Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces
We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In...
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| Format: | Article |
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MDPI AG
2023-12-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/12/1631 |
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| author | Longyuan Wu Xinlong Feng Yinnian He |
| author_facet | Longyuan Wu Xinlong Feng Yinnian He |
| author_sort | Longyuan Wu |
| collection | DOAJ |
| description | We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method. |
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| format | Article |
| id | doaj.art-a74098ee124e479da068aef021eedca6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-08T20:47:59Z |
| publishDate | 2023-12-01 |
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| series | Entropy |
| spelling | doaj.art-a74098ee124e479da068aef021eedca62023-12-22T14:07:26ZengMDPI AGEntropy1099-43002023-12-012512163110.3390/e25121631Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on SurfacesLongyuan Wu0Xinlong Feng1Yinnian He2College of Mathematics and Systems Science, Xinjiang University, Urumqi 830017, ChinaCollege of Mathematics and Systems Science, Xinjiang University, Urumqi 830017, ChinaCollege of Mathematics and Systems Science, Xinjiang University, Urumqi 830017, ChinaWe present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.https://www.mdpi.com/1099-4300/25/12/1631surface convection–reaction–diffusion equationsexplicit–implicit methodTaylor expansionsurface finite elementstability |
| spellingShingle | Longyuan Wu Xinlong Feng Yinnian He Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces Entropy surface convection–reaction–diffusion equations explicit–implicit method Taylor expansion surface finite element stability |
| title | Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces |
| title_full | Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces |
| title_fullStr | Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces |
| title_full_unstemmed | Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces |
| title_short | Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces |
| title_sort | modified characteristic finite element method with second order spatial accuracy for solving convection dominated problem on surfaces |
| topic | surface convection–reaction–diffusion equations explicit–implicit method Taylor expansion surface finite element stability |
| url | https://www.mdpi.com/1099-4300/25/12/1631 |
| work_keys_str_mv | AT longyuanwu modifiedcharacteristicfiniteelementmethodwithsecondorderspatialaccuracyforsolvingconvectiondominatedproblemonsurfaces AT xinlongfeng modifiedcharacteristicfiniteelementmethodwithsecondorderspatialaccuracyforsolvingconvectiondominatedproblemonsurfaces AT yinnianhe modifiedcharacteristicfiniteelementmethodwithsecondorderspatialaccuracyforsolvingconvectiondominatedproblemonsurfaces |