Astron. Astrophys. 319, 1025-1035 (1997)

Appendix A: numerical details

A.1. Finite element solver

The GSS equation is solved by means of the method of finite elements. The original code was introduced by Camenzind (1987) for relativistic astrophysical MHD applications. Haehnelt (1990) extended the procedure for Kerr metrics. The further evolution (however in special relativity) by Fendt (1994) and Fendt et al. (1995), now enables the code for an integration throughout the singular surface of the light cylinder and the calculation of smooth, global solutions. For the present investigation the latest version of the code (Fendt 1994) was re-arranged for the application in Kerr geometry.

In the finite element approach the integration region G is discretized in a set of isoparametric curvilinear 8-node elements of the serendipity class (Schwarz 1984). Within each element the flux function is expanded as

denote the magnetic flux at the nodal point i of the element and are rectilinear coordinates on the normalized element.

For a solution, the stream equation is multiplied by a test function N (Galerkin ansatz) and integrated over the 2D plasma domain G applying Green's identity. We end up with the weak form of the GSS equation,

where n now denotes the unit vector perpendicular to the boundary , dA and ds the area and boundary elements, and J the source term of the R.H.S. of Eq. (12). With Eq. (A1) the integral equation corresponds to a matrix equation

with the integrals on each grid element

and

(Haehnelt 1990). Each component of Eq. (A3) corresponds to the force equilibrium between neighbouring nodal points of each element. Inversion of matrix equation (A3) eventually gives the solution for each nodal point. The expansion (A1) provides the solution in any point .

Fig. 4. Example of a numerical grid applied for the finite element code. The element boundaries must follow the shape of the two light surfaces

A.2. Boundary conditions

With the model assumptions discussed above, the computations have to satisfy the following boundary conditions.

- Rotational axis: .

- Light surfaces: Here, the regularity condition, Eq. (15), has to be satisfied. In the finite element approach this regularity condition is automatically satisfied. Like the homogeneous Neumann condition the regularity condition is a natural boundary condition on in the sense that the surface integral (s. Eq. (A5)) does not contribute (see Fendt et al. 1995).

- Disk surface: Here, satisfies a Dirichlet condition, corresponding to a finite flux distribution,

For , . The parameters , E, n are chosen such that the foot points of the flux surfaces are concentrated to the innermost region. This is required by the assumption of a rigid rotation of the magnetosphere.

- Jet boundary: Along the jet boundary asymptotically collimating to a cylindrical shape we fix

by definition. As mentioned before, the shape of the jet boundary has to be well adjusted (and has to be found in an iterative way) in order to satisfy the regularity requirement.

- Asymptotic boundary : We assume that the jet has been collimated into a cylindrical shape. In this region with a distance of about M from the black hole, the geometry is very close to Minkowski space. We use either homogeneous Neumann conditions or the non-linear analytic solution of the special relativistic, asymptotic jet equilibrium Eq. (17) as Dirichlet condition. When the outer and inner domain are calculated separately, then Dirichlet conditions are required at the upper boundary. Otherwise it would not be possible to fix the flux in this domain.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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