def compute_smallbsquared(gammaDD, betaU, alpha, smallb4U):
global smallbsquared
import BSSN.ADMBSSN_tofrom_4metric as AB4m
AB4m.g4DD_ito_BSSN_or_ADM("ADM", gammaDD, betaU, alpha)
smallbsquared = sp.sympify(0)
for mu in range(4):
for nu in range(4):
smallbsquared += AB4m.g4DD[mu][nu] * smallb4U[mu] * smallb4U[nu]
def compute_TEM4UD(gammaDD, betaU, alpha, TEM4UU):
global TEM4UD
# Next compute T^mu_nu = T^{mu delta} g_{delta nu}, needed for S_tilde flux.
# First we'll need g_{alpha nu} in terms of ADM quantities:
import BSSN.ADMBSSN_tofrom_4metric as AB4m
AB4m.g4DD_ito_BSSN_or_ADM("ADM", gammaDD, betaU, alpha)
TEM4UD = ixp.zerorank2(DIM=4)
for mu in range(4):
for nu in range(4):
for delta in range(4):
TEM4UD[mu][nu] += TEM4UU[mu][delta] * AB4m.g4DD[delta][nu]
def compute_smallb4U_with_driftvU_for_FFE(gammaDD,betaU,alpha, u4U,B_notildeU, sqrt4pi):
global smallb4_with_driftv_for_FFE_U
import BSSN.ADMBSSN_tofrom_4metric as AB4m
AB4m.g4DD_ito_BSSN_or_ADM("ADM",gammaDD,betaU,alpha)
u4D = ixp.zerorank1(DIM=4)
for mu in range(4):
for nu in range(4):
u4D[mu] += AB4m.g4DD[mu][nu]*u4U[nu]
smallb4_with_driftv_for_FFE_U = ixp.zerorank1(DIM=4)
# b^0 = 0
smallb4_with_driftv_for_FFE_U[0] = 0
# b^i = B^i / [alpha * u^0 * sqrt(4 pi)]
for i in range(3):
smallb4_with_driftv_for_FFE_U[i+1] = B_notildeU[i] / (alpha*u4U[0]*sqrt4pi)
def compute_smallb4U(gammaDD, betaU, alpha, u4U, B_notildeU, sqrt4pi):
global smallb4U
import BSSN.ADMBSSN_tofrom_4metric as AB4m
AB4m.g4DD_ito_BSSN_or_ADM("ADM", gammaDD, betaU, alpha)
u4D = ixp.zerorank1(DIM=4)
for mu in range(4):
for nu in range(4):
u4D[mu] += AB4m.g4DD[mu][nu] * u4U[nu]
smallb4U = ixp.zerorank1(DIM=4)
u4_dot_B_notilde = sp.sympify(0)
for i in range(3):
u4_dot_B_notilde += u4D[i + 1] * B_notildeU[i]
# b^0 = (u_j B^j)/[alpha * sqrt(4 pi)]
smallb4U[0] = u4_dot_B_notilde / (alpha * sqrt4pi)
# b^i = [B^i + (u_j B^j)]/[alpha * u^0 * sqrt(4 pi)]
for i in range(3):
smallb4U[i + 1] = (B_notildeU[i] + u4_dot_B_notilde * u4U[i + 1]) / (alpha * u4U[0] * sqrt4pi)
def GiRaFFE_Higher_Order():
#Step 1.0: Set the spatial dimension parameter to 3.
par.set_parval_from_str("grid::DIM", 3)
DIM = par.parval_from_str("grid::DIM")
# Step 1.1: Set the finite differencing order to 4.
#par.set_parval_from_str("finite_difference::FD_CENTDERIVS_ORDER", 4)
thismodule = "GiRaFFE_NRPy"
# M_PI will allow the C code to substitute the correct value
M_PI = par.Cparameters("#define", thismodule, "M_PI", "")
# ADMBase defines the 4-metric in terms of the 3+1 spacetime metric quantities gamma_{ij}, beta^i, and alpha
gammaDD = ixp.register_gridfunctions_for_single_rank2("AUX",
"gammaDD",
"sym01",
DIM=3)
betaU = ixp.register_gridfunctions_for_single_rank1("AUX", "betaU", DIM=3)
alpha = gri.register_gridfunctions("AUX", "alpha")
# GiRaFFE uses the Valencia 3-velocity and A_i, which are defined in the initial data module(GiRaFFEfood)
ValenciavU = ixp.register_gridfunctions_for_single_rank1("AUX",
"ValenciavU",
DIM=3)
AD = ixp.register_gridfunctions_for_single_rank1("EVOL", "AD", DIM=3)
# B^i must be computed at each timestep within GiRaFFE so that the Valencia 3-velocity can be evaluated
BU = ixp.register_gridfunctions_for_single_rank1("AUX", "BU", DIM=3)
# <a id='step3'></a>
#
# ## Step 1.2: Build the four metric $g_{\mu\nu}$, its inverse $g^{\mu\nu}$ and spatial derivatives $g_{\mu\nu,i}$ from ADM 3+1 quantities $\gamma_{ij}$, $\beta^i$, and $\alpha$
#
# $$\label{step3}$$
# \[Back to [top](#top)\]
#
# Notice that the time evolution equation for $\tilde{S}_i$
# $$
# \partial_t \tilde{S}_i = - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}
# $$
# contains $\partial_i g_{\mu \nu} = g_{\mu\nu,i}$. We will now focus on evaluating this term.
#
# The four-metric $g_{\mu\nu}$ is related to the three-metric $\gamma_{ij}$, index-lowered shift $\beta_i$, and lapse $\alpha$ by
# $$
# g_{\mu\nu} = \begin{pmatrix}
# -\alpha^2 + \beta^k \beta_k & \beta_j \\
# \beta_i & \gamma_{ij}
# \end{pmatrix}.
# $$
# This tensor and its inverse have already been built by the u0_smallb_Poynting__Cartesian.py module ([documented here](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)), so we can simply load the module and import the variables.
# Step 1.2: import u0_smallb_Poynting__Cartesian.py to set
# the four metric and its inverse. This module also sets b^2 and u^0.
import u0_smallb_Poynting__Cartesian.u0_smallb_Poynting__Cartesian as u0b
u0b.compute_u0_smallb_Poynting__Cartesian(gammaDD, betaU, alpha,
ValenciavU, BU)
betaD = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
betaD[i] += gammaDD[i][j] * betaU[j]
# We will now pull in the four metric and its inverse.
import BSSN.ADMBSSN_tofrom_4metric as AB4m # NRPy+: ADM/BSSN <-> 4-metric conversions
AB4m.g4DD_ito_BSSN_or_ADM("ADM")
g4DD = AB4m.g4DD
AB4m.g4UU_ito_BSSN_or_ADM("ADM")
g4UU = AB4m.g4UU
# Next we compute spatial derivatives of the metric, $\partial_i g_{\mu\nu} = g_{\mu\nu,i}$, written in terms of the three-metric, shift, and lapse. Simply taking the derivative of the expression for $g_{\mu\nu}$ above, we find
# $$
# g_{\mu\nu,l} = \begin{pmatrix}
# -2\alpha \alpha_{,l} + \beta^k_{\ ,l} \beta_k + \beta^k \beta_{k,l} & \beta_{i,l} \\
# \beta_{j,l} & \gamma_{ij,l}
# \end{pmatrix}.
# $$
#
# Notice the derivatives of the shift vector with its indexed lowered, $\beta_{i,j} = \partial_j \beta_i$. This can be easily computed in terms of the given ADMBase quantities $\beta^i$ and $\gamma_{ij}$ via:
# \begin{align}
# \beta_{i,j} &= \partial_j \beta_i \\
# &= \partial_j (\gamma_{ik} \beta^k) \\
# &= \gamma_{ik} \partial_j\beta^k + \beta^k \partial_j \gamma_{ik} \\
# \beta_{i,j} &= \gamma_{ik} \beta^k_{\ ,j} + \beta^k \gamma_{ik,j}.
# \end{align}
#
# Since this expression mixes Greek and Latin indices, we will need to store the expressions for each of the three spatial derivatives as separate variables.
