A maximum principle of the Fourier spectral method for diffusion equations

Junseok Kim, Soobin Kwak, Hyun Geun Lee, Youngjin Hwang, Seokjun Ham

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)5396-5405
Number of pages10
JournalElectronic Research Archive
Issue number9
Publication statusPublished - 2023

Bibliographical note

Funding Information:
The first author (J. S. Kim) expresses thanks for the support from the BK21 FOUR program. The authors would like to express sincere thanks to the reviewers for their valuable suggestions and insightful comments to improve the paper.

Publisher Copyright:
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)


  • diffusion equation
  • Fourier spectral method
  • maximum principle

ASJC Scopus subject areas

  • General Mathematics


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