Experimental Design Factors

Numerical computations can quickly become infeasibly costly as the number dimensions increase.

At a fixed resolution, the amount of resources required scales exponentially with the number of dimensions. However, many physical behaviours of interest can be studied even when rotational symmetries are imposed. By exploiting these symmetries, we can reduce the effective dimension-ality of the problem, which in turn significantly reduces the computational cost involved. One obvious way to proceed is to express the line element in a system of coordinates in which the symmetries are manifest. The downside of this method, however, is the emergence of coordinate singularities at the axes of rotational symmetries, which can become problematic for numerical evolution unless they are carefully handled. Furthermore, the evolution equations will be different

for each component, resulting in a more complex code. Another method, which was implemented in [161], is to perform dimensional reduction on the Einstein equation itself, then perform thedC1 decomposition on the resulting modified equation. Thecartoon method[162] is an alternative ap-proach which allows us to impose symmetries on anydC1decomposition of the Einstein equation while remaining in the relatively simple Cartesian coordinates. This was originally proposed for axisymmetricd C1simulations. The method involves setting up a 2D Cartesian grid to hold the dimensionally reduced data, along with an extra layer of 2D grid above it. Data on these adjacent layers are not evolved using the equations, but are instead filled with values obtained by rotating the actual data grid up, thus imposing the required symmetry. With this extra grid layer, it is possible to calculate derivatives using finite differences as usual. Besides the additional resources required to hold the extra grid, the downside of this approach is that the symmetry is only approximately imposed: the derivatives obtained via finite differences along the reduced dimensions are not what one would get if the symmetries were imposed analytically.

In our work, we employ the so-calledmodified Cartoon method [160, 33], where we continue to work in Cartesian coordinates, but the derivatives along the symmetry-reduced dimensions are now calculated using expressions obtained by imposing the symmetries analytically. We begin by considering a 4-dimensional spacelike slice†with Cartesian coordinates.x; y; z; w/and impose aU.1/symmetry on the.z; w/plane. In order to impose the rotational symmetry, we change to polar coordinates on this plane, thus

zDcos ; w Dsin; (6.37)

where is the coordinate on the orbits of ourU.1/symmetry. Therefore, WD@=@ is a Killing vector on†. We now choose to perform our computation on the planew0, and we denote this subsurface by†0. Our setup is now effectively 3+1, and we use the indexi D1; 2; 3to range over the remaining coordinates.x; y; z/. In terms of these Cartesian coordinates, our Killing vector has components

Dz @w w @z: (6.38)

TheU.1/symmetry can now be expressed as the requirement that the Lie derivativesL of all physical quantities on† vanish and setting w D 0. Expressed in Cartesian coordinates, these conditions can be rearranged to obtain variousw-derivatives in terms of grid derivatives on †0.

The resulting relations for the components required for standard CCZ4 can be found in [33], and a complete list all components is given in [163]. For scalar quantitiesF, we have

@wF D0

@i@wF D0

@w@wF D @zF z : For vector quantitiesV^a, we have

./ @wVⁱ D ıⁱ_zV^w

z @wV^w D V^z

z ./ @j@wVⁱ D ıⁱ_z

@jV^w

z ı_j^zV^w z²

@j@wV^w D @jV^z z ıjz

V^z z²

@w@wVⁱ D @zVⁱ

z ıⁱ_zV^z

z² ./ @w@wV^w D @zV^w z

V^w z² : For second-rank tensor quantitiesTab, we have

./ @wTij D ıjz

Tiw

z ıiz

Twj

z

@wTiw D Tiz

z ıiz

Tww

z ./ @wTww D 2

zTzw

./ @_k@wTij D 1

z ıiz@_kTwj Cıjz@_kTiw C 1

z²ı_kz ıizTwj CıjzTiw

@_k@wTiw D 1

z.@_kTiz ıiz@_kTww/ 1

z² .ı_kzTiz ıizı_kzTww/ ./ @_k@wTww D 2

z@_kTzw

2

z²ı_kzTzw

@w@wTij D 1 z@zTij

1

z² ıizTzj CıjzTiz 2 ıizıjzTww ./ @_w@_wT_iw D 1

z@_zT_iw 1

z².T_iwC3 ı_izT_zw/

@w@wTww D 1

z@zTwwC 1

z².Tzz Tww/

More generally, ind spatial dimensions we can take our symmetry group to beSO.n/for any 2 n d. To fix notation, we take n D d 2 and denote the Cartesian coordinates on † by .x; y; z; w¹; : : : ; w^{n 1}/. As before, the SO.n/ acts on the .z; w¹; : : : ; w^{n 1}/ hyperplane, and our computational domain is the 3-dimensional subsurface †0 obtained by setting all the

w^A coordinates to zero. For n > 2, the SO.n/ contains U.1/ subgroups corresponding to rotations in the.w^A; w^B/planes for anyA¤B. The Killing vector generating this subgroup is

.AB/WDw^A@B w^B@A. For any vectorV and second rank tensorT, we find that

.AB/LV^B D0)V^AD0

.AB/LTiB D0)TiAD0

.AB/LTBB D0)TAB D0

.AB/LTAB D0)TAA DTBB

Therefore, vector quantities do not carry any extra cartoon components, while tensor quantities only carry a single diagonal cartoon component denoted byTww. Next, for eachw^A, we consider the Killing vector ^.A/ WDz @A w^A@z. We now proceed in the same way as in theU.1/case to obtain the same expressions as before, except that those marked with./now vanish. Additionally, we find that expressions involving two different cartoon indices vanish, i.e. those with two w indices on the LHS should be replaced by A; B and the RHS multipled by a factor of ı_AB or ı_B^A as appropriate. For example,@j@wV^w ! @j@BV^A D ı_B^A

1

z@jV^z _z¹2ıjzV^z

. The only expression with a somewhat nontrivial change is the one for@w@wTww, which now reads

@_C@_DT_AB D 1

zı_ABı_CD@_zT_wwC 1

z².ı_ACı_BDCı_ADı_BC/ .T_zz T_ww/ :

We can now apply these expressions to thedC1CCZ4 equations to obtain what is essentially the 3C1CCZ4 equations with additional terms added involving the cartoon components. From an implementational point of view, this is simply achieved by adding a few extra grid variables for the cartoon components to the code which implements the equations (6.16)-(6.26) (the ‘RHS code’).

One obvious potential difficulty is the apparent singular behaviour of some cartoon expressions at z D 0. While it is possible to regularise the expressions and treat the evaluation atz D 0as a special case, sinceGRChombois a cell-centered code, we never actually have grid points atzD0 and therefore we can straightforwardly implement the cartoon expressions.