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 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.