4.4 The FO-CCZ4 formulation
4.4.2 The FO-CCZ4 system of equations
Reduction to first order: auxiliary variables definition
In order to reduce any system of equations from second to first order form, derivatives of the original variables have to be promoted to new auxiliary vari-ables, which will be then promoted to independently evolved variables. Start-ing from the original CCZ4 system we introduce the followStart-ing 33auxiliary variables, which correspond to first spatial derivatives of the metric terms:
Ai :=∂ilnα=∂iα
α (4.14a)
Bki :=∂kβi (4.14b)
Dkij:= 1
2∂kγ˜ij (4.14c)
Pi :=∂ilnφ= ∂iφ
φ . (4.14d)
The following second order ordering constraints (Gundlach and Martin-Garcia, 2006) are an immediate consequence of (4.14) and the symmetry of second-order derivatives (note that the spacetime functions are assumed to be twice continuously differentiable, since the spacetime is described as a differ-entiable manifold in general relativity):
Aki :=∂kAi−∂iAk = 0, Bikl:=∂kBli−∂lBki = 0,
Dklij:=∂kDlij−∂lDkij= 0, Pki :=∂kPi−∂iPk= 0. (4.15) A˜ij is by construction trace-free, therefore the following additional con-straint holds:γ˜ijA˜ij = 0; and thus
Tk:=∂k
˜γijA˜ij
=∂k˜γijA˜ij+ ˜γij∂kA˜ij = 0. (4.16) These relations will be employed later in order to enforce strong hyperbolicity of the system. The time evolution equations for the auxiliary quantities are obtained by the standard procedure of applying the time derivative operator
∂tto equations (4.14), exchanging the order of the space and time derivatives on the right-hand side of the resulting equations and making use of the PDEs (4.13a), (4.13c) and the evolution equations for the lapse and shift to eliminate the time derivatives from the left hand side.
In principle many different first-order formulations of the CCZ4 system are possible, since any non purely algebraic term in the original second-order sys-tem can be written as a combination of conservative terms and non-conservative products (see (Gundlach and Martin-Garcia,2006;Hilditch and Richter,2015) for a parametric study of such families of systems). Two extreme cases are of particular interest: the first being a fully conservative form of the equations, in which all terms are cast in a flux-conservative form (seee.g. Alic et al.(2009) as an example for the first-order Z4 system); and a second formulation, fully non-conservative, similar to the ideas outlined inAlcubierre(2008), in which by making maximum use of the first-order ordering constraints the variables defining the 4-metric (α, βi, φ and γ˜ij) are evolved simply by a nonlinear
system of ordinary differential equations (ODEs) and the rest of the dynam-ics is encoded in conservative products. The coefficients of these non-conservative products are only functions ofα,βi,φand˜γij and no terms in-volving derivatives of these variables appear. The dynamical variables of the FO-CCZ4 system (including the auxiliary variables involved in the formula-tion of the Gamma-driver shift condiformula-tion) are then: A˜ij,K,Θ,Γˆi,bi(bi being the auxiliary field used to formulate the Gamma-driver condition) and the aux-iliary variablesAk,Bki,Pk andDkij. In this work we elected upon experimen-tation to follow the second approach, so that our final system of 58 evolution equations is in fully non conservative form and consist of 11 ODEs and 47 PDEs. The form of the system and the very special structure it assumes are presented in section4.4.3.
The FO-CCZ4 equations
The final first-order form of the FO-CCZ4 system is non-conservative and ap-pears in the following form
∂Q
∂t +A1(Q)∂Q
∂x1
+A2(Q)∂Q
∂x2
+A3(Q)∂Q
∂x3
=S(Q), (4.17)
where the symbols shorthands for the state vectorQ, the system matricesAi and the purely algebraic source termsS(Q)respectively.
From now on the discussion is restricted to the vacuum case for simplic-ity,i.e. assume the stress-energy tensor to be identically null in the following.
The neglected matter terms can be trivially added to the equations as needed since they are purely algebraic terms; as such they do not enter the principal part of the system, and can be neglected in computing its eigenstructure and determining its hyperbolicity. For the same reason they also do not pose any issues to the numerical discretization. Therefore our final non-conservative
first-order FO-CCZ4 system reads as follows:
with the PDEs for the auxiliary variables given by:
In the preceding equations those terms that have been added to the PDEs to obtain an approximate symmetrization of the sparsity pattern are indicated in red (see the discussion in section4.4.3and figure4.1). Note that the function g(α)in equation (4.18b) controls the type of slicing gauge condition which is emplyed. In particularg(α) = 1leads to harmonic slicing and whileg(α) = 2/αto the so-called1 + logslicing conditionBona et al.(1995).
