Branden Fung: Proposed 449 project

Numerical stability of critical collapse in the Schrodinger-Poisson system

Table of Contents

      
 
 
 
 

1 Motivation and Overview

  1. Critical gravitational collapse: interesting link between black hole formation and phase transitions (such as ferromagnetic transition in spin systems).
  2. 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.
  3. 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.)
  4. 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.
  5. Solutions near "critical" point, i.e. near the blow-up threshold are often self-similar (scale invariant).
  6. 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.
  7. 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

  1. Quickly repeat and verify key calculations performed previously in two BSc theses:
    1. Andrew Inwood
    2. David Shinkaruk
  2. 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
  3. Want to extend computations to 2 + 1 in one (or both) of two possible ways:
    1. 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 \(1/r\)
    2. 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
  4. Possible extension: fully 3 + 1 calculation in \(t,x,y,z\) coordinates

3 Significance & Novelty

  1. Few calculations of critical phenomena beyond spherical symmetry.
  2. Very few in 3 + 1 context.
  3. May provide hints/predictions of stability properties in general relativistic case when symmetry restrictions are relaxed

4 Techniques

  1. Finite differencing with implicit methods for the Schrodinger equation and fast solvers (multigrid) for the Poisson equation
  2. Self-similar solutions require adaptive mesh refinement (AMR)
  3. 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.
  4. 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.
  5. Will use MATLAB's C-code generating facilities (possibly augmented with hand-coded pieces) for large scale calculations.