In scientific and engineering modelling, computers are often used to
solve partial differential equations, the mathematical foundation    
for many physical phenomena, for example the propagation of waves.
Such modelling can be done in relatively straightforward ways
("finite difference" or basic "finite element" methods), but a more
sophisticated class of techniques, "spectral methods," yield high
accuracy for some problems. In our work we adopt a refined spectral
method that features sparsity; in effect, the first step toward a
solution is taken through the mathematical structure itself, without
requiring significant computer computation. Our sparse spectral
approach had earlier been shown effective for some ordinary differential
equations, but not for the much more difficult challenge of partial
differential equations.

It is often best to develop new mathematical/computational methods in
the context of a specific physical problem. In our case the problem
involves binary neutron stars. A binary pair of such stars emits
gravitational waves and spirals inward, a process that is described
by the partial differential equations of Einstein's general theory of
relativity. The evolution in time of the inspiral can currently be
computed well with highly developed computational infrastructure. What
is missing is a realistic starting point, the initial configuration.

The partial differential equations that must be solved for the initial
configuration have proved to be a stubborn challenge, and researchers
have typically used a solution that is mathematically convenient, but
far from appropriate physically. This starting point evolves into a
correct solution, but it means that the evolution computation must be
carried out for a longer time than would be necessary for a more
physically correct starting solution.

We have endeavored to use sparse spectral methods to solve for a
physically correct initial approximation. The magnitude of the
challenge has required that the problem be broken into a sequence
of ever more advanced projects. The first of these was the solution
not of the Einstein equations, but of Newton's equations for the
initial binary configuration. The main simplification in this model
is that there are no waves. Solving this "simple" problem turned
out, as anticipated, to reveal many important details and to set up
much of the infrastructure for a more realistic model. Indeed, many
unanticipated problems developed in the Newtonian model, in
particular, those involving the stellar surface, that were not
particularly "Einsteinian."

The outcome of this work is a highly precise set of computer
programs that solves for initial neutron star structure. We have
not solved the Einstein equations themselves, but rather an
approximation ("post-Newtonian/post-Minkowskian") to Einstein's
equations. This approximation should be more than adequate for an
improved initial configuration. We have, therefore, partially achieved
our goals for this work. Next steps include looking at the physics
inherent in the initial solutions themselves, and collaborating with
researchers who need initial configurations for their codes that
evolve these initial configurations.

Last Modified: 11/28/2016
Modified by: Stephen R Lau