A maximum principle of the Fourier spectral method for diffusion equations

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

doi:10.3934/era.2023273

Electronic Research Archive

2023, Volume 31, Issue 9: 5396-5405. doi: 10.3934/era.2023273

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Research article

A maximum principle of the Fourier spectral method for diffusion equations

1.
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
2.
Department of Mathematics, Kwangwoon University, Seoul 01897, Republic of Korea

Received: 04 May 2023 Revised: 07 July 2023 Accepted: 24 July 2023 Published: 31 July 2023

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.
- maximum principle,
- Fourier spectral method,
- diffusion equation
Citation: Junseok Kim, Soobin Kwak, Hyun Geun Lee, Youngjin Hwang, Seokjun Ham. A maximum principle of the Fourier spectral method for diffusion equations[J]. Electronic Research Archive, 2023, 31(9): 5396-5405. doi: 10.3934/era.2023273

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Abstract

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.

References

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