Numerical Relativity - List of Formalisms

various hyperbolic representations of the ADM 3+1 initial value problem equations

original ADM, non-hyperbolic
From Alcubierre page 180 or so, with amendment for non-vacuum K taken from eqn 2.5.8
Also from Baumgarte & Shapiro, eqns 2.108, 2.106
1: $\frac{d}{dt} \alpha = -\alpha^2 f K$
6: $\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
6: $\frac{d}{dt} K_{ij} = -D_i D_j \alpha + \alpha (R_{ij} + K K_{ij} - 2 K_{ik} {K^k}_j) + 4 \pi \alpha (\gamma_{ij} (S - \rho) - 2 S_{ij})$

original ADM, from Alcuberre's "Introduction to 3+1 Numerical Relativity" section 5.4
weakly hyperbolic
source-only variables: (7 total, to be propagated with time waves)
1: $\frac{d}{dt} \alpha = -\alpha^2 f K$
6: $\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
hyperbolic variables ("up to principle part"): (27 total)
3: $\partial_0 a_i + \alpha (f K)_{,i} = 0$
18: $\partial_0 d_{ijk} + \alpha K_{jk,i} = 0$
6: $\partial_0 K_{ij} + \alpha {\Lambda^k}_{ij,k} = 0$
expanded:
$\partial_t a_i - \beta^r \delta^m_i a_{m,r} + \alpha f \gamma^{mn} \delta^r_i K_{mn,r} = 2 \alpha f {d_i}^{jk} K_{jk} - f' \alpha^2 a_i K $
$\partial_t d_{ijk} - \beta^r \delta^m_i \delta^n_j \delta^p_k d_{mnp,r} + \alpha \delta^m_j \delta^n_k \delta^r_i K_{mn,r} = 0 $
$\partial_t K_{ij} + \alpha \delta^m_{(i} \delta^r_{j)} a_{m,r} + \alpha ( \gamma^{mr} \delta^n_i \delta^p_j + \gamma^{np} \delta^m_{(i} \delta^r_{j)} - 2 \gamma^{mn} \delta^p_{(i} \delta^r_{j)} ) d_{mnp,r} - \beta^r \delta^m_i \delta^n_j K_{mn,r} = 2 \alpha d_{lij} {d_k}^{kl} + 2 \alpha {d_{(i}}^{kl} (d_{j)kl} - 2 d_{kl|j)}) $
matrix form:
$\left[\matrix{ a_i \\ d_{ijk} \\ K_{ij} }\right] + \left[\matrix{ -\beta^r \delta^m_i & 0 & \alpha f \gamma^{mn} \delta^r_i \\ 0 & -\beta^r \delta^m_i \delta^n_j \delta^p_k & \alpha \delta^m_j \delta^n_k \delta^r_i \\ \alpha \delta^m_{(i} \delta^r_{j)} & \alpha ( \gamma^{mr} \delta^n_{(i} \delta^p_{j)} + \gamma^{np} \delta^m_{(i} \delta^r_{j)} - 2 \gamma^{mn} \delta^p_{(i} \delta^r_{j)} ) & -\beta^r \delta^m_i \delta^n_j }\right] \left[\matrix{ a_m \\ d_{mnp} \\ K_{mn} }\right]_{,r} = \left[\matrix{ 2 \alpha f {d_i}^{jk} K_{jk} - f' \alpha^2 a_i K \\ 0 \\ 2 \alpha (d_{lij} {d_k}^{kl} + {d_{(i}}^{kl} (d_{j)kl} - 2 d_{kl|j)})) }\right]$
eigenvalues and eigenfields in x-direction:
15 time waves: $\lambda = -\beta^x$ for $w = a_{q}, d_{qij}, a_x - f {d_{xm}}^m$ for $p,q \ne x$
10 light waves: ${\lambda^l}_{iq\pm} = -\beta^x + \alpha \sqrt{\gamma^{xx}}$ for ${w^l}_{iq\pm} = \sqrt{\gamma^{xx}} K_{iq} \pm {\Lambda^x}_{iq}$ (note the book says $\sqrt{\gamma^{xx}} K_{pq} \mp {\Lambda^x}_{pq}$, but I get bad results with this.
2 gauge waves: ${\lambda^f}_\pm = -\beta^x + \alpha \sqrt{f \gamma^{xx}}$ for ${w^f}_\pm = \sqrt{f \gamma^{xx}} K \pm \Lambda^x$ (once again, book says $\mp$)
proof (for the $x$-direction flux matrix):
2: $a_{q,t} - \beta^x a_{q,x} = 0$, so $\lambda = -\beta^x$ and $w = a_q$. (note that $a_{x,t} - \beta^x a_{x,x} = -\alpha (f K)_{,x} \ne 0$).
12: $d_{qij,t} - \beta^x d_{qij,x} = 0$, so $\lambda = -\beta^x$ and $w = d_{qij}$. (note that $d_{xij,t} - \beta^x d_{xij,x} = -\alpha K_{ij,x} \ne 0$).
$K_{ij,t} - \beta^x K_{ij,x} = -\alpha {\Lambda^k}_{ij,k}$

