Branden Fung: Proposed 449 project
Numerical stability of critical collapse in the Schrodinger-Poisson system
1
Motivation and Overview
- Critical gravitational collapse: interesting link between black hole
formation and phase transitions (such as ferromagnetic transition in spin
systems).
- Mathematically, belongs to class of problems in which time-dependent PDEs
exhibit blow up, i.e. where, dependent on initial data, solutions
can develop singularities in a finite time.
- Many examples of this in physics,
in fluid dynamics for example (although in the case of fluids
singularities never actually form, presumably due to some
underlying microphysics---such as viscosity, that arises when the
flow is sufficiently close to singular.)
- Mathematical physicists also very interested in the issue, which is
closely related to global existence (or non-existence) of smooth solutions to
PDEs: for the case of the Navier-Stokes
equations, this is the crux of one of the seven
Millennium problems
which is worth $1,000,000 for whomever first proves global existence, or
provides a counter example.
- Solutions near "critical" point, i.e. near the blow-up threshold are often
self-similar (scale invariant).
- Want to study this type of behaviour in model of Newtonian gravitational
collapse where the governing PDEs are much simpler than those for the
general relativistic case.
- Model couples matter field governed by the Schrodinger equation with
non-linearities to Newtonian gravity, governed by the Poisson equation: hence
the Schrodinger-Poisson system.
2
Project Aim
- Quickly repeat and verify key calculations performed previously in two BSc theses:
- Andrew Inwood
- David Shinkaruk
- Those calculations were carried out with a restriction to spherical symmetry:
\( (t,r) \) coordinates, i.e. 1 space +
1 time dimension \(\equiv\) a 1 + 1 calculation
- Want to extend computations to 2 + 1 in one (or both) of two possible ways:
- In \(t, x, y\) (Cartesian) coordinates: different physics than
spherically-symmetric case
- e.g. as \(r\to\infty\), gravitational potential goes like \(\ln r\) rather than
\(1/r\)
- In \(t, \rho, z\) (cylindrical) coordinates: same basic physics, solutions are
now constrained to be axial-symmetric,
and can investigate stability of critical solutions (found by Inwood and
Shinkaruk) with respect to non-spherical perturbations
- Possible extension: fully 3 + 1 calculation in \(t,x,y,z\) coordinates
3
Significance & Novelty
- Few calculations of critical phenomena beyond spherical symmetry.
- Very few in 3 + 1 context.
- May provide hints/predictions of stability properties in general relativistic
case when symmetry restrictions are relaxed
4
Techniques
- Finite differencing with implicit methods for the Schrodinger equation and
fast solvers (multigrid) for the Poisson equation
- Self-similar solutions require adaptive mesh refinement (AMR)
- 2 + 1 and especially 3 + 1 calculations will be computationally expensive,
so aim to run in parallel on Compute Canada High Performance Computing (HPC)
facilities.
- Will begin with MATLAB code, ideal for rapid prototyping of numerical
analysis code, as well as production runs for 1 + 1 and some of the 2 + 1
computations.
- Will use MATLAB's C-code generating facilities (possibly augmented with
hand-coded pieces) for large scale calculations.