This is a wave equation solver in spherical symmetry that demonstrates the use of parity-limited Chebyshev polynomials
A wave equation solver in spherical symmetry that demonstrates
the use of parity-limited Chebyshev polynomials
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:
