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Astron. Astrophys. 327, 1-7 (1997)

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2. The integration and boundary conditions

Eq. (2) is an elliptic linear second order differential equation with variable coefficients. From the physical point of view, it would be preferable to carry out the integration as an initial value problem, to start with an initial configuration and calculate the shape and density of the cloud at the final step, at [FORMULA]. However, elliptic differential equations cannot usually be integrated this way, and a unique solution does not exist. Our first attempts to treat the problem as an initial value problem were indeed very unstable, confirming this fact. Probably, this intrinsic instability of the equation is somewhat associated with a physical complexity. It was therefore necessary to look for a boundary value solution. In this case, a unique solution does exist, but as we must assume the final geometry of the cloud, the predictive possibilities are completely lost. Nevertheless, the evolution of the cloud can be followed by means of very stable methods and some combinations of free parameters and boundary conditions can be rejected if they provide physically implausible solutions.

We chose "Simultaneous Over-Relaxation" method (SOR) with Chebyshev acceleration (Press et al. 1989; Holt, 1984; Smith, 1985 and others).

The equation is written as


or equivalently


with obvious definitions of the coefficients [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Subindex l denotes time and subindex j the spatial variable r. The iterative method calculates a new [FORMULA] map from a previous step [FORMULA] map by


where [FORMULA] is the residual calculated by


and [FORMULA] is the relaxation parameter. When using Chebyshev acceleration, [FORMULA] is estimated at every iterative step. The network is divided into white and black points as in a chess-board. The value of [FORMULA] in white points is calculated from the previous step values of [FORMULA] in black points, and at the next step [FORMULA] in black points are calculated from [FORMULA] in white points. The relaxation parameter is calculated with the series




if the size of the net is [FORMULA].

Convergence was usually obtained in less than 800 steps, and the solution is very stable.

We need to take boundary conditions in space and in time (see Fig. 1). Far from the flux tube (at about [FORMULA]) we would have [FORMULA], for instance for [FORMULA]. The other space-boundary could be either a van Neumann condition, [FORMULA], or again [FORMULA] in the opposite direction. Because of the symmetry of the flux tube they must be equivalent, with the latter condition being more time and memory demanding. We have tried both and obtained the same result. This was one way to test the stability of the SOR.

[FIGURE] Fig. 1. Diagram of the boundary conditions

With respect to the time-boundaries, we have at [FORMULA] two possibilities. Either [FORMULA], which we call "homogeneity" or [FORMULA], which we call "isocurvature". In the first case, it is implicitly assumed that no inhomogeneities are initially present: these are subsequently produced by magnetic field structures. As the presence of magnetic fields introduces a metric perturbation, the isocurvature condition assumes that this energy density excess is initially compensated by an under-concentration of the dominant particles, so that the curvature is initially constant. We have numerically found that the two conditions produce different behaviours only in the very first time steps and that the evolution coincides through most of the period considered. This is discussed below. We have not begun at [FORMULA] exactly but at [FORMULA] (remembering that t varies between 0 and 1). On the one hand [FORMULA] may introduce some instability, as discussed below, as foreseen in the theory. On the other hand [FORMULA] is Big Bang time, which is beyond our scope.

At [FORMULA], we have adopted [FORMULA], i.e. with [FORMULA] being a gaussian. The parameter [FORMULA] was adopted such that [FORMULA] was [FORMULA], because this would be a typical value of [FORMULA] at [FORMULA], in order to reproduce the present inhomogeneity field.

Results for low and intermediate scales were numerically found by the above described procedure. For the larger scales, it is shown later that the solution can be theoretically found.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998