3.2 Numerical methods

3.2.1 Symplectic methods

Symplectic numerical methods have proven useful in studies of the approach to the singularity in cosmological models [38Jump To The Next Citation Point]. Symplectic ODE and PDE [101189] 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 [42Jump To The Next Citation Point].

For a field q(t) and its conjugate momentum p(t), the Hamiltonian operator splits into kinetic and potential energy sub-Hamiltonians. Thus, for an arbitrary potential V (q),

H = 1p2 + V (q) = H1 (p) + H2 (q). (6 ) 2
If the vector X = (p,q) defines the variables at time t, then the time evolution is given by
dX--= {H, X } ≡ AX, (7 ) dt PB
where {} P B is the Poisson bracket. The usual exponentiation yields an evolution operator
AΔt A1(Δt∕2) A2Δt A1(Δt∕2) 3 e = e e e + O (Δt ) (8 )
for A = A1 + A2 being the generator of the time evolution. Higher order accuracy may be obtained by a better approximation to the evolution operator [234235]. This method is useful when exact solutions for the sub-Hamiltonians are known. For the given H, variation of H1 yields the solution
q = q0 + p0Δt, p = p0, (9 )
while that of H2 yields
| dV || q = q0, p = p0 − ---|| Δt. (10 ) dq q0
Note that H2 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 t + Δt using the exact solutions to the sub-Hamiltonians according to the prescription given by the approximate evolution operator (8View Equation). Extension to more degrees of freedom and to fields is straightforward [42Jump To The Next Citation Point27Jump To The Next Citation Point].

3.2.2 Other methods

Symplectic methods can achieve convergence far beyond that of their formal accuracy if the full solution is very close to the exact solution from one of the sub-Hamiltonians. Examples where this is the case are given in [39Jump To The Next Citation Point36Jump To The Next Citation Point]. On the other hand, because symplectic algorithms are a type of operator splitting, suboperators can be subject to instabilities that are suppressed by the full operator. An example of this may be found in [41Jump To The Next Citation Point]. Other types of PDE solvers are more effective for such spacetimes. One currently popular method is iterative Crank–Nicholson (see [237]) which is, in effect, an implicit method without matrix inversion. It was first applied to numerical cosmology by Garfinkle [111Jump To The Next Citation Point] and was recently used in this context to evolve T2 symmetric cosmologies [41Jump To The Next Citation Point].

As pointed out in [42Jump To The Next Citation Point36Jump To The Next Citation Point41Jump To The Next Citation Point43Jump To The Next Citation Point], spiky features in collapsing inhomogeneous cosmologies will cause any fixed spatial resolution to become inadequate. Such evolutions are therefore candidates for adaptive mesh refinement such as that implemented by Hern and Stuart [143Jump To The Next Citation Point142].


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