Project A7;     (2011 - 2014)

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Pseudo-spectral methods for the Einstein equations on hyperboloidal slices

    Principal Investigators: Marcus Ansorg

This Project is concerned with the development of novel numerical techniques for the solution of constraint and time evolution equations in general relativity. Three aspects are central: (i) the incorporation of future null infinity, (ii) the use of pseudo-spectral methods for spatial directions, and (iii) the introduction of appropriate coordinates related to conformal mappings known from complex analysis.

In recent years, the realm of Numerical relativity has seen tremendous development. The computation of more than 10 orbits of inspiraling binary black holes has become feasible, and the emitted gravitational waves have been studied in great detail. In the upcoming years, the research in this area will be concerned with refined studies to explore the available parameter space. In particular, large (or even extremely large) mass ratios of the two constituents are of interest, as such binary objects are among the most relevant sources for detection of gravitational waves.

This sophisticated exploration requires, especially in view of large mass ratios, the development of novel numerical algorithms that go beyond the techniques which are used at the present stage. In particular, very high accuracy would be desirable as well as specifically adapted methods, in order to deal with the issues of the resolution of strong gradient regimes in the vicinity of the small mass constituent.

In project A7, these points are addressed in terms of a combination of three aspects:

  1. The incorporation of future null infinity.

    The outer computational boundary is placed at future null infinity (abbreviated by scri^+). This boundary contains all points which are approached asymptotically by null rays (light rays and gravitational waves) which can escape to infinity. Hence this concept permits the complete investigation of the outgoing radiation and is therefore an important ingredient in the development of novel numerical algorithms.

    The inclusion of future null infinity is performed by an appropriate conformal compactification, through which scri^+ is put at the exterior boundary of a finite computational domain. As a consequence, Einstein's field equations are to be considered on hyperboloidal slices which are space-like in the interior of this domain but become asymptotically null at the outer boundary. In this setting, the field equations split into constraint and evolution equations, which both have to be treated according to the specific mathematical and geometric conditions given at scri^+.

  2. The use of pseudo-spectral methods.

    Pseudo-spectral methods have the remarkable capability of providing exponential convergence rate when the underlying problem admits a smooth solution. They have been used widely for the solution of elliptic equations in relativity, in particular for the construction of equilibrium models of rotating neutron stars, fluid rings and central-black-hole-fluid-ring systems, as well as for the highly accurate computation of binary initial data. In the context of dynamical binary black hole evolutions, pseudo-spectral methods applied to the spatial directions yield the most accurate wave forms to date.

    An important requirement for the applicability of pseudo-spectral methods is the regularity of the underlying solution. In the context of future null infinity it has been shown that (i) regular initial data can be constructed on hyperboloidal slices, and (ii) the time evolution of such data remains regular, in particular at scri^+.

    In order to realize extremely accurate solutions, project A7 uses pseudo- spectral expansions with respect to spatial directions. In particular, constraint equations on the initial hyperboloidal slice are treated in terms of pseudo-spectral methods. But also for the time evolution equations we expand the field quantities, considered in the spatial directions, with respect to a pseudo-spectral scheme.

  3. The introduction of specifically adapted coordinates.

    In a number of contributions in relativity, pseudo-spectral methods have proven to work particularly well when considered in specifically adapted coordinate systems. The computational domain may or may not possess specific boundary points at which the underlying field equations degenerate. In project A7 we explore the applicability of coordinate transformations related to conformal mappings (known from complex analysis) and their compatibility with pseudo-spectral methods in the treatment of the Einstein equations on hyperboloidal slices. As a specific feature, the coordinates to be used in the schemes are adapted to both future null infinity and inner boundaries like black hole horizons such that they coincide exactly with numerical domain boundaries.


  Marcus Ansorg   Professor, 2011
  Rodrigo Panosso Macedo   Postdoc, 2011
  David Schinkel   Student, 2011
  Erik Buchholz   Student, 2011