Bona, Masso, Seidel, Stela "New Formalism for Numerical Relativity" 1995
and "First order hyperbolic formalism for Numerical Relativity" 1997
and whatever paper around that time in which $V_k$ has no flux term.
source-only variables: (7 total)
1: $\frac{d}{dt} \alpha = -\alpha^2 f K$
6: $\gamma_{ij,t} = -2 \alpha K_{ij}$
other vars:
${b_i}^k = \frac{1}{2} {\beta^k}_{,i}$
hyperbolic variables: (30 total)
3: $a_{k,t} + (-\beta^r a_k + \alpha Q \delta^r_k)_{,r} = (2 {b_k}^r - \alpha tr s \delta^r_k) a_r$
18: $d_{kij,t} + (-\beta^r d_{kij} + \alpha \delta^r_k (K_{ij} - s_{ij}))_{,r} = (2 {b_k}^r - \alpha tr s \delta^r_k) d_{rij}$
6: $K_{ij,t} + (-\beta^r K_{ij} + \alpha {\Lambda^r}_{ij})_{,r} = \alpha S_{ij}$
3: $V_{k,t} + (-\beta^r V_k + \alpha ({s^r}_k - tr s \delta^r_k))_{,r} = \alpha P_k$
other state variables:
${{b_k}^i}_{k,t} + (\beta^r {b_k}^i + \alpha Q^i \delta^r_k)_{,r} = (2 {b_k}^r - \alpha tr s \delta^r_k) {b_r}^i$
other variables:
$s_{ij} = 2 b_{(ij)} / \alpha$

Bona-Masso ADM formalism with $V_i$ momentum constraint, without lapse, from Alcubierre "The appearance of coordinate shocks in hyperbolic formalisms of General Realtivity" 1997. https://arxiv.org/abs/gr-qc/9609015
source-only variables: (7 total)
1: $\alpha_{,t} = -\alpha^2 f K$
6: $\gamma_{ij,t} = -2 \alpha K_{ij}$
hyperbolic variables: (30 total)
3: $a_{k,t} + (a f K)_{,k} = 0$
18: $d_{kij,t} + (\alpha K_{ij})_{,k} = 0$
6: $K_{ij,t} + (\alpha {\Lambda^k}_{ij})_{,k} = \alpha S_{ij}$
3: $V_{k,t} = \alpha P_k$
constraints: (3 total)
$V_i = {d_{im}}^m - {d^m}_{mi}$
eigenvalues and eigenfields in x-direction: (37 total, notice that the source-only variables are advected with the timelike waves)
25 time waves: $\lambda = 0$ for $w = \alpha, \gamma_{ij}, a_{q}, d_{qij}, V_i, a_x - f {{d_x}^m}_m$, for indexes $q \ne x$
10 light waves: ${\lambda^l}_{iq\pm} = \pm \alpha \sqrt{\gamma^{xx}}$ for ${w^l}_{iq\pm} = K_{iq} \pm \sqrt{\gamma^{xx}} (d_{xiq} + \delta^x_i V_{q} / \gamma^{xx})$
2 gauge waves: ${\lambda^f}_\pm = \pm \alpha \sqrt{f \gamma^{xx}}$ for ${w^f}_\pm = \sqrt{f} K \pm \sqrt{\gamma^{xx}} (a_x + 2 V^x / \gamma^{xx})$

