Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations
This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-10-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/10/1735 |
| _version_ | 1827704647703330816 |
|---|---|
| author | Cheng-Yu Ku Jing-En Xiao Chih-Yu Liu |
| author_facet | Cheng-Yu Ku Jing-En Xiao Chih-Yu Liu |
| author_sort | Cheng-Yu Ku |
| collection | DOAJ |
| description | This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in the space–time domain, which is a combination of <i>n</i>–dimensional Euclidean space and time into an <i>n</i> + 1–dimensional manifold. Because the initial and boundary conditions were applied on the space–time domain boundaries, we converted the transient problem into an inverse boundary value problem. Additionally, all partial derivatives of the proposed TRPSF are a series of continuous functions, which are nonsingular and smooth. Solutions were approximated by solving the system of simultaneous equations formulated from the boundary, source, and internal collocation points. Numerical examples including stationary and nonstationary convection–diffusion problems were employed. The numerical solutions revealed that the proposed space–time meshless approach may achieve more accurate numerical solutions than those obtained by using the conventional radial basis function (RBF) with the time-marching scheme. Furthermore, the numerical examples indicated that the TRPSF is more stable and accurate than other RBFs for solving the convection–diffusion equation. |
| first_indexed | 2024-03-10T15:44:48Z |
| format | Article |
| id | doaj.art-d6c5f68a303a4334a415894060ce0493 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:44:48Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d6c5f68a303a4334a415894060ce04932023-11-20T16:33:02ZengMDPI AGMathematics2227-73902020-10-01810173510.3390/math8101735Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion EquationsCheng-Yu Ku0Jing-En Xiao1Chih-Yu Liu2Department of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanCenter of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanDepartment of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanThis article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in the space–time domain, which is a combination of <i>n</i>–dimensional Euclidean space and time into an <i>n</i> + 1–dimensional manifold. Because the initial and boundary conditions were applied on the space–time domain boundaries, we converted the transient problem into an inverse boundary value problem. Additionally, all partial derivatives of the proposed TRPSF are a series of continuous functions, which are nonsingular and smooth. Solutions were approximated by solving the system of simultaneous equations formulated from the boundary, source, and internal collocation points. Numerical examples including stationary and nonstationary convection–diffusion problems were employed. The numerical solutions revealed that the proposed space–time meshless approach may achieve more accurate numerical solutions than those obtained by using the conventional radial basis function (RBF) with the time-marching scheme. Furthermore, the numerical examples indicated that the TRPSF is more stable and accurate than other RBFs for solving the convection–diffusion equation.https://www.mdpi.com/2227-7390/8/10/1735space–timetransient radial basis functionconvection–diffusion equationmeshlessradial polynomials |
| spellingShingle | Cheng-Yu Ku Jing-En Xiao Chih-Yu Liu Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations Mathematics space–time transient radial basis function convection–diffusion equation meshless radial polynomials |
| title | Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations |
| title_full | Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations |
| title_fullStr | Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations |
| title_full_unstemmed | Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations |
| title_short | Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations |
| title_sort | space time radial basis function based meshless approach for solving convection diffusion equations |
| topic | space–time transient radial basis function convection–diffusion equation meshless radial polynomials |
| url | https://www.mdpi.com/2227-7390/8/10/1735 |
| work_keys_str_mv | AT chengyuku spacetimeradialbasisfunctionbasedmeshlessapproachforsolvingconvectiondiffusionequations AT jingenxiao spacetimeradialbasisfunctionbasedmeshlessapproachforsolvingconvectiondiffusionequations AT chihyuliu spacetimeradialbasisfunctionbasedmeshlessapproachforsolvingconvectiondiffusionequations |