3.3 Mixmaster Dynamics3 Singularities in Cosmological Models3.1 Singularities in Spatially Homogeneous

3.2 Symplectic Numerical Methods

Symplectic numerical methods have proven useful in studies of the approach to the singularity in cosmological models [26Jump To The Next Citation Point In The Article]. Symplectic ODE and PDE [75, 141] methods comprise a type of operator splitting. An outline of the method (for one degree of freedom) follows. Details of the application to the Gowdy model (PDE's in one space and one time direction) are given elsewhere [32Jump To The Next Citation Point In The Article].

For a field q (t) and its conjugate momentum p (t) split the Hamiltonian operator into kinetic and potential energy subhamiltonians. Thus,

equation206

If the vector X = (p, q) defines the variables at time t, then the time evolution is given by

equation209

where tex2html_wrap_inline1450 is the Poisson bracket. The usual exponentiation yields an evolution operator

  equation215

for tex2html_wrap_inline1452 the generator of the time evolution. Higher order accuracy may be obtained by a better approximation to the evolution operator [172, 173]. This method is useful when exact solutions for the subhamiltonians are known. For the given H, variation of tex2html_wrap_inline1456 yields the solution

equation223

while that of tex2html_wrap_inline1458 yields

equation225

Note that tex2html_wrap_inline1458 is exactly solvable for any potential V no matter how complicated, although the required differenced form of the potential gradient may be non-trivial. One evolves from t to tex2html_wrap_inline1466 using the exact solutions to the subhamiltonians according to the prescription given by the approximate evolution operator (8Popup Equation). Extension to more degrees of freedom and to fields is straightforward [32Jump To The Next Citation Point In The Article, 22Jump To The Next Citation Point In The Article].



3.3 Mixmaster Dynamics3 Singularities in Cosmological Models3.1 Singularities in Spatially Homogeneous

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-1998-7
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de