Bona-Masso with $V_i$ momentum constraint, from Alcuberre's "Introduction to 3+1 Numerical Relativity" section 5.5
source-only variables: (7 total)
1: $\frac{d}{dt} \alpha = -\alpha^2 f K$
6: $\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
hyperbolic variables: (30 total)
3: $\partial_0 a_k + \alpha (f K)_{,k} = 0$
18: $\partial_0 d_{kij} + \alpha K_{ij,k} = 0$
6: $\partial_0 K_{ij} + \alpha {\Lambda^k}_{ij,k} = 0$
3: $\partial_0 V_i = 0$
eigenvalues and eigenfields in x-direction:
18 time waves: $\lambda = -\beta^x$ for $w = a_q, d_{qij}, V_i, a_x - f {d_{xm}}^m$
10 light waves: ${\lambda^l}_{iq\pm} = -\beta^x \pm \alpha \sqrt{\gamma^{xx}}$ for ${w^l}_{iq\pm} = \sqrt{\gamma^{xx}} K_{iq} \pm {\Lambda^x}_{iq}$ (book says $\mp$)
2 gauage waves: ${\lambda^f}_\pm = -\beta^x \pm \alpha \sqrt{f \gamma^{xx}}$ for ${w^f}_\pm = \sqrt{f \gamma^{xx}} K \pm \Lambda^x$ (book says $\mp$)

Bona-Masso with $\Gamma^i$ evolution, from Alcuberre's "Introduction to 3+1 Numerical Relativity" section 5.5 starting at eqn 5.5.21
source-only variables: (7 total)
1: $\frac{d}{dt} \alpha = -\alpha^2 f K$
6: $\frac{d}{dt} \gamma_{ij,t} = -2 \alpha K_{ij}$
hyperbolic variables: (30 total)
3: $\partial_0 a_i + \alpha (f K)_{,i} = 0$
18: $\partial_0 d_{ijk} + \alpha K_{jk,i} = 0$
6: $\partial_0 K_{ij} + \alpha {\Lambda^k}_{ij,k} = 0$
3: $\frac{d}{dt} \Gamma^i = \gamma^{lm} {\beta^i}_{,lm} - \alpha \gamma^{il} K_{,l} + \alpha a_l (2 K^{il} - \gamma^{il} K) + 2 \alpha K^{lm} {\Gamma^i}_{lm} - 16 \pi j^i$
eigenvalues and eigenfields in x-direction (match those of the Bona-Masso with $V_i$ momentum constraints):
18 time waves: $\lambda = -\beta^x$ for $w = a_q, d_{qij}, \frac{1}{2} ({{d_i}^m}_m - \Gamma_i), a_x - f {d_{xm}}^m$
10 light waves: ${\lambda^l}_{iq\pm} = -\beta^x \pm \alpha \sqrt{\gamma^{xx}}$ for ${w^l}_{iq\pm} = \sqrt{\gamma^{xx}} K_{iq} \pm {\Lambda^x}_{iq}$ (book says $\mp$)
2 gauage waves: ${\lambda^f}_\pm = -\beta^x \pm \alpha \sqrt{f \gamma^{xx}}$ for ${w^f}_\pm = \sqrt{f \gamma^{xx}} K \pm \Lambda^x$ (book says $\mp$)

