{"title":"对流扩散方程中立方 UAT 和立方 UAH 张力 B 样条 DQM 的比较研究：统计验证","authors":"Manpreet Kaur, Mamta Kapoor","doi":"10.1007/s10910-024-01641-5","DOIUrl":null,"url":null,"abstract":"<div><p>In this work, two numerical techniques are compared for solving one and two-dimensional convection-diffusion equations. First technique is referred as “MCUAT tension B-spline,\" and the second technique is labeled as “MCUAH tension B-spline.\" Various aspects are examined to validate the compatibility of results, including comparisons between numerical and exact solutions and evaluation of different error norms. Present errors are compared with existing literature, presenting a remarkable improvisation. Statistical validation of work is tested via a correlation matrix heatmap generated in Python. The order of convergence of the proposed work is also included. Via an observation of comparison of results, it is claimed that UAT results are slightly better than UAH results. Different aspects of correlation, such as strongly negative correlation and perfect positive correlation, are notified. The present work will introduce new dimensions to the field of numerical techniques.</p></div>","PeriodicalId":648,"journal":{"name":"Journal of Mathematical Chemistry","volume":"62 9","pages":"2090 - 2128"},"PeriodicalIF":1.7000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A comparative study of cubic UAT and cubic UAH tension B-splines DQM for convection-diffusion equation: a statistical validation\",\"authors\":\"Manpreet Kaur, Mamta Kapoor\",\"doi\":\"10.1007/s10910-024-01641-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this work, two numerical techniques are compared for solving one and two-dimensional convection-diffusion equations. First technique is referred as “MCUAT tension B-spline,\\\" and the second technique is labeled as “MCUAH tension B-spline.\\\" Various aspects are examined to validate the compatibility of results, including comparisons between numerical and exact solutions and evaluation of different error norms. Present errors are compared with existing literature, presenting a remarkable improvisation. Statistical validation of work is tested via a correlation matrix heatmap generated in Python. The order of convergence of the proposed work is also included. Via an observation of comparison of results, it is claimed that UAT results are slightly better than UAH results. Different aspects of correlation, such as strongly negative correlation and perfect positive correlation, are notified. The present work will introduce new dimensions to the field of numerical techniques.</p></div>\",\"PeriodicalId\":648,\"journal\":{\"name\":\"Journal of Mathematical Chemistry\",\"volume\":\"62 9\",\"pages\":\"2090 - 2128\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Chemistry\",\"FirstCategoryId\":\"92\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10910-024-01641-5\",\"RegionNum\":3,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Chemistry","FirstCategoryId":"92","ListUrlMain":"https://link.springer.com/article/10.1007/s10910-024-01641-5","RegionNum":3,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
在这项工作中，对求解一维和二维对流扩散方程的两种数值技术进行了比较。第一种技术称为 "MCUAT 张力 B 样条"，第二种技术称为 "MCUAH 张力 B 样条"。为了验证结果的兼容性，对各方面进行了研究，包括数值解与精确解的比较以及不同误差规范的评估。将目前的误差与现有文献进行了比较，发现两者有显著的改进。通过 Python 生成的相关矩阵热图对工作的统计验证进行了测试。此外，还包括拟议工作的收敛顺序。通过对结果对比的观察，我们发现 UAT 的结果略好于 UAH 的结果。报告还指出了相关性的不同方面，如强烈负相关和完全正相关。本工作将为数值技术领域引入新的内容。
A comparative study of cubic UAT and cubic UAH tension B-splines DQM for convection-diffusion equation: a statistical validation
In this work, two numerical techniques are compared for solving one and two-dimensional convection-diffusion equations. First technique is referred as “MCUAT tension B-spline," and the second technique is labeled as “MCUAH tension B-spline." Various aspects are examined to validate the compatibility of results, including comparisons between numerical and exact solutions and evaluation of different error norms. Present errors are compared with existing literature, presenting a remarkable improvisation. Statistical validation of work is tested via a correlation matrix heatmap generated in Python. The order of convergence of the proposed work is also included. Via an observation of comparison of results, it is claimed that UAT results are slightly better than UAH results. Different aspects of correlation, such as strongly negative correlation and perfect positive correlation, are notified. The present work will introduce new dimensions to the field of numerical techniques.
期刊介绍：
The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches.
Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.
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