#
# So, we will first set
# $$ g_{00,l} = \underbrace{-2\alpha \alpha_{,l}}_{\rm Term\ 1} + \underbrace{\beta^k_{\ ,l} \beta_k}_{\rm Term\ 2} + \underbrace{\beta^k \beta_{k,l}}_{\rm Term\ 3} $$
# Step 1.2, cont'd: Build spatial derivatives of the four metric
# Step 1.2.a: Declare derivatives of grid functions. These will be handled by FD_outputC
alpha_dD = ixp.declarerank1("alpha_dD")
betaU_dD = ixp.declarerank2("betaU_dD", "nosym")
gammaDD_dD = ixp.declarerank3("gammaDD_dD", "sym01")
# Step 1.2.b: These derivatives will be constructed analytically.
betaDdD = ixp.zerorank2()
g4DDdD = ixp.zerorank3(DIM=4)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
# \gamma_{ik} \beta^k_{,j} + \beta^k \gamma_{ik,j}
betaDdD[i][j] += gammaDD[i][k] * betaU_dD[k][j] + betaU[
k] * gammaDD_dD[i][k][j]
# Step 1.2.c: Set the 00 components
# Step 1.2.c.i: Term 1: -2\alpha \alpha_{,l}
for l in range(DIM):
g4DDdD[0][0][l + 1] = -2 * alpha * alpha_dD[l]
# Step 1.2.c.ii: Term 2: \beta^k_{\ ,l} \beta_k
for l in range(DIM):
for k in range(DIM):
g4DDdD[0][0][l + 1] += betaU_dD[k][l] * betaD[k]
# Step 1.2.c.iii: Term 3: \beta^k \beta_{k,l}
for l in range(DIM):
for k in range(DIM):
g4DDdD[0][0][l + 1] += betaU[k] * betaDdD[k][l]
# Now we will contruct the other components of $g_{\mu\nu,l}$. We will first construct
# $$ g_{i0,l} = g_{0i,l} = \beta_{i,l}, $$
# then
# $$ g_{ij,l} = \gamma_{ij,l} $$
# Step 1.2.d: Set the i0 and 0j components
for l in range(DIM):
for i in range(DIM):
# \beta_{i,l}
g4DDdD[i + 1][0][l + 1] = g4DDdD[0][i + 1][l + 1] = betaDdD[i][l]
#Step 1.2.e: Set the ij components
for l in range(DIM):
for i in range(DIM):
for j in range(DIM):
# \gamma_{ij,l}
g4DDdD[i + 1][j + 1][l + 1] = gammaDD_dD[i][j][l]
# <a id='step4'></a>
#
# # $T^{\mu\nu}_{\rm EM}$ and its derivatives
#
# Now that the metric and its derivatives are out of the way, let's return to the evolution equation for $\tilde{S}_i$,
# $$
# \partial_t \tilde{S}_i = - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}.
# $$
# We turn our focus now to $T^j_{{\rm EM} i}$ and its derivatives. To this end, we start by computing $T^{\mu \nu}_{\rm EM}$ (from eq. 27 of [Paschalidis & Shapiro's paper on their GRFFE code](https://arxiv.org/pdf/1310.3274.pdf)):
#
# $$\boxed{T^{\mu \nu}_{\rm EM} = b^2 u^{\mu} u^{\nu} + \frac{b^2}{2} g^{\mu \nu} - b^{\mu} b^{\nu}.}$$
#
# Notice that $T^{\mu\nu}_{\rm EM}$ is written in terms of
#
# * $b^\mu$, the 4-component magnetic field vector, related to the comoving magnetic field vector $B^i_{(u)}$
# * $u^\mu$, the 4-velocity
# * $g^{\mu \nu}$, the inverse 4-metric
#
# However, $\texttt{GiRaFFE}$ has access to only the following quantities, requiring in the following sections that we write the above quantities in terms of the following ones:
#
# * $\gamma_{ij}$, the 3-metric
# * $\alpha$, the lapse
# * $\beta^i$, the shift
# * $A_i$, the vector potential
# * $B^i$, the magnetic field (we assume only in the grid interior, not the ghost zones)
# * $\left[\sqrt{\gamma}\Phi\right]$, the zero-component of the vector potential $A_\mu$, times the square root of the determinant of the 3-metric
# * $v_{(n)}^i$, the Valencia 3-velocity
# * $u^0$, the zero-component of the 4-velocity
#
# ## Step 2.0: $u^i$ and $b^i$ and related quantities
# $$\label{step4}$$
# \[Back to [top](#top)\]
#
# We begin by importing what we can from u0_smallb_Poynting__Cartesian.py. We will need the four-velocity $u^\mu$, which is related to the Valencia 3-velocity $v^i_{(n)}$ used directly by $\texttt{GiRaFFE}$ (see also [Duez, et al, eqs. 53 and 56](https://arxiv.org/pdf/astro-ph/0503420.pdf))
# \begin{align}
# u^i &= u^0 (\alpha v^i_{(n)} - \beta^i), \\
# u_j &= \alpha u^0 \gamma_{ij} v^i_{(n)},
# \end{align}
# and $v^i_{(n)}$ is the Valencia three-velocity. These have already been constructed in terms of the Valencia 3-velocity and other 3+1 ADM quantities by the u0_smallb_Poynting__Cartesian.py module, so we can simply import these variables:
# Step 2.0: u^i, b^i, and related quantities
# Step 2.0.a: import the four-velocity, as written in terms of the Valencia 3-velocity
global uD, uU
uD = ixp.register_gridfunctions_for_single_rank1("AUX", "uD")
uU = ixp.register_gridfunctions_for_single_rank1("AUX", "uU")
u4upperZero = gri.register_gridfunctions("AUX", "u4upperZero")
for i in range(DIM):
uD[i] = u0b.uD[i].subs(u0b.u0, u4upperZero)
uU[i] = u0b.uU[i].subs(u0b.u0, u4upperZero)
# We also need the magnetic field 4-vector $b^{\mu}$, which is related to the magnetic field by [eqs. 23, 24, and 31 in Duez, et al](https://arxiv.org/pdf/astro-ph/0503420.pdf):
# \begin{align}
# b^0 &= \frac{1}{\sqrt{4\pi}} B^0_{\rm (u)} = \frac{u_j B^j}{\sqrt{4\pi}\alpha}, \\
# b^i &= \frac{1}{\sqrt{4\pi}} B^i_{\rm (u)} = \frac{B^i + (u_j B^j) u^i}{\sqrt{4\pi}\alpha u^0}, \\
# \end{align}
# where $B^i$ is the variable tracked by the HydroBase thorn in the Einstein Toolkit. Again, these have already been built by the u0_smallb_Poynting__Cartesian.py module, so we can simply import the variables.
