apply boundary conditions using a uniform grid

[Yi-De Liou]

Dear Developer,

I am curious if it is possible to apply boundary conditions using a uniform grid instead of the Chebyshev grid, which inherently exhibits non-uniform spacing. This question arises because I recently encountered a numerical model that treats each simulation node as representing a real crystal lattice array, thereby requiring uniform spacing.

Best regards,
YD

[Keaton Burns]

Hi YD,

This is not possible. However, you should really think of Chebyshev methods as applying equations and constraints to the polynomial expansion of the solution, rather than on the Chebyshev grid. From this point of view, if you have a continuum model for your crystal problem, it should be fine. If the model is fundamentally discrete, though, then spectral methods are likely not the best for your application, unless the boundary conditions are extremely simple and can e.g. be handled with symmetry constraints on sine/cosine modes.

Best,

-Keaton

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[Yi-De Liou]

Hi Keaton,

Thank you for your explanation and suggestion. I plan to start by treating my problem as a continuum model, solving it using Chebyshev basis, and then interpolating the solution onto a Fourier grid to evaluate its validity.

I’m also curious about your comment that "boundary conditions are extremely simple and can, for example, be handled with symmetry constraints on sine/cosine modes." Does this imply that it’s possible to use a Fourier basis to solve the problem while enforcing constraints on the field variables at paired boundaries, such as f(x=0)=f(x=Lx)=0, which would remain periodic along the x-direction?

Best regards,
YD

keaton...@gmail.com 在 2024年11月18日 星期一晚上9:31:04 [UTC+8] 的信中寫道：

[Keaton Burns]

If you’re equations and boundary conditions satisfy a parity/reflection symmetry at the boundaries, you can use just sine or cosine series for the solution and still get spectral accuracy and satisfy the boundary conditions. This generally only happens in simple circumstances where the problem doesn’t really form boundary layers, etc. If you search for “parity-restricted spectral methods”, or similar, you should be able to find papers and more discussions of these techniques.

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