The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations
In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discre...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3469 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850227224553193472 |
|---|---|
| author | Xin Shen Xuehua Yang Haixiang Zhang |
| author_facet | Xin Shen Xuehua Yang Haixiang Zhang |
| author_sort | Xin Shen |
| collection | DOAJ |
| description | In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results. |
| format | Article |
| id | doaj-art-4abe32f00cd749adbb676a88169f5636 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-4abe32f00cd749adbb676a88169f56362025-08-20T02:04:54ZengMDPI AGMathematics2227-73902024-11-011222346910.3390/math12223469The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution EquationsXin Shen0Xuehua Yang1Haixiang Zhang2School of Science, Hunan University of Technology, Zhuzhou 412007, ChinaSchool of Science, Hunan University of Technology, Zhuzhou 412007, ChinaSchool of Science, Hunan University of Technology, Zhuzhou 412007, ChinaIn this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.https://www.mdpi.com/2227-7390/12/22/3469nonlinear evolution equationfinite difference methodalternate direction implicitstabilityconvergence |
| spellingShingle | Xin Shen Xuehua Yang Haixiang Zhang The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations Mathematics nonlinear evolution equation finite difference method alternate direction implicit stability convergence |
| title | The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations |
| title_full | The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations |
| title_fullStr | The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations |
| title_full_unstemmed | The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations |
| title_short | The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations |
| title_sort | high order adi difference method and extrapolation method for solving the two dimensional nonlinear parabolic evolution equations |
| topic | nonlinear evolution equation finite difference method alternate direction implicit stability convergence |
| url | https://www.mdpi.com/2227-7390/12/22/3469 |
| work_keys_str_mv | AT xinshen thehighorderadidifferencemethodandextrapolationmethodforsolvingthetwodimensionalnonlinearparabolicevolutionequations AT xuehuayang thehighorderadidifferencemethodandextrapolationmethodforsolvingthetwodimensionalnonlinearparabolicevolutionequations AT haixiangzhang thehighorderadidifferencemethodandextrapolationmethodforsolvingthetwodimensionalnonlinearparabolicevolutionequations AT xinshen highorderadidifferencemethodandextrapolationmethodforsolvingthetwodimensionalnonlinearparabolicevolutionequations AT xuehuayang highorderadidifferencemethodandextrapolationmethodforsolvingthetwodimensionalnonlinearparabolicevolutionequations AT haixiangzhang highorderadidifferencemethodandextrapolationmethodforsolvingthetwodimensionalnonlinearparabolicevolutionequations |