# Step 2.0.b: import the small b terms
smallb4U = ixp.zerorank1(DIM=4)
smallb4D = ixp.zerorank1(DIM=4)
for mu in range(4):
smallb4U[mu] = u0b.smallb4U[mu].subs(u0b.u0, u4upperZero)
smallb4D[mu] = u0b.smallb4D[mu].subs(u0b.u0, u4upperZero)
smallb2 = u0b.smallb2etk.subs(u0b.u0, u4upperZero)
# <a id='step5'></a>
#
# ## Step 2.1: Construct all components of the electromagnetic stress-energy tensor $T^{\mu \nu}_{\rm EM}$
# $$\label{step5}$$
#
# \[Back to [top](#top)\]
#
# We now have all the pieces to calculate the stress-energy tensor,
# $$T^{\mu \nu}_{\rm EM} = \underbrace{b^2 u^{\mu} u^{\nu}}_{\rm Term\ 1} +
# \underbrace{\frac{b^2}{2} g^{\mu \nu}}_{\rm Term\ 2}
# - \underbrace{b^{\mu} b^{\nu}}_{\rm Term\ 3}.$$
# Because $u^0$ is a separate variable, we could build the $00$ component separately, then the $\mu0$ and $0\nu$ components, and finally the $\mu\nu$ components. Alternatively, for clarity, we could create a temporary variable $u^\mu=\left( u^0, u^i \right)$
# Step 2.1: Construct the electromagnetic stress-energy tensor
# Step 2.1.a: Set up the four-velocity vector
u4U = ixp.zerorank1(DIM=4)
u4U[0] = u4upperZero
for i in range(DIM):
u4U[i + 1] = uU[i]
# Step 2.1.b: Build T4EMUU itself
T4EMUU = ixp.zerorank2(DIM=4)
for mu in range(4):
for nu in range(4):
# Term 1: b^2 u^{\mu} u^{\nu}
T4EMUU[mu][nu] = smallb2 * u4U[mu] * u4U[nu]
for mu in range(4):
for nu in range(4):
# Term 2: b^2 / 2 g^{\mu \nu}
T4EMUU[mu][nu] += smallb2 * g4UU[mu][nu] / 2
for mu in range(4):
for nu in range(4):
# Term 3: -b^{\mu} b^{\nu}
T4EMUU[mu][nu] += -smallb4U[mu] * smallb4U[nu]
# <a id='step6'></a>
#
# # Step 2.2: Derivatives of the electromagnetic stress-energy tensor
# $$\label{step6}$$
#
# \[Back to [top](#top)\]
#
# If we look at the evolution equation, we see that we will need spatial derivatives of $T^{\mu\nu}_{\rm EM}$. When confronted with derivatives of complicated expressions, it is generally convenient to declare those expressions as gridfunctions themselves, allowing NRPy+ to take finite-difference derivatives of the expressions. This can even reduce the truncation error associated with the finite differences, because the alternative is to use a function of several finite-difference derivatives, allowing more error to accumulate than the extra gridfunction will introduce. While we will use that technique for some of the subexpressions of $T^{\mu\nu}_{\rm EM}$, we don't want to rely on it for the whole expression; doing so would require us to take the derivative of the magnetic field $B^i$, which is itself found by finite-differencing the vector potential $A_i$. Thus $B^i$ cannot be *consistently* defined in ghost zones. To potentially reduce numerical errors induced by inconsistent finite differencing, we will differentiate $T^{\mu\nu}_{\rm EM}$ term-by-term so that finite-difference derivatives of $A_i$ appear.
#
# We will now now take these spatial derivatives of $T^{\mu\nu}_{\rm EM}$, applying the chain rule until it is only in terms of basic gridfunctions and their derivatives: $\alpha$, $\beta^i$, $\gamma_{ij}$, $A_i$, and the four-velocity $u^i$. Along the way, we will also set up useful temporary variables representing the steps of the chain rule. (Notably, *all* of these quantities will be written in terms of $A_i$ and its derivatives):
#
# * $B^i$ (already computed in terms of $A_k$, via $B^i = \epsilon^{ijk} \partial_j A_k$),
# * $B^i_{,l}$,
# * $b^i$ and $b_i$ (already computed),
# * $b^i_{,k}$,
# * $b^2$ (already computed),
# * and $\left(b^2\right)_{,j}$.
#
# (The variables not already computed will not be seen by the ETK, as they are written in terms of $A_i$ and its derivatives; they simply help to organize the NRPy+ code.)
#
# So then,
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} &= \partial_j (g_{\mu i} T^{\mu j}_{\rm EM}) \\
# &= \partial_j \left[g_{\mu i} \left(b^2 u^j u^\mu + \frac{b^2}{2} g^{j\mu} - b^j b^\mu\right)\right] \\
# &= \underbrace{g_{\mu i,j} T^{\mu j}_{\rm EM}}_{\rm Term\ A} + g_{\mu i} \left( \underbrace{\partial_j \left(b^2 u^j u^\mu \right)}_{\rm Term\ B} + \underbrace{\partial_j \left(\frac{b^2}{2} g^{j\mu}\right)}_{\rm Term\ C} - \underbrace{\partial_j \left(b^j b^k\right)}_{\rm Term\ D} \right) \\
# \end{align}
# Following the product and chain rules for each term, we find that
# \begin{align}
# {\rm Term\ B} &= \partial_j (b^2 u^j u^\mu) \\
# &= \partial_j b^2 u^j u^\mu + b^2 \partial_j u^j u^\mu + b^2 u^j \partial_j u^\mu \\
# &= \underbrace{\left(b^2\right)_{,j} u^j u^\mu + b^2 u^j_{,j} u^\mu + b^2 u^j u^{\mu}_{,j}}_{\rm To\ Term\ 2\ below} \\
# {\rm Term\ C} &= \partial_j \left(\frac{b^2}{2} g^{j\mu}\right) \\
# &= \frac{1}{2} \left( \partial_j b^2 g^{j\mu} + b^2 \partial_j g^{j\mu} \right) \\
# &= \underbrace{\frac{1}{2} \left(b^2\right)_{,j} g^{j\mu} + \frac{b^2}{2} g^{j\mu}_{\ ,j}}_{\rm To\ Term\ 3\ below} \\
# {\rm Term\ D} &= \partial_j (b^j b^\mu) \\
# &= \underbrace{b^j_{,j} b^\mu + b^j b^\mu_{,j}}_{\rm To\ Term\ 2\ below}\\
# \end{align}
#
# So,
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} &= g_{\mu i,j} T^{\mu j}_{\rm EM} \\
# &+ g_{\mu i} \left(\left(b^2\right)_{,j} u^j u^\mu +b^2 u^j_{,j} u^\mu + b^2 u^j u^{\mu}_{,j} + \frac{1}{2}\left(b^2\right)_{,j} g^{j\mu} + \frac{b^2}{2} g^{j\mu}_{\ ,j} + b^j_{,j} b^\mu + b^j b^\mu_{,j}\right);
# \end{align}
# We will rearrange this once more, collecting the $b^2$ terms together, noting that Term A will become Term 1:
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} =& \
# \underbrace{g_{\mu i,j} T^{\mu j}_{\rm EM}}_{\rm Term\ 1} \\
# & + \underbrace{g_{\mu i} \left( b^2 u^j_{,j} u^\mu + b^2 u^j u^\mu_{,j} + \frac{b^2}{2} g^{j\mu}_{\ ,j} + b^j_{,j} b^\mu + b^j b^\mu_{,j} \right)}_{\rm Term\ 2} \\
# & + \underbrace{g_{\mu i} \left( \left(b^2\right)_{,j} u^j u^\mu + \frac{1}{2} \left(b^2\right)_{,j} g^{j\mu} \right).}_{\rm Term\ 3} \\
# \end{align}
#
# <a id='table2'></a>
#
# **List of Derivatives**
# $$\label{table2}$$
#
# Note that this is in terms of the derivatives of several other quantities:
#
# * [Step 2.2.a](#capitalBideriv): $B^i_{,l}$: Since $b^i$ is itself a function of $B^i$, we will first need the derivatives $B^i_{,l}$ in terms of the evolved quantity $A_i$ (the vector potential).
# * [Step 2.2.b](#bideriv): $b^i_{,k}$: Once we have $B^i_{,l}$ we can evaluate derivatives of $b^i$, $b^i_{,k}$
# * [Step 2.2.c](#b2deriv): The derivative of $b^2 = g_{\mu\nu} b^\mu b^\nu$, $\left(b^2\right)_{,j}$
# * [Step 2.2.d](#gupijderiv): Derivatives of $g^{\mu\nu}$, $g^{\mu\nu}_{\ ,k}$
# * [Step 2.2.e](#alltogether): Putting it together: $\partial_j T^{j}_{{\rm EM} i}$
# * [Step 2.2.e.i](#alltogether1): Putting it together: Term 1
# * [Step 2.2.e.ii](#alltogether2): Putting it together: Term 2
# * [Step 2.2.e.iii](#alltogether3): Putting it together: Term 3
# <a id='capitalBideriv'></a>
#
# ## Step 2.2.a: Derivatives of $B^i$
#
# $$\label{capitalbideriv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# First, we will build the derivatives of the magnetic field. Since $b^i$ is a function of $B^i$, we will start from the definition of $B^i$ in terms of $A_i$, $B^i = \frac{[ijk]}{\sqrt{\gamma}} \partial_j A_k$. We will first apply the product rule, noting that the symbol $[ijk]$ consists purely of the integers $-1, 0, 1$ and thus can be treated as a constant in this process.
