Solving Einstein's Equations for Binary Black Hole Spacetimes

Basic Numerical Methods II
A few groups (Caltech/Cornell, Meudon) use spectral methods.
Represent functions as finite sums: u(x, t) = ∑ N−1
k=0 ũk(t)e ikx .
Choose grid points x n to allow efficient (and exact) inversion of the
series: ũ k (t) = ∑ N−1
n=0 w n u(x n , t)e −ikxn .
Obtain derivative formulas by differentiating the series:
∂ x u(x n , t) = ∑ N−1
k=0 ũk(t)∂ x e ikxn = ∑ N−1
m=0 D n m u(x m , t).
Errors in spectral methods are dominated by the size of ũ N .
Estimate the errors (e.g. for Fourier series of smooth functions):
ũ N = 1
2π
∫ π
−π
u(x)e −iNx dx ≤ 1 ∣ ∣∣∣
N p max d p ∣
u(x) ∣∣∣
dx p .
Lee Lindblom (Caltech) Binary Black Holes UW Milwaukee 10/14/2011 18 / 32