{"title":"基于样条的 2D+t 时空方法求解转移","authors":"Logan Larose, Jude T. Anderson, David M. Williams","doi":"arxiv-2409.11639","DOIUrl":null,"url":null,"abstract":"This work introduces a new solution-transfer process for slab-based\nspace-time finite element methods. The new transfer process is based on\nHsieh-Clough-Tocher (HCT) splines and satisfies the following requirements: (i)\nit maintains high-order accuracy up to 4th order, (ii) it preserves a discrete\nmaximum principle, (iii) it enforces mass conservation, and (iv) it constructs\na smooth, continuous surrogate solution in between space-time slabs. While many\nexisting transfer methods meet the first three requirements, the fourth\nrequirement is crucial for enabling visualization and boundary condition\nenforcement for space-time applications. In this paper, we derive an error\nbound for our HCT spline-based transfer process. Additionally, we conduct\nnumerical experiments quantifying the conservative nature and order of accuracy\nof the transfer process. Lastly, we present a qualitative evaluation of the\nvisualization properties of the smooth surrogate solution.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":null,\"url\":null,\"abstract\":\"This work introduces a new solution-transfer process for slab-based\\nspace-time finite element methods. The new transfer process is based on\\nHsieh-Clough-Tocher (HCT) splines and satisfies the following requirements: (i)\\nit maintains high-order accuracy up to 4th order, (ii) it preserves a discrete\\nmaximum principle, (iii) it enforces mass conservation, and (iv) it constructs\\na smooth, continuous surrogate solution in between space-time slabs. While many\\nexisting transfer methods meet the first three requirements, the fourth\\nrequirement is crucial for enabling visualization and boundary condition\\nenforcement for space-time applications. In this paper, we derive an error\\nbound for our HCT spline-based transfer process. Additionally, we conduct\\nnumerical experiments quantifying the conservative nature and order of accuracy\\nof the transfer process. Lastly, we present a qualitative evaluation of the\\nvisualization properties of the smooth surrogate solution.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"45 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.11639\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11639","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Spline-based solution transfer for space-time methods in 2D+t
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.