# \begin{align}
# B^i_{,l} &= \partial_l \left( \frac{[ijk]}{\sqrt{\gamma}} \partial_j A_k \right) \\
# &= [ijk] \partial_l \left( \frac{1}{\sqrt{\gamma}}\right) \partial_j A_k + \frac{[ijk]}{\sqrt{\gamma}} \partial_l \partial_j A_k \\
# &= [ijk]\left(-\frac{\gamma_{,l}}{2\gamma^{3/2}}\right) \partial_j A_k + \frac{[ijk]}{\sqrt{\gamma}} \partial_l \partial_j A_k \\
# \end{align}
# Now, we will substitute back in for the definition of the Levi-Civita tensor: $\epsilon^{ijk} = [ijk] / \sqrt{\gamma}$. Then we will substitute the magnetic field $B^i$ back in.
# \begin{align}
# B^i_{,l} &= -\frac{\gamma_{,l}}{2\gamma} \epsilon^{ijk} \partial_j A_k + \epsilon^{ijk} \partial_l \partial_j A_k \\
# &= -\frac{\gamma_{,l}}{2\gamma} B^i + \epsilon^{ijk} A_{k,jl}, \\
# \end{align}
#
# Thus, the expression we are left with for the derivatives of the magnetic field is:
# \begin{align}
# B^i_{,l} &= \underbrace{-\frac{\gamma_{,l}}{2\gamma} B^i}_{\rm Term\ 1} + \underbrace{\epsilon^{ijk} A_{k,jl}}_{\rm Term\ 2}, \\
# \end{align}
# where $\epsilon^{ijk} = [ijk] / \sqrt{\gamma}$ is the antisymmetric Levi-Civita tensor and $\gamma$ is the determinant of the three-metric.
#
# Step 2.2: Derivatives of the electromagnetic stress-energy tensor
# We already have a handy function to define the Levi-Civita symbol in WeylScalars
import WeylScal4NRPy.WeylScalars_Cartesian as weyl
# Initialize the Levi-Civita tensor by setting it equal to the Levi-Civita symbol
LeviCivitaSymbolDDD = weyl.define_LeviCivitaSymbol_rank3()
LeviCivitaTensorDDD = ixp.zerorank3()
LeviCivitaTensorUUU = ixp.zerorank3()
global gammaUU, gammadet
gammaUU = ixp.register_gridfunctions_for_single_rank2(
"AUX", "gammaUU", "sym01")
gammadet = gri.register_gridfunctions("AUX", "gammadet")
gammaUU, gammadet = ixp.symm_matrix_inverter3x3(gammaDD)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
LeviCivitaTensorDDD[i][j][
k] = LeviCivitaSymbolDDD[i][j][k] * sp.sqrt(gammadet)
LeviCivitaTensorUUU[i][j][
k] = LeviCivitaSymbolDDD[i][j][k] / sp.sqrt(gammadet)
AD_dD = ixp.declarerank2("AD_dD", "nosym")
# Step 2.2.a: Construct the derivatives of the magnetic field.
gammadet_dD = ixp.declarerank1("gammadet_dD")
AD_dDD = ixp.declarerank3("AD_dDD", "sym12")
# The other partial derivatives of B^i
BUdD = ixp.zerorank2()
for i in range(DIM):
for l in range(DIM):
# Term 1: -\gamma_{,l} / (2\gamma) B^i
BUdD[i][l] = -gammadet_dD[l] * BU[i] / (2 * gammadet)
for i in range(DIM):
for l in range(DIM):
for j in range(DIM):
for k in range(DIM):
# Term 2: \epsilon^{ijk} A_{k,jl}
BUdD[i][
l] += LeviCivitaTensorUUU[i][j][k] * AD_dDD[k][j][l]
# <a id='bideriv'></a>
#
# ## Step 2.2.b: Derivatives of $b^i$
# $$\label{bideriv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# Now, we will code the derivatives of the spatial components of $b^{\mu}$, $b^i$:
# $$
# b^i_{,k} = \frac{1}{\sqrt{4 \pi}} \frac{\left(\alpha u^0\right) \left(B^i_{,k} + u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k}\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\alpha u^0\right)}{\left(\alpha u^0\right)^2}.
# $$
#
# We should note that while $b^\mu$ is a four-vector (and the code reflects this: $\text{smallb4U}$ and $\text{smallb4U}$ have $\text{DIM=4}$), we only need the spatial components. We will only focus on the spatial components for the moment.
#
#
# Let's go into a little more detail on where this comes from. We start from the definition $$b^i = \frac{B^i + (u_j B^j) u^i}{\sqrt{4\pi}\alpha u^0};$$ we then apply the quotient rule:
# \begin{align}
# b^i_{,k} &= \frac{\left(\sqrt{4\pi}\alpha u^0\right) \partial_k \left(B^i + (u_j B^j) u^i\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\sqrt{4\pi}\alpha u^0\right)}{\left(\sqrt{4\pi}\alpha u^0\right)^2} \\
# &= \frac{1}{\sqrt{4 \pi}} \frac{\left(\alpha u^0\right) \partial_k \left(B^i + (u_j B^j) u^i\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\alpha u^0\right)}{\left(\alpha u^0\right)^2} \\
# \end{align}
# Note that $\left( \alpha u^0 \right)$ is being used as its own gridfunction, so $\partial_k \left(a u^0\right)$ will be finite-differenced by NRPy+ directly. We will also apply the product rule to the term $\partial_k \left(B^i + (u_j B^j) u^i\right) = B^i_{,k} + u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k}$. So,
# $$ b^i_{,k} = \frac{1}{\sqrt{4 \pi}} \frac{\left(\alpha u^0\right) \left(B^i_{,k} + u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k}\right) - \left(B^i + (u_j B^j) u^i\right) \partial_k \left(\alpha u^0\right)}{\left(\alpha u^0\right)^2}. $$
#
# It will be easier to code this up if we rearrange these terms to group together the terms that involve contractions over $j$. Doing that, we find
# $$
# b^i_{,k} = \frac{\overbrace{\alpha u^0 B^i_{,k} - B^i \partial_k (\alpha u^0)}^{\rm Term\ Num1} + \overbrace{\left( \alpha u^0 \right) \left( u_{j,k} B^j u^i + u_j B^j_{,k} u^i + u_j B^j u^i_{,k} \right)}^{\rm Term\ Num2.a} - \overbrace{\left( u_j B^j u^i \right) \partial_k \left( \alpha u^0 \right) }^{\rm Term\ Num2.b}}{\underbrace{\sqrt{4 \pi} \left( \alpha u^0 \right)^2}_{\rm Term\ Denom}}.
# $$
global u0alpha
u0alpha = gri.register_gridfunctions("AUX", "u0alpha")
u0alpha = alpha * u4upperZero
u0alpha_dD = ixp.declarerank1("u0alpha_dD")
uU_dD = ixp.declarerank2("uU_dD", "nosym")
uD_dD = ixp.declarerank2("uD_dD", "nosym")
# Step 2.2.b: Construct derivatives of the small b vector
# smallbUdD represents the derivative of smallb4U
smallbUdD = ixp.zerorank2()
for i in range(DIM):
for k in range(DIM):
# Term Num1: \alpha u^0 B^i_{,k} - B^i \partial_k (\alpha u^0)
smallbUdD[i][k] += u0alpha * BUdD[i][k] - BU[i] * u0alpha_dD[k]
for i in range(DIM):
for k in range(DIM):
for j in range(DIM):
# Term Num2.a: terms that require contractions over k, and thus an extra loop.
# ( \alpha u^0 ) ( u_{j,k} B^j u^i
# + u_j B^j_{,k} u^i
# + u_j B^j u^i_{,k} )
smallbUdD[i][k] += u0alpha * (uD_dD[j][k] * BU[j] * uU[i] +
uD[j] * BUdD[j][k] * uU[i] +
uD[j] * BU[j] * uU_dD[i][k])
for i in range(DIM):
for k in range(DIM):
for j in range(DIM):
#Term 2.b (More contractions over k): ( u_j B^j u^i ) ( \alpha u^0 ),k
smallbUdD[i][k] += -(uD[j] * BU[j] * uU[i]) * u0alpha_dD[k]
for i in range(DIM):
for k in range(DIM):
# Term Denom: Divide the numerator by sqrt(4 pi) * (alpha u^0)^2
smallbUdD[i][k] /= sp.sqrt(4 * M_PI) * u0alpha * u0alpha
# <a id='b2deriv'></a>
#
# ## Step 2.2.c: Derivative of $b^2$
# $$\label{b2deriv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# Here, we will take the derivative of $b^2 = g_{\mu\nu} b^\mu b^\nu$. Using the product rule,