Nagy-Ortiz-Reula (NOR) from Alcubierre's "Introduction to 3+1 Numerical Relativity" section 5.5 starting at eqn 5.5.23
adds parameters $\eta, \xi$, where $\eta = 0, \xi = 2$ reduces to Bona-Masso, and $\eta = -\frac{1}{4}, \xi = 0$ reduces to ADM
source-only variables: (7 total)
1: $\frac{d}{dt} \alpha = -\alpha^2 f K$
6: $\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
hyperbolic variables: (30 total)
3: $\partial_0 a_i + \alpha (f K)_{,i} = 0$
18: $\partial_0 d_{ijk} + \alpha K_{jk,i} = 0$
TODO rewrite the following in terms of state variables:
6: $\frac{d}{dt} K_{ij} = -\alpha_{,ij} + {\Gamma^k}_{ij} \alpha_{,k} + \alpha R_{ij} + \alpha K K_{ij} - 2 \alpha K_{ik} {K^k}_j + \eta \alpha \gamma_{ij} H$
3: $\frac{d}{dt} \Gamma^i = \gamma^{lm} {\beta^i}_{,lm} - D_l (\alpha (2 K^{il} - \gamma^{il} K)) + 2 \alpha K^{lm} {\Gamma^i}_{lm} + \alpha \xi M^i$
other variables:
$\vec{s}^x =$ vector normal to the flux surface
$(s^x)_i = \delta^x_i / \sqrt{\gamma^{xx}}$, $(s^x)^i = \gamma^{xi} / \sqrt{\gamma^{xx}}$
$h_{ij} = \gamma_{ij} - (s^x)_i (s^x)_j$
$\hat{K} = h^{ij} K_{ij} = K - K^{xx} / \gamma^{xx}$
$F = (1 + 3 \eta) (2 - \xi) / (f - 1 - 2 \eta (2 - \xi))$
eigenvalues and eigenfields:
18 time waves: $\lambda = -\beta^x$ for $w = a_q, d_{qij}, a_x - f {d_{xm}}^m, \Gamma^i + (\xi - 2) {d_m}^{mi} + (1 - \xi) {d^{im}}_m$
6 light waves: ${\lambda^{light}}_\pm = -\beta^x \pm \alpha \sqrt{\gamma^{xx}}$ for ${w^{light}}_{pq\pm} = \sqrt{\gamma^{xx}} (K_{pq} - \frac{1}{2} h_{pq} \hat{K}) \mp ({\Lambda^x}_{pq} - \frac{1}{2} h_{pq} {\hat\Lambda}^x)$
2 trace waves: ${\lambda^{trace}}_\pm = -\beta^x \pm \alpha \sqrt{ \gamma^{xx} (1 + 2 \eta (2 - \xi)) }$ for ${w^{trace}}_\pm = \sqrt{ \gamma^{xx} (1 + 2 \eta (2 - \xi)) } \hat{K} \mp {\hat\Lambda}^x$
2 longitudinal waves: ${\lambda^{long}}_\pm = -\beta^x \pm \alpha \sqrt{ \gamma^{xx} \xi / 2 }$ for ${w^{long}}_\pm = \sqrt{ \gamma^{xx} \xi / 2 } {K^x}_q \mp {\Lambda^{xx}}_q$
2 gauge waves: ${\lambda^{gauge}}_\pm = -\beta^x \pm \alpha \sqrt{ f \gamma^{xx} }$ for ${w^{gauge}}_\pm = \sqrt{f \gamma^{xx}} (K + F \hat{K}) \mp (\Lambda^x + F {\hat\Lambda}^x)$
NOR is hyperbolic for one of the three conditions:
1. $\eta = 0, \xi = 2, f > 0$
2. $\eta = 0, \xi > 0, f > 0, f \ne 1$
3. $\eta \ne 0, \xi > 0, f > 0, \eta (2 - \xi) > -\frac{1}{2}$

BSSNOK from Alcubierre's "Introduction to 3+1 Numerical Relativity" section 5.6
adds parameters $\xi$, with standard BSSNOK for $\xi = 2$.
note that $\xi > \frac{1}{2}$ for wave speeds to be real.
source-only variables: (7 total)
1: $\frac{d}{dt} \phi = -\frac{1}{6} \alpha K$
6: $\frac{d}{dt} \tilde{\gamma}_{ij} = -2 \alpha \tilde{A}_{ij}$
hyperbolic variables: (30 total)
3: $\frac{d}{dt} a_i = -\alpha Q_{,i}$
3: $\frac{d}{dt} \Phi_i = -\frac{1}{6} \alpha K_{,i}$
15: $\frac{d}{dt} \tilde{d}_{ijk} = -\alpha \tilde{A}_{jk,i}$
1: $\frac{d}{dt} K = -\alpha e^{-4 \phi} \tilde{\gamma}^{mn} a_{n,m}$
5: $\frac{d}{dt} \tilde{A}_{ij} = -\alpha e^{-4 \phi} {\tilde{\Lambda}^k}_{ij,k}$
3: $\frac{d}{dt} \tilde{\Gamma}^i = \alpha ((\xi - 2) \tilde{A}^{ik} - \frac{2}{3} \xi \tilde{\gamma}^{ik} K)_{,k}$
constraints:
${\tilde{A}^k}_k = 0$ needs to be enforced
eigenvalues and eigenfields:
18 time waves: $\lambda = -\beta^x$ for $w = a_q, \Phi_q, \tilde{d}_{qij}, a_x - 6 f \Phi_x, \tilde{\Gamma}^i + (\xi - 2) {\tilde{d}_m}^{mi} - 4 \xi \tilde{\gamma}^{ik} \Phi_k$
2 gauge waves: ${\lambda^{gauge}}_\pm = -\beta^x \pm \alpha e^{-2 \phi} \sqrt{f \tilde{\gamma}^{xx} }$ for ${w^{gauge}}_\pm = e^{-2\phi} \sqrt{f \tilde{\gamma}^{xx}} K \mp a^x$
4 longitudinal waves: ${\lambda^{long}}_\pm = -\beta^x \pm \alpha e^{-2\phi} \sqrt{\tilde{\gamma}^{xx} \xi / 2}$ for ${w^{long}}_\pm = e^{2\phi} \sqrt{ \tilde{\gamma}^{xx} \xi / 2 } {\tilde{A}^x}_q \mp {\tilde{\Lambda}^{xx}}_q$
4 light waves: ${\lambda^{light}}_\pm = -\beta^x \pm \alpha e^{-2\phi} \sqrt{\tilde{\gamma}^{xx}}$ for ${w^{light}}_\pm = e^{2 \phi} \sqrt{\tilde{\gamma}^{xx}} ( \tilde{A}_{pq} + \frac{1}{2} \tilde{\gamma}_{pq} \tilde{A}^{xx} / \tilde{\gamma}^{xx}) \mp ({\tilde{\Lambda}^x}_{pq} + \frac{1}{2} \tilde{\gamma}_{pq} \tilde{\Lambda}^{xxx} / \tilde{\gamma}^{xx}$
2 transverse-trace waves: ${\lambda^{trace}} = -\beta^x + \alpha e^{-2 \phi} \sqrt{ \tilde{\gamma}^{xx} (2 \xi - 1) / 3 }$ (should this be $\pm$?) for $w = e^{2\phi} \sqrt{ \tilde{\gamma}^{xx} (2\xi - 1)/3} (\tilde{A}^{xx} - \frac{2}{3} \tilde{\gamma}^{xx} K) \mp (\tilde{\Lambda}^{xxx} - \frac{2}{3} \tilde{\gamma}^{xx} \tilde{a}^x)$

