Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation
We collected 20 explicit and stable numerical algorithms for the one-dimensional transient diffusion equation and analytically examined their consistency and convergence properties. Most of the methods used have been constructed recently and their truncation errors are given in this paper for the fi...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-11-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/15/11/425 |
| _version_ | 1797469442170421248 |
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| author | Ádám Nagy János Majár Endre Kovács |
| author_facet | Ádám Nagy János Majár Endre Kovács |
| author_sort | Ádám Nagy |
| collection | DOAJ |
| description | We collected 20 explicit and stable numerical algorithms for the one-dimensional transient diffusion equation and analytically examined their consistency and convergence properties. Most of the methods used have been constructed recently and their truncation errors are given in this paper for the first time. The truncation errors contain the ratio of the time and space steps; thus, the algorithms are conditionally consistent. We performed six numerical tests to compare their performance and try to explain the observed accuracies based on the truncation errors. In one of the experiments, the diffusion coefficient is supposed to change strongly in time, where a nontrivial analytical solution containing the Kummer function was successfully reproduced. |
| first_indexed | 2024-03-09T19:21:28Z |
| format | Article |
| id | doaj.art-dcd6e1a971fd4e36bd91fabc303a4484 |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-03-09T19:21:28Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-dcd6e1a971fd4e36bd91fabc303a44842023-11-24T03:23:19ZengMDPI AGAlgorithms1999-48932022-11-01151142510.3390/a15110425Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion EquationÁdám Nagy0János Majár1Endre Kovács2Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, HungaryInstitute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, HungaryInstitute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, HungaryWe collected 20 explicit and stable numerical algorithms for the one-dimensional transient diffusion equation and analytically examined their consistency and convergence properties. Most of the methods used have been constructed recently and their truncation errors are given in this paper for the first time. The truncation errors contain the ratio of the time and space steps; thus, the algorithms are conditionally consistent. We performed six numerical tests to compare their performance and try to explain the observed accuracies based on the truncation errors. In one of the experiments, the diffusion coefficient is supposed to change strongly in time, where a nontrivial analytical solution containing the Kummer function was successfully reproduced.https://www.mdpi.com/1999-4893/15/11/425truncation errordiffusionheat conductionexplicit time-integrationunconditionally stable numerical methods |
| spellingShingle | Ádám Nagy János Majár Endre Kovács Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation Algorithms truncation error diffusion heat conduction explicit time-integration unconditionally stable numerical methods |
| title | Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation |
| title_full | Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation |
| title_fullStr | Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation |
| title_full_unstemmed | Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation |
| title_short | Consistency and Convergence Properties of 20 Recent and Old Numerical Schemes for the Diffusion Equation |
| title_sort | consistency and convergence properties of 20 recent and old numerical schemes for the diffusion equation |
| topic | truncation error diffusion heat conduction explicit time-integration unconditionally stable numerical methods |
| url | https://www.mdpi.com/1999-4893/15/11/425 |
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