# \begin{align}
# \left(b^2\right)_{,j} &= \partial_j \left( g_{\mu\nu} b^\mu b^\nu \right) \\
# &= g_{\mu\nu,j} b^\mu b^\nu + g_{\mu\nu} b^\mu_{,j} b^\nu + g_{\mu\nu} b^\mu b^\nu_{,j} \\
# &= g_{\mu\nu,j} b^\mu b^\nu + 2 g_{\mu\nu} b^\mu_{,j} b^\nu.
# \end{align}
# We have already defined the spatial derivatives of the four-metric $g_{\mu\nu,j}$ in [this section](#step3); we have also defined the spatial derivatives of spatial components of $b^\mu$, $b^i_{,k}$ in [this section](#bideriv). Notice the above expression requires spatial derivatives of the *zeroth* component of $b^\mu$ as well, $b^0_{,j}$, which we will now compute. Starting with the definition, and applying the quotient rule:
# \begin{align}
# b^0 &= \frac{u_k B^k}{\sqrt{4\pi}\alpha}, \\
# \rightarrow b^0_{,j} &= \frac{1}{\sqrt{4\pi}} \frac{\alpha \left( u_{k,j} B^k + u_k B^k_{,j} \right) - u_k B^k \alpha_{,j}}{\alpha^2} \\
# &= \frac{\alpha u_{k,j} B^k + \alpha u_k B^k_{,j} - \alpha_{,j} u_k B^k}{\sqrt{4\pi} \alpha^2}.
# \end{align}
# We will first code the numerator, and then divide through by the denominator.
# Step 2.2.c: Construct the derivative of b^2
# First construct the derivative b^0_{,j}
# This four-vector will make b^2 simpler:
smallb4UdD = ixp.zerorank2(DIM=4)
# Fill in the zeroth component
for j in range(DIM):
for k in range(DIM):
# The numerator: \alpha u_{k,j} B^k
# + \alpha u_k B^k_{,j}
# - \alpha_{,j} u_k B^k
smallb4UdD[0][j + 1] += alpha * uD_dD[k][j] * BU[k] + alpha * uD[
k] * BUdD[k][j] - alpha_dD[j] * uD[k] * BU[k]
for j in range(DIM):
# Divide through by the denominator: \sqrt{4\pi} \alpha^2
smallb4UdD[0][j + 1] /= sp.sqrt(4 * M_PI) * alpha * alpha
# At this point, both $b^0_{\ ,j}$ and $b^i_{\ ,j}$ have been computed, but one exists inconveniently in the $4\times 4$ component $\verb|smallb4UdD[][]|$ and the other in the $3\times 3$ component $\verb|smallbUdD[][]|$. So that we can perform full implied sums over $g_{\mu\nu} b^\mu_{,j} b^\nu$ more conveniently, we will now store all information from $\verb|smallbUdD[i][j]|$ into $\verb|smallb4UdD[i+1][j+1]|$:
# Now, we'll fill out the rest of the four-vector with b^i_{,j} that we derived above.
for i in range(DIM):
for j in range(DIM):
smallb4UdD[i + 1][j + 1] = smallbUdD[i][j]
# Using 4-component (Greek-indexed) quantities, we can now complete our construction of
# $$\left(b^2\right)_{,j} = g_{\mu\nu,j} b^\mu b^\nu + 2 g_{\mu\nu} b^\mu_{,j} b^\nu:$$
smallb2_dD = ixp.zerorank1()
for j in range(DIM):
for mu in range(4):
for nu in range(4):
# g_{\mu\nu,j} b^\mu b^\nu
# + 2 g_{\mu\nu} b^\mu_{,j} b^\nu
smallb2_dD[j] += g4DDdD[mu][nu][j + 1] * smallb4U[
mu] * smallb4U[nu] + 2 * g4DD[mu][nu] * smallb4UdD[mu][
j + 1] * smallb4U[nu]
# <a id='gupijderiv'></a>
#
# ## Step 2.2.d: Derivatives of $g^{\mu\nu}$
# $$\label{gupijderiv}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# We will need derivatives of the inverse four-metric, as well. Let us begin with $g^{00}$: since $g^{00} = -1/\alpha^2$ ([Gourgoulhon, eq. 4.49](https://arxiv.org/pdf/gr-qc/0703035.pdf)), $$g^{00}_{\ ,k} = \frac{2 \alpha_{,k}}{\alpha^3}$$
#
# Step 2.2.d: Construct derivatives of the components of g^{\mu\nu}
g4UUdD = ixp.zerorank3(DIM=4)
for k in range(DIM):
# 2 \alpha_{,k} / \alpha^3
g4UUdD[0][0][k + 1] = 2 * alpha_dD[k] / alpha**3
# Now, we will code the $g^{i0}_{\ ,k}$ and $g^{0j}_{\ ,k}$ components. According to [Gourgoulhon, eq. 4.49](https://arxiv.org/pdf/gr-qc/0703035.pdf), $g^{i0} = g^{0i} = \beta^i/\alpha^2$, so $$g^{i0}_{\ ,k} = g^{0i}_{\ ,k} = \frac{\alpha^2 \beta^i_{,k} - 2 \beta^i \alpha \alpha_{,k}}{\alpha^4}$$ by the quotient rule. So, we'll code
# $$
# g^{i0} = g^{0i} =
# \underbrace{\frac{\beta^i_{,k}}{\alpha^2}}_{\rm Term\ 1}
# - \underbrace{\frac{2 \beta^i \alpha_{,k}}{\alpha^3}}_{\rm Term\ 2}
# $$
for k in range(DIM):
for i in range(DIM):
# Term 1: \beta^i_{,k} / \alpha^2
g4UUdD[i + 1][0][k +
1] = g4UUdD[0][i +
1][k +
1] = betaU_dD[i][k] / alpha**2
for k in range(DIM):
for i in range(DIM):
# Term 2: -2 \beta^i \alpha_{,k} / \alpha^3
g4UUdD[i + 1][0][k + 1] += -2 * betaU[i] * alpha_dD[k] / alpha**3
g4UUdD[0][i + 1][k + 1] += -2 * betaU[i] * alpha_dD[k] / alpha**3
# We will also need derivatives of the spatial part of the inverse four-metric: since $g^{ij} = \gamma^{ij} - \frac{\beta^i \beta^j}{\alpha^2}$ ([Gourgoulhon, eq. 4.49](https://arxiv.org/pdf/gr-qc/0703035.pdf)),
# \begin{align}
# g^{ij}_{\ ,k} &= \gamma^{ij}_{\ ,k} - \frac{\alpha^2 \partial_k (\beta^i \beta^j) - \beta^i \beta^j \partial_k \alpha^2}{(\alpha^2)^2} \\
# &= \gamma^{ij}_{\ ,k} - \frac{\alpha^2\beta^i \beta^j_{,k}+\alpha^2\beta^i_{,k} \beta^j-2\beta^i \beta^j \alpha \alpha_{,k}}{\alpha^4}. \\
# &= \gamma^{ij}_{\ ,k} - \frac{\alpha\beta^i \beta^j_{,k}+\alpha\beta^i_{,k} \beta^j-2\beta^i \beta^j \alpha_{,k}}{\alpha^3} \\
# g^{ij}_{\ ,k} &= \underbrace{\gamma^{ij}_{\ ,k}}_{\rm Term\ 1} - \underbrace{\frac{\beta^i \beta^j_{,k}}{\alpha^2}}_{\rm Term\ 2} - \underbrace{\frac{\beta^i_{,k} \beta^j}{\alpha^2}}_{\rm Term\ 3} + \underbrace{\frac{2\beta^i \beta^j \alpha_{,k}}{\alpha^3}}_{\rm Term\ 4}. \\
# \end{align}
#
gammaUU_dD = ixp.declarerank3("gammaUU_dD", "sym01")
# The spatial derivatives of the spatial components of the four metric:
# Term 1: \gamma^{ij}_{\ ,k}
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j + 1][k + 1] = gammaUU_dD[i][j][k]
# Term 2: - \beta^i \beta^j_{,k} / \alpha^2
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j +
1][k +
1] += -betaU[i] * betaU_dD[j][k] / alpha**2
# Term 3: - \beta^i_{,k} \beta^j / \alpha^2
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j +
1][k +
1] += -betaU_dD[i][k] * betaU[j] / alpha**2
# Term 4: 2\beta^i \beta^j \alpha_{,k}\alpha^3
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
g4UUdD[i + 1][j + 1][
k + 1] += 2 * betaU[i] * betaU[j] * alpha_dD[k] / alpha**3
