{"title":"An efficient spline-based DQ method for 2D/3D Riesz space-fractional convection–diffusion equations","authors":"Xiaogang Zhu, Yaping Zhang","doi":"10.1016/j.jocs.2024.102364","DOIUrl":null,"url":null,"abstract":"<div><p>This paper proposes an efficient spline-based DQ method for the 2D and 3D convection–diffusion equations (CDEs) with Riesz fractional derivative in space, which have been widely used to describe the anomalous solute transport in complex media. Firstly, a spline-based differential quadrature (DQ) formula is developed to approximate the Riesz derivative by using cubic B-splines as trial functions, which allows us to approximate the fractional derivatives with high accuracy and small computational cost. We then utilize it to discretize the fractional derivatives in the governing equation and a cubic B-spline DQ scheme is further established by applying the finite difference (FD) scheme to the resulting system of ordinary differential equations. A brief implementation of the proposed DQ method is also presented. To examine the effectiveness of this spline-based DQ method, numerical tests are finally done on some benchmark problems and the simulation of rotating Gaussian hill in convection-dominated flow governed by fractional derivatives. The advantages in computational accuracy and efficiency are illustrated by comparing the results with the other algorithms in open literature.</p></div>","PeriodicalId":48907,"journal":{"name":"Journal of Computational Science","volume":null,"pages":null},"PeriodicalIF":3.1000,"publicationDate":"2024-06-14","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/S1877750324001571","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
This paper proposes an efficient spline-based DQ method for the 2D and 3D convection–diffusion equations (CDEs) with Riesz fractional derivative in space, which have been widely used to describe the anomalous solute transport in complex media. Firstly, a spline-based differential quadrature (DQ) formula is developed to approximate the Riesz derivative by using cubic B-splines as trial functions, which allows us to approximate the fractional derivatives with high accuracy and small computational cost. We then utilize it to discretize the fractional derivatives in the governing equation and a cubic B-spline DQ scheme is further established by applying the finite difference (FD) scheme to the resulting system of ordinary differential equations. A brief implementation of the proposed DQ method is also presented. To examine the effectiveness of this spline-based DQ method, numerical tests are finally done on some benchmark problems and the simulation of rotating Gaussian hill in convection-dominated flow governed by fractional derivatives. The advantages in computational accuracy and efficiency are illustrated by comparing the results with the other algorithms in open literature.
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).