Kidder-Scheel-Teukolsky (KST) family, from Alcubierre's "Introduction to 3+1 Numerical Relativity" section 5.7
$\xi, \eta, \chi$ free parameters subject to $1 - \xi + 2 \chi \ne 0$, $1 + 2 \chi + 4 \eta (1 - \xi + 2 \chi) > 0$, $\xi - \chi / 2 > 0$, $f > 0$.
Optimally: $\eta = -\frac{1}{3}, \xi = 1 + \chi / 2, \chi \ne 0$
source-only variables: (7 total)
1: $\frac{d}{dt} \alpha = -\alpha^2 f K$
6: $\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
hyperbolic variables: (27 total)
3: $\partial_0 a_i + \alpha (f K)_{,i} = 0$
18: $\partial_0 d_{ijk} + \alpha K_{jk,i} = \alpha (\xi \gamma_{i(j} M_{k)} + \chi \gamma_{jk} M_i)$
6: $\partial_0 K_{ij} + \alpha ({\Lambda^k}_{ij,k} + 2 \eta \gamma_{ij} V^k) = 0$
eigenvalues and eigenfields:
15 time waves: $\lambda = -\beta^x$ for $w = a_q, d_{qij} - \frac{1}{1 - \xi + 2 \chi} (\xi \gamma_{q(i} V_{j)} + \chi \gamma_{ij} V_q), a_x - f ( {d_{xm}}^m - \frac{\xi + 3 \chi}{1 - \xi + 2 \chi} V_x )$
2 longitudinal waves: $\lambda_\pm = -\beta^x \pm \alpha \sqrt{\gamma^{xx} (\xi - \chi / 2)}$ for $w_\pm = \sqrt{\gamma^{xx} (\xi - \chi / 2)} {K^x}_q \mp {\Lambda^{xx}}_q$
4 transverse-traceless light waves: ${\lambda^{light}}_\pm = -\beta^x \pm \alpha \sqrt{\gamma^{xx}}$ for $w_\pm = \sqrt{\gamma^{xx}} (K_{pq} - \frac{1}{2} h_{pq} \hat{K}) \mp ({\Lambda^x})_{pq} - \frac{1}{2} h_{pq} \hat{\Lambda}^x)$
2 surface-trace waves: ${\lambda^{trace}}_\pm = -\beta^x \pm \alpha \sqrt{\gamma^{xx} (1 + 2 \chi + 4 \eta (1 - \xi + 2 \chi))}$ for $w_\pm = \sqrt{\gamma^{xx} (1 + 2 \chi + 4 \eta (1 - \xi + 2 \chi))} \hat{K} \mp \hat{\Lambda}^x$
2 gauge waves: ${\lambda^{gauge}}_\pm = -\beta^x \pm \alpha \sqrt{f \gamma^{xx}}$ for $w_\pm = \sqrt{f \gamma^{xx}} (K + F \hat{K}) \mp (\Lambda^x + F \hat{\Lambda}^x)$