# <a id='alltogether'></a>
#
# ## Step 2.2.e: Putting it together:
# $$\label{alltogether}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# So, we can now put it all together, starting from the expression we derived above in [Step 2.2](#step6):
# \begin{align}
# \partial_j T^{j}_{{\rm EM} i} =& \
# \underbrace{g_{\mu i,j} T^{\mu j}_{\rm EM}}_{\rm Term\ 1} \\
# & + \underbrace{g_{\mu i} \left( b^2 u^j_{,j} u^\mu + b^2 u^j u^\mu_{,j} + \frac{b^2}{2} g^{j\mu}_{\ ,j} + b^j_{,j} b^\mu + b^j b^\mu_{,j} \right)}_{\rm Term\ 2} \\
# & + \underbrace{g_{\mu i} \left( \left(b^2\right)_{,j} u^j u^\mu + \frac{1}{2} \left(b^2\right)_{,j} g^{j\mu} \right).}_{\rm Term\ 3} \\
# \end{align}
#
# <a id='alltogether1'></a>
#
# ### Step 2.2.e.i: Putting it together: Term 1
# $$\label{alltogether1}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# We will now construct this term by term. Term 1 is straightforward: $${\rm Term\ 1} = \gamma_{\mu i,j} T^{\mu j}_{\rm EM}.$$
# Step 2.2.e: Construct TEMUDdD_contracted itself
# Step 2.2.e.i
TEMUDdD_contracted = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 1: g_{\mu i,j} T^{\mu j}_{\rm EM}
TEMUDdD_contracted[i] += g4DDdD[mu][i + 1][j +
1] * T4EMUU[mu][j +
1]
# We'll need derivatives of u4U for the next part:
u4UdD = ixp.zerorank2(DIM=4)
u4upperZero_dD = ixp.declarerank1(
"u4upperZero_dD"
) # Note that derivatives can't be done in 4-D with the current version of NRPy
for i in range(DIM):
u4UdD[0][i + 1] = u4upperZero_dD[i]
for i in range(DIM):
for j in range(DIM):
u4UdD[i + 1][j + 1] = uU_dD[i][j]
# <a id='alltogether2'></a>
#
# ### Step 2.2.e.ii: Putting it together: Term 2
# $$\label{alltogether2}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# We will now add $${\rm Term\ 2} = g_{\mu i} \left( \underbrace{b^2 u^j_{,j} u^\mu}_{\rm Term\ 2a} + \underbrace{b^2 u^j u^\mu_{,j}}_{\rm Term\ 2b} + \underbrace{\frac{b^2}{2} g^{j\mu}_{\ ,j}}_{\rm Term\ 2c} + \underbrace{b^j_{,j} b^\mu}_{\rm Term\ 2d} + \underbrace{b^j b^\mu_{,j}}_{\rm Term\ 2e} \right)$$ to $\partial_j T^{j}_{{\rm EM} i}$. These are the terms that involve contractions over $k$ (but no metric derivatives like Term 1 had).
#
# Step 2.2.e.ii
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2a: g_{\mu i} b^2 u^j_{,j} u^\mu
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2 * uU_dD[j][j] * u4U[mu]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2b: g_{\mu i} b^2 u^j u^\mu_{,j}
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2 * uU[j] * u4UdD[mu][j + 1]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2c: g_{\mu i} b^2 g^{j \mu}_{,j} / 2
TEMUDdD_contracted[i] += g4DD[mu][i + 1] * smallb2 * g4UUdD[
j + 1][mu][j + 1] / 2
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2d: g_{\mu i} b^j_{,j} b^\mu
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallbUdD[j][j] * smallb4U[mu]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 2e: g_{\mu i} b^j b^\mu_{,j}
TEMUDdD_contracted[i] += g4DD[mu][i + 1] * smallb4U[
j + 1] * smallb4UdD[mu][j + 1]
# <a id='alltogether3'></a>
#
# ### Step 2.2.e.iii: Putting it together: Term 3
# $$\label{alltogether3}$$
#
# \[Back to [List of Derivatives](#table2)\]
#
# Now, we will add $${\rm Term\ 3} = g_{\mu i} \left( \underbrace{\left(b^2\right)_{,j} u^j u^\mu}_{\rm Term\ 3a} + \underbrace{\frac{1}{2} \left(b^2\right)_{,j} g^{j\mu}}_{\rm Term\ 3b} \right).$$
# Step 2.2.e.iii
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 3a: g_{\mu i} ( b^2 )_{,j} u^j u^\mu
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2_dD[j] * uU[j] * u4U[mu]
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
# Term 3b: g_{mu i} ( b^2 )_{,j} g^{j\mu} / 2
TEMUDdD_contracted[i] += g4DD[mu][
i + 1] * smallb2_dD[j] * g4UU[j + 1][mu] / 2
#
# # Evolution equation for $\tilde{S}_i$
# <a id='step7'></a>
#
# ## Step 3.0: Construct the evolution equation for $\tilde{S}_i$
# $$\label{step7}$$
#
# \[Back to [top](#top)\]
#
# Finally, we will return our attention to the time evolution equation (from eq. 13 of the [original paper](https://arxiv.org/pdf/1704.00599.pdf)),
# \begin{align}
# \partial_t \tilde{S}_i &= - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} \\
# &= -T^j_{{\rm EM} i} \partial_j (\alpha \sqrt{\gamma}) - \alpha \sqrt{\gamma} \partial_j T^j_{{\rm EM} i} + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} \\
# &= \underbrace{-g_{i\mu} T^{\mu j}_{\rm EM} \partial_j (\alpha \sqrt{\gamma})
# }_{\rm Term\ 1} - \underbrace{\alpha \sqrt{\gamma} \partial_j T^j_{{\rm EM} i}}_{\rm Term\ 2} + \underbrace{\frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}}_{\rm Term\ 3} .
# \end{align}
# We will first take derivatives of $\alpha \sqrt{\gamma}$, then construct each term in turn.
# Step 3.0: Construct the evolution equation for \tilde{S}_i
# Here, we set up the necessary machinery to take FD derivatives of alpha * sqrt(gamma)
global alpsqrtgam
alpsqrtgam = gri.register_gridfunctions("AUX", "alpsqrtgam")
alpsqrtgam = alpha * sp.sqrt(gammadet)
alpsqrtgam_dD = ixp.declarerank1("alpsqrtgam_dD")
global Stilde_rhsD
Stilde_rhsD = ixp.zerorank1()
# The first term: g_{i\mu} T^{\mu j}_{\rm EM} \partial_j (\alpha \sqrt{\gamma})
for i in range(DIM):
for j in range(DIM):
for mu in range(4):
Stilde_rhsD[i] += -g4DD[i + 1][mu] * T4EMUU[mu][
j + 1] * alpsqrtgam_dD[j]
# The second term: \alpha \sqrt{\gamma} \partial_j T^j_{{\rm EM} i}
for i in range(DIM):
Stilde_rhsD[i] += -alpsqrtgam * TEMUDdD_contracted[i]
# The third term: \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} / 2
for i in range(DIM):
for mu in range(4):
for nu in range(4):
Stilde_rhsD[i] += alpsqrtgam * T4EMUU[mu][nu] * g4DDdD[mu][nu][
i + 1] / 2
# # Evolution equations for $A_i$ and $\Phi$
# <a id='step8'></a>
#
# ## Step 4.0: Construct the evolution equations for $A_i$ and $[\sqrt{\gamma}\Phi]$
# $$\label{step8}$$
#
# \[Back to [top](#top)\]
#
# We will also need to evolve the vector potential $A_i$. This evolution is given as eq. 17 in the [$\texttt{GiRaFFE}$](https://arxiv.org/pdf/1704.00599.pdf) paper:
# $$\boxed{\partial_t A_i = \epsilon_{ijk} v^j B^k - \partial_i (\underbrace{\alpha \Phi - \beta^j A_j}_{\rm AevolParen}),}$$
# where $\epsilon_{ijk} = [ijk] \sqrt{\gamma}$ is the antisymmetric Levi-Civita tensor, the drift velocity $v^i = u^i/u^0$, $\gamma$ is the determinant of the three metric, $B^k$ is the magnetic field, $\alpha$ is the lapse, and $\beta$ is the shift.
