Parity-Mixed Methods Applied to Advection-Diffusion
Parity-Mixed Methods Applied to Advection-Diffusion
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This project is a presentation on recent advances in Parity-Mixing methods for fast FFT based PDE solvers. Discussed how these parity mixing methods effectively reduce aliasing and can be easily applied to multi-dimensional FFT based PDE solvers.
This project is a presentation on recent advances in Parity-Mixing methods for fast FFT based PDE solvers. Discussed how these parity mixing methods effectively reduce aliasing and can be easily applied to multi-dimensional FFT based PDE solvers.
Aliasing is the process by which high frequency wave numbers affect lower frequency wave numbers. When applying spectral or pseudo-spectral solvers, such as those based on FFT implementations, it is important to take proper anti-aliasing steps to avoid an incorrect numerical solution.
Parity-mixing is a particular form of aliasing errors that occurs when trying to represent sine basis functions in terms of a cosine basis expansion in non-periodic domains (or vice versa). This happens because in non-periodic domains sine and cosine basis functions no longer have mutual orthogonality, causing spectrum broadening and aliasing errors.
For example if we had a grid spacing for our solution of the advection diffusion equation to the left and implemented a pseudo-spectral method using DST, the first derivative term would end up being represented in a cosine basis. When applying the spectral solver this would cause parity mixing errors to appear. Doing a brute force calculation of the proper spectral coefficients is of order O(N^2) which defeats the point of a fast solver.
Normally these forms of errors result in needing to scrap the spectral solver and result to a slower method (IMEX, finite element, etc.). However the work of Vasil et al.* shows how we can prevent these effects by introducing a helper function that eliminates spectrum broadening errors for at least the first 3/4 of the frequencies.
Here we see the errors between doing a naive DST transform on a cosine basis function (blue), e.g. cos(jx), and using the parity-mixing method (red) to the brute force calculation. Here 'j' represents the frequency of the cosine basis function.
Now we can see parity mixing can help a numerical solution of the previous form of the advection-diffusion equation. The examples below show how the naive method aliases out any advection effects, but the parity aware method produces an accurate numerical solution while allowing us to use fast spectral methods.
Naive Method
Parity-Mixing Aware
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