DOIUrl":null,"url":null,"abstract":"<div><div>Magnetohydrodynamics (MHD) has extensive applications in diverse fields, making the study of three-dimensional (3D) MHD problems crucial. For MHD flows, when the Hartmann (<span><math><mrow><mi>H</mi><mi>a</mi></mrow></math></span>) number is large, leading to a convection-dominated regime where convection terms overcome diffusion. In such scenarios, standard Galerkin methods fail to suppress non-physical oscillations in solutions, as they lack inherent stabilization mechanisms for strong convection. This paper introduces the variational multiscale element free Galerkin (VMEFG) method to solve 3D steady MHD equations. The VMEFG method inherits the advantage of the element free Galerkin (EFG) method in avoiding the complex meshing process, which is particularly challenging for complex 3D problems. Moreover, compared with the EFG method, it shows enhanced stability in dealing with convection-dominant problems and can automatically generate stabilized parameters, outperforming other stabilization techniques. To verify the numerical stability and accuracy of the VMEFG method, several numerical experiments on various domains including cubic, annular cubic, spherical, and annular spherical domains were conducted and compared with EFG solutions and existing literature results. The results indicate that the VMEFG method can effectively avoid numerical oscillations and maintain stability for 3D MHD problems at high <span><math><mrow><mi>H</mi><mi>a</mi></mrow></math></span> number, providing a reliable and efficient solution for such problems.</div></div>","PeriodicalId":48907,"journal":{"name":"Journal of Computational Science","volume":"91 ","pages":"Article 102683"},"PeriodicalIF":3.7000,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1877750325001607","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Magnetohydrodynamics (MHD) has extensive applications in diverse fields, making the study of three-dimensional (3D) MHD problems crucial. For MHD flows, when the Hartmann () number is large, leading to a convection-dominated regime where convection terms overcome diffusion. In such scenarios, standard Galerkin methods fail to suppress non-physical oscillations in solutions, as they lack inherent stabilization mechanisms for strong convection. This paper introduces the variational multiscale element free Galerkin (VMEFG) method to solve 3D steady MHD equations. The VMEFG method inherits the advantage of the element free Galerkin (EFG) method in avoiding the complex meshing process, which is particularly challenging for complex 3D problems. Moreover, compared with the EFG method, it shows enhanced stability in dealing with convection-dominant problems and can automatically generate stabilized parameters, outperforming other stabilization techniques. To verify the numerical stability and accuracy of the VMEFG method, several numerical experiments on various domains including cubic, annular cubic, spherical, and annular spherical domains were conducted and compared with EFG solutions and existing literature results. The results indicate that the VMEFG method can effectively avoid numerical oscillations and maintain stability for 3D MHD problems at high number, providing a reliable and efficient solution for such problems.
期刊介绍:
Computational Science is a rapidly growing multi- and interdisciplinary field that uses advanced computing and data analysis to understand and solve complex problems. It has reached a level of predictive capability that now firmly complements the traditional pillars of experimentation and theory.
The recent advances in experimental techniques such as detectors, on-line sensor networks and high-resolution imaging techniques, have opened up new windows into physical and biological processes at many levels of detail. The resulting data explosion allows for detailed data driven modeling and simulation.
This new discipline in science combines computational thinking, modern computational methods, devices and collateral technologies to address problems far beyond the scope of traditional numerical methods.
Computational science typically unifies three distinct elements:
• Modeling, Algorithms and Simulations (e.g. numerical and non-numerical, discrete and continuous);
• Software developed to solve science (e.g., biological, physical, and social), engineering, medicine, and humanities problems;
• Computer and information science that develops and optimizes the advanced system hardware, software, networking, and data management components (e.g. problem solving environments).