# The scalar electric potential $\Phi$ is also evolved by eq. 19:
# $$\boxed{\partial_t [\sqrt{\gamma} \Phi] = -\partial_j (\underbrace{\alpha\sqrt{\gamma}A^j - \beta^j [\sqrt{\gamma} \Phi]}_{\rm PevolParenU[j]}) - \xi \alpha [\sqrt{\gamma} \Phi],}$$
# with $\xi$ chosen as a damping factor.
#
# ### Step 4.0.a: Construct some useful auxiliary gridfunctions for the other evolution equations
#
# After declaring a some needed quantities, we will also define the parenthetical terms (underbrace above) that we need to take derivatives of. That way, we can take finite-difference derivatives easily. Note that we use $A^j = \gamma^{ij} A_i$, while $A_i$ (with $\Phi$) is technically a four-vector; this is justified, however, since $n_\mu A^\mu = 0$, where $n_\mu$ is a normal to the hypersurface, $A^0=0$ (according to Sec. II, subsection C of [this paper](https://arxiv.org/pdf/1110.4633.pdf)).
# Step 4.0: Construct the evolution equations for A_i and sqrt(gamma)Phi
# Step 4.0.a: Construct some useful auxiliary gridfunctions for the other evolution equations
xi = par.Cparameters(
"REAL", thismodule, "xi", 0.1
) # The (dimensionful) Lorenz damping factor. Recommendation: set to ~1.5/max(delta t).
# Define sqrt(gamma)Phi as psi6Phi
psi6Phi = gri.register_gridfunctions("EVOL", "psi6Phi")
Phi = psi6Phi / sp.sqrt(gammadet)
# We'll define a few extra gridfunctions to avoid complicated derivatives
global AevolParen, PevolParenU
AevolParen = gri.register_gridfunctions("AUX", "AevolParen")
PevolParenU = ixp.register_gridfunctions_for_single_rank1(
"AUX", "PevolParenU")
# {\rm AevolParen} = \alpha \Phi - \beta^j A_j
AevolParen = alpha * Phi
for j in range(DIM):
AevolParen += -betaU[j] * AD[j]
# {\rm PevolParenU[j]} = \alpha\sqrt{\gamma} \gamma^{ij} A_i - \beta^j [\sqrt{\gamma} \Phi]
for j in range(DIM):
PevolParenU[j] = -betaU[j] * psi6Phi
for i in range(DIM):
PevolParenU[j] += alpha * sp.sqrt(gammadet) * gammaUU[i][j] * AD[i]
AevolParen_dD = ixp.declarerank1("AevolParen_dD")
PevolParenU_dD = ixp.declarerank2("PevolParenU_dD", "nosym")
# ### Step 4.0.b: Construct the actual evolution equations for $A_i$ and $[\sqrt{\gamma}\Phi]$
#
# Now to set the evolution equations ([eqs. 17 and 19](https://arxiv.org/pdf/1704.00599.pdf)), recalling that the drift velocity $v^i = u^i/u^0$:
# \begin{align}
# \partial_t A_i &= \epsilon_{ijk} v^j B^k - \partial_i (\alpha \Phi - \beta^j A_j) \\
# &= \epsilon_{ijk} \frac{u^j}{u^0} B^k - {\rm AevolParen\_dD[i]} \\
# \partial_t [\sqrt{\gamma} \Phi] &= -\partial_j \left(\left(\alpha\sqrt{\gamma}\right)A^j - \beta^j [\sqrt{\gamma} \Phi]\right) - \xi \alpha [\sqrt{\gamma} \Phi] \\
# &= -{\rm PevolParenU\_dD[j][j]} - \xi \alpha [\sqrt{\gamma} \Phi]. \\
# \end{align}
# Step 4.0.b: Construct the actual evolution equations for A_i and sqrt(gamma)Phi
global A_rhsD, psi6Phi_rhs
A_rhsD = ixp.zerorank1()
psi6Phi_rhs = sp.sympify(0)
for i in range(DIM):
A_rhsD[i] = -AevolParen_dD[i]
for j in range(DIM):
for k in range(DIM):
A_rhsD[i] += LeviCivitaTensorDDD[i][j][k] * (
uU[j] / u4upperZero) * BU[k]
psi6Phi_rhs = -xi * alpha * psi6Phi
for j in range(DIM):
psi6Phi_rhs += -PevolParenU_dD[j][j]
def stress_energy_source_terms_ito_T4UU_and_ADM_or_BSSN_metricvars(
inputvars, custom_T4UU=None):
# Step 1: Check if rfm.reference_metric() already called. If not, BSSN
# quantities are not yet defined, so cannot proceed!
if rfm.have_already_called_reference_metric_function == False:
print(
"BSSN_source_terms_ito_T4UU(): Must call reference_metric() first!"
)
sys.exit(1)
# Step 2.a: Define gamma4DD[mu][nu] = g_{mu nu} + n_{mu} n_{nu}
alpha = sp.symbols("alpha", real=True)
zero = sp.sympify(0)
n4D = [-alpha, zero, zero, zero]
AB4m.g4DD_ito_BSSN_or_ADM(inputvars)
gamma4DD = ixp.zerorank2(DIM=4)
for mu in range(4):
for nu in range(4):
gamma4DD[mu][nu] = AB4m.g4DD[mu][nu] + n4D[mu] * n4D[nu]
# Step 2.b: If expression for components of T4UU not given, declare T4UU here
if custom_T4UU == None:
T4UU = ixp.declarerank2("T4UU", "sym01", DIM=4)
else:
T4UU = custom_T4UU
# Step 2.c: Define BSSN source terms
global SDD, SD, S, rho
# Step 2.c.i: S_{ij} = gamma_{i mu} gamma_{j nu} T^{mu nu}
SDD = ixp.zerorank2()
for i in range(3):
for j in range(3):
for mu in range(4):
for nu in range(4):
SDD[i][j] += gamma4DD[i + 1][mu] * gamma4DD[
j + 1][nu] * T4UU[mu][nu]
# Step 2.c.ii: S_{i} = -gamma_{i mu} n_{nu} T^{mu nu}
SD = ixp.zerorank1()
for i in range(3):
for mu in range(4):
for nu in range(4):
SD[i] += -gamma4DD[i + 1][mu] * n4D[nu] * T4UU[mu][nu]
# Step 2.c.iii: S = gamma^{ij} S_{ij}
if inputvars == "ADM":
gammaDD = ixp.declarerank2("gammaDD", "sym01")
gammaUU, dummydet = ixp.symm_matrix_inverter3x3(gammaDD) # Set gammaUU
elif inputvars == "BSSN":
import BSSN.ADM_in_terms_of_BSSN as AitoB # NRPy+: ADM quantities in terms of BSSN quantities
AitoB.ADM_in_terms_of_BSSN()
gammaUU = AitoB.gammaUU
S = zero
for i in range(3):
for j in range(3):
S += gammaUU[i][j] * SDD[i][j]
# Step 2.c.iv: rho = n_{mu} n_{nu} T^{mu nu}
rho = zero
for mu in range(4):
for nu in range(4):
rho += n4D[mu] * n4D[nu] * T4UU[mu][nu]
return SDD, SD, S, rho
def compute_u0_smallb_Poynting__Cartesian(gammaDD=None,
betaU=None,
alpha=None,
ValenciavU=None,
BU=None):
if gammaDD is None: # use "is None" instead of "==None", as the former is more correct.
