4.3 Evolution equations4 Numerical Issues4.1 Why use the conformal

4.2 Construction of initial data 

In order to start the evolution one has to determine the initial data. So far, the numerical methods used for their construction are based entirely on Theorem  7 . It is assumed that the extrinsic curvature of the hyperboloidal initial surface is pure trace. Then the Yamabe equation is solved for a conformal factor which chooses from a given conformal class of Riemannian metrics the one which has constant negative curvature. Finally, the other initial data are obtained from Equations (34Popup Equation).

There are two problems with this kind of procedure. First, the Yamabe equation degenerates on the boundary. This is a difficult problem if one tries to prove existence of solutions of this equation because the loss of ellipticity on the boundary means that one cannot simply appeal to known theorems. However, numerically this has not been a serious obstacle. The Yamabe equation is solved in a more or less standard way by using a Richardson iteration scheme to reduce the solution of the non-linear equation to a series of inversions of a linear operator.

The degeneracy of the equation on the boundary forces the solution and its derivative by regularity to have certain well defined boundary values. This and the global nature of the elliptic equations suggests to use pseudo-spectral (or collocation) methods. Such methods are well suited for problems for which it is known beforehand that the solution will be sufficiently regular. Then, the solution can be expanded into a series of certain basis functions which are globally analytic. Therefore, the regularity conditions on the boundary are already built into the method. We refer readers interested in these methods to a recent review paper by Bonazzola et al. [23].

The more difficult problem in the determination of the initial data is the fact that the curvature components are obtained by successive division by tex2html_wrap_inline3721 . Since tex2html_wrap_inline3721 vanishes on the boundary one has to compute a value which is of the form 0/0. Although the theorem tells us that this is well defined (provided that some boundary conditions are satisfied), it still poses a numerical problem, because a straightforward implementation of l'Hôpital's rule looses accuracy. There have been two methods to overcome this problem. The first one [40] is again based on pseudo-spectral methods. Essentially, in this approach the multiplication with the function tex2html_wrap_inline3721 is expressed as a linear operation between the expansion coefficients of tex2html_wrap_inline3721 and the other factor. The pseudo-inversion of this linear operator (in the sense of finding its Moore-Penrose inverse [106]) then corresponds to division by tex2html_wrap_inline3721 . Although this methods seems to work well, it is not generally applicable because it depends heavily on the geometric setup.

A more general method has been devised by Hübner [82Jump To The Next Citation Point In The Article]. Here, the problem of dividing by tex2html_wrap_inline3721 has been reformulated into a problem for solving a second-order PDE for the quotient. This PDE is similar to the Yamabe equation in the sense that it also degenerates on the boundary where tex2html_wrap_inline3721 vanishes, but it is linear. Both methods have been applied in 2D test cases.

4.3 Evolution equations4 Numerical Issues4.1 Why use the conformal

image Conformal Infinity
Jörg Frauendiener
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