Gauge drivers for the generalized harmonic Einstein equations

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LINDBLOM et al. PHYSICAL REVIEW D 77, 084001 (2008) acteristic fields of this system. The constraint C H ia propagates at the speed 1 1 n k N k , while C H ija propagates H at the speed n k N k . This constraint evolution system is strongly (and also symmetric) hyperbolic. The constraint evolution system is also homogeneous in the constraints; i.e., the right sides of Eqs. (41) and (42) are proportional to the constraints. This implies, for example, that these constraints will remain satisfied within the domain of dependence of the subset of the initial surface on which they are satisfied. E. Boundary conditions Boundary conditions are needed for any of the characteristic fields having incoming (i.e., negative) characteristic speeds on the boundary. Some of these boundary conditions can be determined by the need to prevent the influx of constraint violations, while others can be chosen to control the particular gauge condition being imposed at the boundary. In analogy with the scalar field system [5], the needed constraint preserving boundary conditions for this system are d t Z H1 a d t Z H2 ia D t Z H1 a 1 H 1 n k N k n i C H ia ; (43) D t Z H2 ia n k N k P i j n l C H jla ; (44) where d t Z H1 a @ t H a and d t Z H2 ia P k i @ H t ka represent the constraint field projections of the time derivatives of the dynamical fields, while D t Z H1 a and D t Z H2 ia represent the constraint field projections of the right sides of the evolution equations for these fields. The characteristic fields Ua H need boundary conditions whenever the corresponding speeds v N n k N k are negative. Since v
GAUGE DRIVERS FOR THE GENERALIZED HARMONIC ... PHYSICAL REVIEW D 77, 084001 (2008) A. Slicing conditions One of oldest gauge conditions used in numerical relativity is maximal slicing [6], where the trace of the extrinsic curvature of the t constant hypersurfaces vanishes: K 0. More generally constant curvature slicings are sometimes used, K K 0 , where K 0 is constant on each slice (but may be a specified function of time). This gauge condition can be written in the form G^t 0, where 1 G^t K 0 K K 0 2 gij ij g ij t c ijc: (49) Using Eq. (45) with Eqs. (48) and (49), we obtain F^t 1 2 ta t b ab 1K 0 1 2 1 1 g ij ij 1 1 t a g ij ija: (50) The choice of the arbitrary slicing gauge parameter, 1 1, gives a very simple expression for the constant curvature target gauge source function F^t , but other choices may be more stable or more effective. Perhaps the most widely used slicing conditions are various members of the family introduced by Bona and Massó [7]. These gauge conditions are evolution equations for the lapse N having the general form @ t N N k @ k N N 2 f N K K 0 ; (51) where f N is an arbitrary function of the lapse. The original gauge condition [7] uses 1. The particular case f N 2=N corresponds to the widely used one-pluslog slicing condition [8–11]. An expression for the general form of these gauge conditions in terms of the first-order GH fields is given by G^t K 0 g ij t a ija 1 2 gij ij 1 2f N ta t b 1 2Nf N Ni t a t b iab: (52) Using this condition in Eq. (45) results in the needed target gauge source functions for these Bona-Massó slicing conditions: F^t 1 f N 2f N t a t b ab 1K 0 1 1 t a g ij ija ab 1 1 2Nf N Nk t a t b kab 1 2 1 1 g ij ij: (53) The slicing gauge parameter choice 1 1 makes this expression for the Bona-Massó gauge condition particularly simple; however, any choice with 1 0 is allowed. B. Shift conditions The simplest shift condition (from our perspective) is referred to as freezing [12]. This condition fixes the trace of the Christoffel symbol associated with the conformal spatial metric ~g ij g g ij , where g detg ij and is a constant. (Often is chosen to be 1 3 so that det~g ij 1, but any value is allowed.) The relevant trace of this conformal connection is defined by The 3 ~i 3 ~i jk ~g jk g g ik g jl 1 2 g ij g kl -freezing shift condition simply requires that @ t 3 ~i jkl: (54) 0: (55) For our purposes this must be translated into a condition on the spacetime metric and its first derivatives. This is accomplished by integrating Eq. (55) to obtain 3 ~i 3 ~i 0 , where 3 ~i 0 is the trace evaluated at the initial time. This condition can be expressed in terms of firstorder GH fields as 3 G i g g ~j ij 0 i l g jk 1 i j g kl 2 jkl: (56) Using Eqs. (45) and (48), this gauge condition is easily transformed into the needed target gauge source function: F i 1 2 1 2 1 g jk ijk 1 2 ta t b iab t a ai 2g g 3 ij ~j 0 2 1 g jk jki; (57) where the shift gauge parameter 2 0 can be chosen freely. As a modest generalization we might also want to consider -fixing conditions for which 3 ~i is specified as a function of time. For example we might want to set 3 ~i 3 t ~i 0 e t . This can be done by replacing 3 ~i 0 with the desired 3 ~i t in Eqs. (56) and (57). The most commonly used shift conditions in the numerical relativity community are the -driver conditions. The simplest of these can be written as the following evolution equations for the shift [10], @ t N i B i ; (58) @ t B i 2B i @ t 3 ~i ; (59) where 3 ~i is the trace of the conformal spatial connection, Eq. (54), and and 2 are adjustable constants. The 3 parameter is usually set to 4 on the basis of causality arguments [9,10]. But these arguments do not apply when the lapse and shift are evolved with the GH Einstein equations, so we leave as an adjustable parameter. Unfortunately this shift condition is not of the form G i G i x; ; @ , which is required by our gauge-driver system, because the right side of Eq. (59) depends on second derivatives of the spacetime metric. This particular -driver condition, Eqs. (58) and (59), can be transformed however into the more useful form @ t N i 3 ~i 2 i ; (60) 084001-7
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Gauge drivers for the generalized harmonic Einstein equations

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