A maximum principle of the Fourier spectral method for diffusion equations
In this study, we investigate a maximum principle of the Fourier spectral method (FSM) for diffusion equations. It is well known that the FSM is fast, efficient and accurate. The maximum principle holds for diffusion equations: A solution satisfying the diffusion equation has the maximum value under...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-07-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023273?viewType=HTML |
| _version_ | 1797657990215499776 |
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| author | Junseok Kim Soobin Kwak Hyun Geun Lee Youngjin Hwang Seokjun Ham |
| author_facet | Junseok Kim Soobin Kwak Hyun Geun Lee Youngjin Hwang Seokjun Ham |
| author_sort | Junseok Kim |
| collection | DOAJ |
| description | In this study, we investigate a maximum principle of the Fourier spectral method (FSM) for diffusion equations. It is well known that the FSM is fast, efficient and accurate. The maximum principle holds for diffusion equations: A solution satisfying the diffusion equation has the maximum value under the initial condition or on the boundary points. The same result can hold for the discrete numerical solution by using the FSM when the initial condition is smooth. However, if the initial condition is not smooth, then we may have an oscillatory profile of a continuous representation of the initial condition in the FSM, which can cause a violation of the discrete maximum principle. We demonstrate counterexamples where the numerical solution of the diffusion equation does not satisfy the discrete maximum principle, by presenting computational experiments. Through numerical experiments, we propose the maximum principle for the solution of the diffusion equation by using the FSM. |
| first_indexed | 2024-03-11T17:52:39Z |
| format | Article |
| id | doaj.art-ff58ecd931704388815f4250293d47b6 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-03-11T17:52:39Z |
| publishDate | 2023-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-ff58ecd931704388815f4250293d47b62023-10-18T01:18:37ZengAIMS PressElectronic Research Archive2688-15942023-07-013195396540510.3934/era.2023273A maximum principle of the Fourier spectral method for diffusion equationsJunseok Kim0Soobin Kwak1Hyun Geun Lee2Youngjin Hwang3Seokjun Ham41. Department of Mathematics, Korea University, Seoul 02841, Republic of Korea1. Department of Mathematics, Korea University, Seoul 02841, Republic of Korea2. Department of Mathematics, Kwangwoon University, Seoul 01897, Republic of Korea1. Department of Mathematics, Korea University, Seoul 02841, Republic of Korea1. Department of Mathematics, Korea University, Seoul 02841, Republic of KoreaIn this study, we investigate a maximum principle of the Fourier spectral method (FSM) for diffusion equations. It is well known that the FSM is fast, efficient and accurate. The maximum principle holds for diffusion equations: A solution satisfying the diffusion equation has the maximum value under the initial condition or on the boundary points. The same result can hold for the discrete numerical solution by using the FSM when the initial condition is smooth. However, if the initial condition is not smooth, then we may have an oscillatory profile of a continuous representation of the initial condition in the FSM, which can cause a violation of the discrete maximum principle. We demonstrate counterexamples where the numerical solution of the diffusion equation does not satisfy the discrete maximum principle, by presenting computational experiments. Through numerical experiments, we propose the maximum principle for the solution of the diffusion equation by using the FSM.https://www.aimspress.com/article/doi/10.3934/era.2023273?viewType=HTMLmaximum principlefourier spectral methoddiffusion equation |
| spellingShingle | Junseok Kim Soobin Kwak Hyun Geun Lee Youngjin Hwang Seokjun Ham A maximum principle of the Fourier spectral method for diffusion equations Electronic Research Archive maximum principle fourier spectral method diffusion equation |
| title | A maximum principle of the Fourier spectral method for diffusion equations |
| title_full | A maximum principle of the Fourier spectral method for diffusion equations |
| title_fullStr | A maximum principle of the Fourier spectral method for diffusion equations |
| title_full_unstemmed | A maximum principle of the Fourier spectral method for diffusion equations |
| title_short | A maximum principle of the Fourier spectral method for diffusion equations |
| title_sort | maximum principle of the fourier spectral method for diffusion equations |
| topic | maximum principle fourier spectral method diffusion equation |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023273?viewType=HTML |
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