A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model
In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate...
| Main Authors: | , , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi-Wiley
2022-01-01
|
| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2022/9367638 |
| _version_ | 1797245052454436864 |
|---|---|
| author | Ali Hasan Ali Ahmed Shawki Jaber Mustafa T. Yaseen Mohammed Rasheed Omer Bazighifan Taher A. Nofal |
| author_facet | Ali Hasan Ali Ahmed Shawki Jaber Mustafa T. Yaseen Mohammed Rasheed Omer Bazighifan Taher A. Nofal |
| author_sort | Ali Hasan Ali |
| collection | DOAJ |
| description | In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods. |
| first_indexed | 2024-04-24T19:20:46Z |
| format | Article |
| id | doaj.art-6f4903f06f0e4f0ea5c30b71170cbc5c |
| institution | Directory Open Access Journal |
| issn | 1099-0526 |
| language | English |
| last_indexed | 2024-04-24T19:20:46Z |
| publishDate | 2022-01-01 |
| publisher | Hindawi-Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj.art-6f4903f06f0e4f0ea5c30b71170cbc5c2024-03-26T00:00:06ZengHindawi-WileyComplexity1099-05262022-01-01202210.1155/2022/9367638A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation ModelAli Hasan Ali0Ahmed Shawki Jaber1Mustafa T. Yaseen2Mohammed Rasheed3Omer Bazighifan4Taher A. Nofal5Department of MathematicsDepartment of Mechanical and Materials EngineeringDepartment of Business AdministrationApplied Sciences DepartmentDepartment of MathematicsDepartment of MathematicIn this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.http://dx.doi.org/10.1155/2022/9367638 |
| spellingShingle | Ali Hasan Ali Ahmed Shawki Jaber Mustafa T. Yaseen Mohammed Rasheed Omer Bazighifan Taher A. Nofal A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model Complexity |
| title | A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model |
| title_full | A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model |
| title_fullStr | A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model |
| title_full_unstemmed | A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model |
| title_short | A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model |
| title_sort | comparison of finite difference and finite volume methods with numerical simulations burgers equation model |
| url | http://dx.doi.org/10.1155/2022/9367638 |
| work_keys_str_mv | AT alihasanali acomparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT ahmedshawkijaber acomparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT mustafatyaseen acomparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT mohammedrasheed acomparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT omerbazighifan acomparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT taheranofal acomparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT alihasanali comparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT ahmedshawkijaber comparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT mustafatyaseen comparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT mohammedrasheed comparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT omerbazighifan comparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel AT taheranofal comparisonoffinitedifferenceandfinitevolumemethodswithnumericalsimulationsburgersequationmodel |