A wave equation solver in spherical symmetry that demonstrates the use of parity-limited Chebyshev polynomials
- Jonah Miller (jonah.maxwell.miller@gmail.com)
Spectral methods are a highly efficient approach to solving partial differential equations because (for smooth problems), the error converges exponentially. This repo demonstrates how to use a particular family of spectral methods, parity-limited Chebyshev-pseudospectral methods, to solve problems in spherical symmetry, where there is a coordinate singularity at the origin. In a Chebyshev-pseudospectral method, functions are assumed to be interpolating polynomials which interpolate known values at known locations, called colocation points.
The secret to handling the origin is that the coordinate singularity can be resolved if one knows the symmetry properties of functions about the origin, r=0. Therefore, we split our Chebyshev polynomials into two families: the odd-parity and even-parity families. Functions can be represented by either family, but not both. Then differential operators map functions between the families. For example, differentiating an odd-parity function makes it even, and vice versa. By writing the Poisosn operator (which has a coordinate singularity) in this way, the singularity can be resolved. Here are the errors of different differential operators as a function of the number of modes per family:
