Continued development of Cactus

Our work in this area is mainly centered in the following topics:

1. Refinement of Evolution Methods: Our starting point has been the Bona-Massó FOFCH (First Order Flux Conservative Hyperbolic) system of equations[2], where the evolution has been split using Strang Splitting' into one step for the evolution of the fluxes (transport) plus two steps for that corresponding to the sources. In order to avoid instabilities in areas of large gradients during transport (e.g.: in the case of BH those resulting from choosing a singularity-avoiding slicing), we are currently implementing advanced methods in CFD (Computational Fluid Dynamics) in the 3D case; namely: Flux Limiter methods[3] where the hyperbolicity of the Bona-Massó system is explicitly taken into account.

   In order to do this, we previously had to develop a new numerical algorithm in order to implement those methods. This new algorithm is essentially based on the computation of the fluxes in the interphases and proceed then by dealing with one direction at a time, neglecting the fluxes in the non-relevant directions and evaluating the eigen-fluxes and limiting each way in that direction. We tested the algorithm, with complete success, first in the case of the 3D wave-equation (since it displays certain similarities with one specific problem that also appears in the case of Einstein's field equations (EFEs)). When applied to EFEs, new problems appeared -as expected- but they have been overcome almost completely at the time of writing this.

   Next step will be to implement both the Bona-Massó evolution system and the above algorithm based on flux limiter methods, as Cactus thorns.

   As a side development in this area, the case of a 2D code with a view towards dealing with axially symmetric systems was investigated and coordinate singularities appeared when approaching the axis of symmetry (in quite a different way than those occurring in the case 1D corresponding to spherical symmetry when the center is approached); this gave rise to a detailed study of the geometric properties of regions close to the axis, with an emphasis on its characterization via different admissible sets of coordinates; deriving normal shift-free forms for the metric and finding the leading power dependence on the radial coordinate of the metric potentials [4], thus allowing for an explicit a priori' cancellation in both members of the equations of the terms responsible for the numerical instabilities.

   Independently, but also in the axisymmetric context, the evolution of matter in axially symmetric black holes has also been looked into [5].

2. Formulation of equations: Starting again from the Bona-Massó FOFCH, we are currently re-writing the equations by re-writing in turn some of the source terms (in particular those corresponding to V, which are related to the momentum constrain, and those corresponding to the extrinsic curvature K), since we have shown that some of the terms appearing in those sources are responsible for the increase at every point of the numerical instabilities in the constrains. By re-writing appropriately those terms we make them vanish over the boundaries, thus achieving a greater stability in the evolution of the sources.