In order to obtain the shift advection terms in the evolution equations for the auxiliary variables, the identities (4.15) have been employed. In fact it re-sults to be very important to use the second-order ordering constraints (4.15) in an appropriate way in order to guarantee strong hyperbolicity: a naive first-order formulation of the original CCZ4 system will general only result in a weakly, rather than strongly, hyperbolic system (see Gundlach and Martin-Garcia (2006) for a detailed discussion on the use of second-order ordering constraints in second order in space first order in time hyperbolic systems).
More over we have found that the use of first and second-order ordering con-straints alone is in fact not sufficient to get a strongly hyperbolic system, but that it is necessary to derive the PDE (4.19c) for the evolution of theDkijfield from (4.13a) by exploitingA˜ijbeing trace-free. This is done via the use of the constraintTk, by adding equation (4.16) to equation (4.19c). These additional terms in the FO-CCZ4 equations, related to the constraints (4.15) and (4.16) are the ones highlighted in red in equations (4.18a)-(4.19d), as mentioned above.
Note finally that a number of additional constants have been introduced in the FO-CCZ4 system, mostly for reasons of convenience. These are:
• τ is a relaxation time introduced to enforce the algebraic constraints on the determinant ofγ˜ij and on the trace ofA˜ij weakly (i.e. without the need to explicitly enforce these constraints after each evolution timestep in a numerical implementation; see the discussion inAlic et al.(2012)).
• eis a cleaning speed for the Hamiltonian constraint. This has been in-troduced following the ideas of the generalized Lagrangian multiplier
(GLM) approach ofDedner et al.(2002). In practice, this constant affects the speed at which violation of the Hamiltonian constraint are propa-gated away from their source. Note that the cleaning of the constraints is a non-physical process, and as suche >1(i.e. superluminal propagation speeds) is allowed; this choice would lead to faster constraint propaga-tion, helping in reducing constraint violations; and is in fact also needed for strong hyperbolicity for certain gauge choices.
• µ >0appears in equation (4.19b) and allows to adjust the relative contri-bution of second-order ordering constraints.
• sappears in the PDEs controlling the shift evolution. It simply allows to switch on or off the evolution of the shift vector,i.e. ifs = 0the gauge condition for the shift reduces to ∂tβi = 0, while for s = 1the usual Gamma driver condition is recovered.
• callows to turn off some of the algebraic source terms of the system.
Note that instead of evolving the lapseαand the conformal factorφ, their logarithmsln(α)andln(φ)are evolved. While not a standard choice, this is a very simple way of preserving the positivity of the respective variables, even at the discrete level. Note also that at a black hole puncture location the lapse would vanish and its logarithm diverge. Therefore a positive lower limit on their values is imposed in our numerical implementation. Since the numerical method we selected to dicretize the FO-CCZ4 system is a DG scheme, in which the solution in every computational element is represented by an interpolat-ing polynomial as described in the previous chapter, in an element surround-ing the puncture the DG polynomial might if fact reach values lower than this lower limit due to the Runge phenomenon. However it would in any case not diverge.