Z4, from Alcubierre's "Introduction to 3+1 Numerical Relativity" section 5.8
$m = $ free parameter of Bona-Masso slicing generalization, must be $m = 2$ for $f = 1$, so $m = 2$ is preferred.
source-only variables: (7 total)
1: $\frac{d}{dt} \alpha = -\alpha^2 f(\alpha) \cdot (K - m \Theta)$
6: $\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
hyperbolic variables:
$\frac{d}{dt} K_{ij} = -D_i \alpha_j + \alpha ( R_{ij} + 2 D_{(i} Z_{j)} - 2 K_{im} {K^m}_j + (K - 2 \Theta) K_{ij} )$
$\frac{d}{dt} \Theta = \frac{1}{2} \alpha (R + (K - 2 \Theta) K - K_{mn} K^{mn} + 2 D_m Z^m - 2 Z^m (ln \alpha)_{,m})$
$\frac{d}{dt} Z_i = \alpha (D_m {K^m}_i - D_i K + \Theta_{,i} - 2 {K^m}_i Z_m - \Theta (ln \alpha)_{,i})$
other variables:
$\Theta = n_a Z^a$
eigenvalues and eigenfields are not provided.

Bona, Ledvinka, Palenzuela, Zacek, "A symmetry-breaking mechanism for the Z4 general-covariant evolution system" 2003, Appendix B (shift-less). https://arxiv.org/abs/gr-qc/0307067
parameters:
$f = 1$
$\lambda = 2$
$\zeta = 1$ for Ricci decomposition, $\zeta = -1$ for de Donder-Fock decomposition, and for the harmonic, symmetric hyperbolic case.
$n = \frac{4}{3}$ is used to be "quasi-equivalent to BSSN"
source-only variables:
1: $\frac{d}{dt} \alpha = -\alpha^2 (f K - \lambda \Theta)$
6: $\frac{d}{dt} \gamma_{ij} = -2 \alpha K_{ij}$
hyperbolic variables: (31 total)
3: $a_{k,t} + (\alpha (f K - \lambda \Theta))_{,k} = 0$
18: $d_{kij,t} + (\alpha K_{ij})_{,k} = 0$
6: $K_{ij,t} + (\alpha {\lambda^k}_{ij})_{,k} = ...$
1: $\Theta_{,t} + (\alpha V^k)_{,k} = ...$
3: $Z_{i,t} + (\alpha \delta^k_i (K - \Theta) - {K^k}_i)_{,k} = ...$
other variables:
$\Theta = n_a Z^a = \alpha Z^0$
${\lambda^k}_{ij} = {d^k}_{ij} + (1 - \zeta) ( {d_{(ij)}}^k - \delta^k_{(i|} {d_{m|j)}}^m) + \delta^k_{(i} (a_{j)} - {d_{j)m}}^m + 2 V_{j)})$ (eqn 34)
${\lambda^k}_{ij} = -{\Gamma^k}_{ij} - \frac{n}{2} V^k \gamma_{ij} + (1 - \zeta) ( {d_{(ij)}}^k - \delta^k_{(i|} {d_{m|j)}}^m) + \delta^k_{(i} (a_{j)} - {d_{j)m}}^m + 2 V_{j)})$ (eqn 41)
...and the two would be the same if ${d^k}_{ij} = -{\Gamma^k}_{ij} - \frac{n}{2} V^k \gamma_{ij}$
$V_k = {d_{km}}^m - {d^m}_{mk} - Z_k$
$n_i = $ the direction that the flux is sampled in. All eigenvectors below are provided in the x direction, so $n_i = [1,0,0]$
$Z_u = $ ... what is this initialized to? There are plenty of references that $Z_i = 0$ recovers ADM, and that $Z_a$ is the zero vector. Should this be equal to $t_{,t}$?
eigenvalues and eigenfields:
17 time waves: $\lambda = -\beta^x$ for $w = a_q, d_{qij}, a_k - f {d_{km}}^m + \lambda V_k$
10 light waves: $\lambda = -\beta^x \pm \alpha \sqrt{\gamma^{xx}}$ for $w_\pm = (K_{ij} - n_i n_j K) \pm ({\lambda^x}_{ij} - n_i n_j {\lambda^{xk}}_k)$
I see 12 light waves. Are there only supposed to be 2x5 light waves based on $K_{ij}$?
How is this 10? Is this the same as $w_\pm = K_{iq} \pm {\lambda^x}_{iq}$ described for ADM?
Is the $ij$ = $xx$ term the excluded eigenfield, as it is in the ADM case?
2 energy waves: $\lambda = -\beta^x \pm \alpha \sqrt{\gamma^{xx}}$ for $w_\pm = \Theta \pm V^x$
2 gauge waves: $\lambda = -\beta^x \pm \alpha \sqrt{f \gamma^{xx}}$ for $w_\pm = \sqrt{f} (K + \frac{2 - \lambda}{f - 1} \Theta) \pm (a^x + \frac{2 f - \lambda}{f - 1} V^x)$

