Helical factorization of the Helmholtz equation

  In this chapter, I demonstrate how multi-dimensional spectral factorization allows the efficient solution of constant coefficient partial differential equations (PDE's). I begin with the simple example of Poisson's equation: following Claerbout (1998b), I show how the helical coordinate system allows the factorization of the finite-difference stencil into a pair of minimum phase filters that can be inverted rapidly by back-substitution.

Moving to a wave propagation example, I construct a finite-difference approximation to the Helmholtz operator in the $(\omega,x)$domain that describes propagation of a single frequency wave. This filter can also be factored into a pair of of filters, and I show these factors can act as recursive one-way wave propagators that are accurate up to 90$^\circ$.