The following explicit expressions and identities apply to various terms
ap-pearing in the evolution equations:
trA˜ = ˜γijA˜ij, and ˜γ=det(˜γij) (4.20)
∂kγ˜ij =−2˜γin˜γmjDknm:=−2Dkij (4.21) Γ˜kij = ˜γkl(Dijl+Djil−Dlij) (4.22)
∂kΓ˜mij =−2Dkml(Dijl+Djil−Dlij)
+ ˜γml ∂(kDi)jl+∂(kDj)il−∂(kDl)ij
(4.23) Γkij = ˜γkl(Dijl+Djil−Dlij)−γ˜kl(˜γjlPi+ ˜γilPj−γ˜ijPl)
= ˜Γkij−γ˜kl(˜γjlPi+ ˜γilPj−˜γijPl) (4.24)
∂kΓmij =−2Dkml(Dijl+Djil−Dlij) + 2Dkml(˜γjlPi+ ˜γilPj−γ˜ijPl)
−2˜γml(DkjlPi+DkilPj−DkijPl) + ˜γml ∂(kDi)jl+∂(kDj)il−∂(kDl)ij
−˜γml γ˜jl∂(kPi)+ ˜γil∂(kPj)−˜γij∂(kPl)
(4.25) Rmikj =∂kΓmij−∂jΓmik+ ΓlijΓmlk−ΓlikΓmlj (4.26)
Rij =Rmimj (4.27)
∇i∇jα =αAiAj−αΓkijAk+α∂(iAj) (4.28)
∇i∇iα =φ2γ˜ij(∇i∇jα) (4.29)
Γ˜i = ˜γjlΓ˜ijl (4.30)
∂kΓ˜i =−2DkjlΓ˜ijl+ ˜γjl∂k˜Γijl (4.31)
Zi =1
2˜γij
Γˆj−Γ˜j
Zi =1
2φ2(ˆΓi−Γ˜i) (4.32)
∇iZj =Dijl
Γˆl−Γ˜l +1
2γ˜jl
∂iΓˆl−∂iΓ˜l
−ΓlijZl (4.33) R+ 2∇kZk =φ2γ˜ij(Rij+∇iZj+∇jZi). (4.34) Here, we have again made use of the second-order ordering constraints (4.15) bysymmetrizingthe spatial derivatives of the auxiliary variables as fol-lows:
∂(kAi) :=∂kAi+∂iAk
2 , ∂(kPi) := ∂kPi+∂iPk
2
∂(kBij):=∂kBij+∂jBik
2 , ∂(kDl)ij := ∂kDlij+∂lDkij
2 . (4.35)
In general the above expressions contain terms contributing to the purely algebraic source term as well as to the non-conservative products, which need to be separate in the numerical implementation. Note also that the Ricci ten-sorRij is directly calculated from its definition in terms of the Riemann ten-sorRmikj, the Christoffel symbols and their derivativesab definitionem,i.e. we don’t employ the typical split of the Ricci tensor ase.g.used inAlic et al.(2012).
Since in general the identity˜γij = 1 cannot be guaranteed to hold exactly at the discrete level (unless the algebraic constraints are rigorously enforced), the contracted Christoffel symbols˜Γiare also directly computed from their defini-tion.
The use of the second-order ordering constraints (4.15) and the constraint Tk to achieve strong hyperbolicity is motivated by examining the structure of the sparsity pattern of the system matrixA·n = A1n1+A2n2+A3n3. In figure 4.1 the sparsity pattern of the system matrix in the normal direction n = 1/√
3(1,1,1)for the Gamma-driver shift condition and the1 + log slic-ing condition for a randomly perturbed flat Minkowski spacetime is p[lotted, neglecting all matrix entries whose absolute value is below a threshold of10−7. The blue dots represent the original sparsity structure without the use of the second-order ordering constraints (4.15) and the constraint (4.16); while the combination of the blue and the red dots shows the sparsity pattern after those terms have been added to the PDE system. Our approach for finding a suit-able form of the ordering constraints to be added is based on the approximate symmetrization of the sparsity pattern of the system matrix, in order to avoid Jordan blocks, which cannot be diagonalized. Such Jordan blocks are evidently present in the sparsity pattern indicated by the blue dots alone in figure4.1,i.e.
without the use of the ordering constraints.
It is clearly evident from figure4.1that the first 11 varibles of the FO-CCZ4 system, i.e. γ˜ij, α, βi and φ, are only evolved via ODEs and that the entire system does not include spatial derivatives of these variables: all entries in the first 11 rows and columns of the system matrix are zero. However this deos not imply that the FO-CCZ4 system is symmetric hyperbolic (in the sense of Friedrichs(1954)). Further work in this direction would be needed to achieve a symmetric hyperbolic form of FO-CCZ4. Note finally that in order to main-tain the split of the system into 11 pure ODEs (4.37) for the metric variablesα, βi,γ˜ij andφ, and with no spatial derivatives of these quantities appearing in the remaining PDE system (4.38), and 47 PDEs, it is not possible to add damp-ing terms proportional to the first-order orderdamp-ing constraints (4.14) to the sys-tem. These terms would contain spatial derivatives of said variables and may eventually lead to Jordan blocks which cannot be diagonalized. We therefore refrain from adding these terms, in contrast to what has been done inBrown et al.(2012). For the same reason also writing the system in a flux-conservative form is not possible, since the fluxes would in general depend on the 4-metric and spatial derivatives ofα,βi,γ˜ijandφwould appear.