Bona, Palenzuela, "Dynamical shift conditions for the Z4 and BSSN hyperbolic formalisms" 2004. https://arxiv.org/abs/gr-qc/0401019

Alic, Bona-Casas, Bona, Rezzolla, Palenzuela. "Conformal and covariant formulation of the Z4 system with constraint-violation damping". 2012. https://arxiv.org/abs/1106.2254

Bona et al. "Geometrically motivated hyperbolic coordinate condions for numerical relativity- Analysis, issues and implementation". 2005.
see my numerical-relativity-codegen/show_flux_matrix - code and results.
But for consistency's sake:
(taken from the 2008 Yano et al paper that references it)
source terms:
$S({b_k}^i) = {b_k}^l {b_l}^i - {b_l}^l {b_k}^i$
$S(d_{kij}) = {b_k}^l d_{lij} - {b_l}^l d_{kij}$
$S(K_{ij}) = = ...$
$S(\Theta) = = ...$
$S(Z_i) = ...$

Yano, Suzuki, Kuroda "Flux-Vector-Splitting (FVS) method for Z4 formalism and its numerical analysis" 2008
This is almost the same as its main reference, 2005 Bona et al "Geometrically..." except that the Bona paper implementation uses a densitized flux so that the eigenvalues are just $\alpha$, while this paper doesn't, so its eigenvalues are $\alpha \sqrt{\gamma^{xx}}$

