A New Pseudo-Spectral Method Using the Discrete Cosine Transform

Area of Science:

• Numerical Analysis
• Computational Science

Background:

• Pseudo-spectral (PS) methods, based on the Fourier transform, are crucial for estimating derivatives in numerical simulations.
• The standard implementation uses the discrete Fourier transform (DFT), which can introduce oscillations due to periodicity when sequence endpoints differ significantly.

Purpose of the Study:

• To propose a novel pseudo-spectral method that mitigates oscillations caused by sequence periodicity.
• To enhance the accuracy of derivative estimation in numerical methods.

Main Methods:

• Developed a new PS method incorporating symmetric extension.
• Mathematically derived the method using the discrete cosine transform (DCT) and its relationship with the DFT.
• Leveraged DCT's property of treating sequences as symmetrically extended for derivative estimation.

Main Results:

• The proposed method effectively addresses oscillations seen in traditional DFT-based PS methods.
• Demonstrated superior performance of the new method through image interpolation experiments.
• The method accurately estimates derivatives in the transformed domain.

Conclusions:

• The symmetric extension-based PS method using DCT offers a robust alternative to DFT-based approaches.
• This technique has significant potential for improving numerical simulations in fluid dynamics, meteorology, and geophysics.