Bona, Palenzuela-Luque, Bona-Casas, "Elements of Numerical Relativity and Relativistic Hydrodynamics". 2009. Section 4.3.3 "First-order Z4 formalism"
parameters:
$m = 2$ (second gauge parameter)
$\zeta = $ ...
$\zeta' = $ ...
$\eta_\xi = $ constraint preserving parameters (something small)
hyperbolic variables:
$a_i = (ln \alpha)_{,i}$ (eqn 4.1)
${b_i}^j = {\beta^j}_{,i}$ (eqn 4.1). Note no $\frac{1}{2}$ which I only see in earlier papers
$d_{kij} = \frac{1}{2} \gamma_{ij,k}$ (eqn 4.1)
$K_{ij} = $ extrinsic curvature
$\Theta = $ perpendicular to $n^a$
$Z_i = $ perpendicular to $n^a$
other variables:
$d_i = {d_{ik}}^k$ (eqn 4.33)
$e_i = {d^k}_{ki}$ (eqn 4.33)
$V_i = d_i - e_i - Z_i$ (eqn 4.56)
$Q = f (K - m \Theta)$ (eqn 4.4), for $m=2$ this becomes $f (K - 2 \Theta)$ (eqn 4.126)
$Q_i = \alpha (a_i - d_i + 2 V_i)$ (eqn 4.128)
$Q_{ij} = K_{ij} - \frac{1}{2 \alpha} (b_{ij} + b_{ji})$ (eqn 4.92)
${\lambda^k}_{ij} = {d^k}_{ij} + \frac{1}{2} \delta^k_i (a_j - d_j + 2 V_j) + \frac{1}{2} \delta^k_j (a_i - d_i - 2 V_i) - \frac{1 + \zeta}{2} ({d_{ij}}^k + {d_{ji}}^k - \delta^k_i e_j - \delta^k_j e_i)$ (eqn 4.121)
flux terms:
$F^l(a_k) = ...$ maybe $= -\beta^l a_k + \delta^l_k \alpha Q$ by eqn 4.95
$F^l({b_k}^i) = ...$ maybe $= -\beta^l {b_k}^i + \delta^l_k \alpha Q^i$ by eqn 4.96
$F^l(d_{kij} = ...$ maybe $= -\beta^l d_{kij} + \delta^l_k \alpha Q_{ij}$ by eqn 4.97
$F^k(K_{ij}) = -\beta^k K_{ij} + \alpha {\lambda^k}_{ij}$ (eqn 4.118)
$F^k(V_i) = ...$
$F^k(\Theta) = -\beta^k \Theta + \alpha V^k$ (eqn 4.119)
$F^k(Z_i) = -\beta^k Z_i + \alpha (\delta^k_i (K - \Theta) - {K^k}_i) + \zeta' ({b_i}^k - \delta^k_i b)$ (eqn 4.120)
$F^k(Q) = ...$
$F^k(Q_i) = ...$
source terms:
$S(\alpha) = \alpha \beta^k a_k - \alpha^2 Q $ (eqn 4.88) $= \alpha \beta^k a_k - \alpha^2 f (K - 2 \Theta)$
$S(\beta^i) = \beta^k {b_k}^i - \alpha Q^i$ (eqn 4.88)
$S(\gamma_{ij}) = 2 \beta^k d_{kij} + b_{ji} + b_{ij} - 2 \alpha K_{ij}$ (eqn 4.86)
$S(a_i) = ...$ maybe $= {b_i}^k a_k - {b_k}^k a_i - \eta_a (a_i - (ln \alpha)_{,i})$ by eqn 4.95, 4.101, 4.7
$S({b_k}^i) = ...$ maybe $= {b_k}^l {b_l}^i - {b_l}^l {b_k}^i - \eta_b ( {b_k}^i - {\beta^i}_{,k} )$ by eqn 4.96, 4.101, 4.83
$S(d_{kij}) = ...$ maybe $= {b_k}^l d_{lij} - {b_l}^l d_{kij} - \eta_d (d_{kij} - \frac{1}{2} \gamma_{ij,k})$ by eqn 4.97, 4.101, 4.8
$S(K_{ij}) = -K_{ij} {b_k}^k + K_{ik} {b_j}^k + K_{jk} {b_i}^k + \alpha ( \frac{1}{2} (1 + \zeta) ( - a_k {\Gamma^k}_{ij} + \frac{1}{2} (a_i d_j + a_j d_i)) + \frac{1}{2} (1 - \zeta) (a_k {d^k}_{ij} - \frac{1}{2} (a_j (2 e_i - d_i) + a_i (2 e_j - d_j)) + 2 ({d_{ir}}^m {d^r}_{mj} + {d_{jr}}^m {d^r}_{mi}) - 2 e_k ({d_{ij}}^k + {d_{ji}}^k) ) + (d_k + a_k - 2 Z_k) {\Gamma^k}_{ij} - {\Gamma^k}_{mj} {\Gamma^m}_{ki} - (a_i Z_j + a_j Z_i) - 2 {K^k}_i K_{kj} + (K - 2 \Theta) K_{ij} ) - 8 \pi \alpha (S_{ij} - \frac{1}{2} (S - \tau) \gamma_{ij})$ (eqn 4.130)
$S(Z_i) = -Z_i {b_k}^k + Z_k {b_i}^k - 8 \pi \alpha S_i + \alpha (a_i (K - 2 \Theta) - a_k {K^k}_i - {K^k}_r {\Gamma^r}_{ki} + {K^k}_i (d_k - 2 Z_k))$ (eqn 4.131)
$S(\Theta) = -\Theta {b_k}^k + \frac{1}{2} \alpha (2 a_k (d^k - e^k - 2 Z^k) + {d_k}^{rs} {\Gamma^k}_{rs} - d^k (d_k - 2 Z_k) - {K^k}_r {K^r}_k + K (K - 2 \Theta)) - 8 \pi \alpha \tau$ (eqn 4.132)
eigenvalues and eigenfields:
transverse waves (2+6+2*6=20): $\lambda = -\beta^x$ for $w = a_p, {b_p}^i, d_{pij}$ for $p \ne x$ (between eqns 4.122 and 4.123)
light-cone waves (3+2+1=6): $\lambda = -\beta^x \pm \alpha$ for $w = F^x(d_{xpq}) \pm F^x(K_{pq}), -F^x(Z_p) \pm F^x(K_{xp}), F^x(V_x) \pm F^x(\Theta)$ (eqns 4.123, 4.124, 4.125)
lapse-related waves (1): $\lambda = -\beta^x \pm \sqrt{f} \alpha$ for $w = F^x(a_x) \pm \sqrt{f} F^x(Q)$ (eqn 4.127)
shift-related waves (3): $\lambda = -\beta^x \pm \alpha$ for $w = F^x(b_{xi}) \pm F^x(Q_i)$ (eqn 4.129)
questions that remain:
What are the flux terms $F^k(Q)$, $F^k(Q_i)$, and $F^k(V_i)$ especially when these aren't evolved variables...
If it's like the Bona-Masso ADM formalism shown in the 2008 Alcubierre book then $F^k(V_i) = -\beta^k V_i$
What is the flux term $F^k(b_{ij})$ compared to the given (inferred, and many pages from its usage) $F^k({b_i}^j)$? Just extract the metric terms and subtract? Or was that a typo -- should it be $F^k({b_i}^k)$ above?
This formalism is strangely lacking any $\gamma^{xx}$ terms in the eigenvalues. Maybe the variables are scaled by the spatial volume $\gamma$?