# Declare these generically if uninitialized.
gammaDD = ixp.declarerank2("gammaDD", "sym01")
betaU = ixp.declarerank1("betaU")
alpha = sp.sympify("alpha")
ValenciavU = ixp.declarerank1("ValenciavU")
BU = ixp.declarerank1("BU")
# Set spatial dimension = 3
DIM = 3
thismodule = __name__
# Step 1.a: Compute the 4-metric $g_{\mu\nu}$ and its inverse
# $g^{\mu\nu}$ from the ADM 3+1 variables, using the
# BSSN.ADMBSSN_tofrom_4metric NRPy+ module
AB4m.g4DD_ito_BSSN_or_ADM("ADM", gammaDD, betaU, alpha)
g4DD = AB4m.g4DD
AB4m.g4UU_ito_BSSN_or_ADM("ADM", gammaDD, betaU, alpha)
g4UU = AB4m.g4UU
# Step 1.b: Our algorithm for computing $u^0$ is as follows:
#
# Let
# R = gamma_{ij} v^i_{(n)} v^j_{(n)} > 1 - 1 / Gamma_MAX.
# Then the velocity exceeds the speed limit (set by the
# maximum Lorentz Gamma, Gamma_MAX), and adjust the
# 3-velocity $v^i$ as follows:
#
# v^i_{(n)} = \sqrt{(1 - 1/Gamma_MAX)/R} * v^i_{(n)}
#
# After this rescaling, we are then guaranteed that if
# R is recomputed, it will be set to its ceiling value
# R = 1 - 1 / Gamma_MAX,
#
# Then $u^0$ can be safely computed via
# u^0 = 1 / (alpha \sqrt{1-R}).
# Step 1.b.i: Compute R = 1 - 1/max(Gamma)
R = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
R += gammaDD[i][j] * ValenciavU[i] * ValenciavU[j]
# Step 1.b.ii: Output C code for computing u^0
GAMMA_SPEED_LIMIT = par.Cparameters("REAL", thismodule,
"GAMMA_SPEED_LIMIT",
10.0) # Default value based on
# IllinoisGRMHD.
# GiRaFFE default = 2000.0
Rmax = 1 - 1 / (GAMMA_SPEED_LIMIT * GAMMA_SPEED_LIMIT)
rescaledValenciavU = ixp.zerorank1()
for i in range(DIM):
rescaledValenciavU[i] = ValenciavU[i] * sp.sqrt(Rmax / R)
rescaledu0 = 1 / (alpha * sp.sqrt(1 - Rmax))
regularu0 = 1 / (alpha * sp.sqrt(1 - R))
global computeu0_Cfunction
computeu0_Cfunction = """
/* Function for computing u^0 from Valencia 3-velocity. */
/* Inputs: ValenciavU[], alpha, gammaDD[][], GAMMA_SPEED_LIMIT (C parameter) */
/* Output: u0=u^0 and velocity-limited ValenciavU[] */\n\n"""
computeu0_Cfunction += outputC(
[R, Rmax], ["const double R", "const double Rmax"],
"returnstring",
params="includebraces=False,CSE_varprefix=tmpR,outCverbose=False")
computeu0_Cfunction += "if(R <= Rmax) "
computeu0_Cfunction += outputC(
regularu0,
"u0",
"returnstring",
params="includebraces=True,CSE_varprefix=tmpnorescale,outCverbose=False"
)
computeu0_Cfunction += " else "
computeu0_Cfunction += outputC(
[
rescaledValenciavU[0], rescaledValenciavU[1],
rescaledValenciavU[2], rescaledu0
], ["ValenciavU0", "ValenciavU1", "ValenciavU2", "u0"],
"returnstring",
params="includebraces=True,CSE_varprefix=tmprescale,outCverbose=False")
# ## Step 1.c: Compute u_j from u^0, the Valencia 3-velocity,
# and g_{mu nu}
# The basic equation is
# u_j &= g_{\mu j} u^{\mu} \\
# &= g_{0j} u^0 + g_{ij} u^i \\
# &= \beta_j u^0 + \gamma_{ij} u^i \\
# &= \beta_j u^0 + \gamma_{ij} u^0 \left(\alpha v^i_{(n)} - \beta^i\right) \\
# &= u^0 \left(\beta_j + \gamma_{ij} \left(\alpha v^i_{(n)} - \beta^i\right) \right)\\
# &= \alpha u^0 \gamma_{ij} v^i_{(n)} \\
global u0
u0 = par.Cparameters(
"REAL", thismodule, "u0", 1e300
) # Will be overwritten in C code. Set to crazy value to ensure this.
global uD
uD = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
uD[j] += alpha * u0 * gammaDD[i][j] * ValenciavU[i]
# ## Step 1.d: Compute $b^\mu$ from above expressions.
# \sqrt{4\pi} b^0 = B^0_{\rm (u)} &= \frac{u_j B^j}{\alpha} \\
# \sqrt{4\pi} b^i = B^i_{\rm (u)} &= \frac{B^i + (u_j B^j) u^i}{\alpha u^0}\\
# $B^i$ is related to the actual magnetic field evaluated in IllinoisGRMHD, $\tilde{B}^i$ via
#
# $$B^i = \frac{\tilde{B}^i}{\gamma},$$
#
# where $\gamma$ is the determinant of the spatial 3-metric.
#
# Pulling this together, we currently have available as input:
# + $\tilde{B}^i$
# + $\gamma$
# + $u_j$
# + $u^0$,
# with the goal of outputting now $b^\mu$ and $b^2$:
M_PI = par.Cparameters("#define", thismodule, "M_PI", "")
# uBcontraction = u_i B^i
global uBcontraction
uBcontraction = sp.sympify(0)
for i in range(DIM):
uBcontraction += uD[i] * BU[i]
# uU = 3-vector representing u^i = u^0 \left(\alpha v^i_{(n)} - \beta^i\right)
global uU
uU = ixp.zerorank1()
for i in range(DIM):
uU[i] = u0 * (alpha * ValenciavU[i] - betaU[i])
global smallb4U
smallb4U = ixp.zerorank1(DIM=4)
smallb4U[0] = uBcontraction / (alpha * sp.sqrt(4 * M_PI))
for i in range(DIM):
smallb4U[1 + i] = (BU[i] + uBcontraction * uU[i]) / (alpha * u0 *
sp.sqrt(4 * M_PI))
# Step 2: Compute the Poynting flux vector S^i
#
# The Poynting flux is defined in Eq. 11 of [Kelly *et al*](https://arxiv.org/pdf/1710.02132.pdf):
# S^i = -\alpha T^i_{\rm EM\ 0} = \alpha\left(b^2 u^i u_0 + \frac{1}{2} b^2 g^i{}_0 - b^i b_0\right)
# We start by computing
# g^\mu{}_\delta = g^{\mu\nu} g_{\nu \delta},
# and then the rest of the Poynting flux vector can be immediately computed from quantities defined above:
# S^i = \alpha T^i_{\rm EM\ 0} = -\alpha\left(b^2 u^i u_0 + \frac{1}{2} b^2 g^i{}_0 - b^i b_0\right)
# Step 2.a.i: compute g^\mu_\delta:
g4UD = ixp.zerorank2(DIM=4)
for mu in range(4):
for delta in range(4):
for nu in range(4):
g4UD[mu][delta] += g4UU[mu][nu] * g4DD[nu][delta]
# Step 2.a.ii: compute b_{\mu}
global smallb4D
smallb4D = ixp.zerorank1(DIM=4)
for mu in range(4):
for nu in range(4):
smallb4D[mu] += g4DD[mu][nu] * smallb4U[nu]
# Step 2.a.iii: compute u_0 = g_{mu 0} u^{mu} = g4DD[0][0]*u0 + g4DD[i][0]*uU[i]
u_0 = g4DD[0][0] * u0
for i in range(DIM):
u_0 += g4DD[i + 1][0] * uU[i]
# Step 2.a.iv: compute b^2, setting b^2 = smallb2etk, as gridfunctions with base names ending in a digit
# are forbidden in NRPy+.
global smallb2etk
smallb2etk = sp.sympify(0)
for mu in range(4):
smallb2etk += smallb4U[mu] * smallb4D[mu]
# Step 2.a.v: compute S^i
global PoynSU
PoynSU = ixp.zerorank1()
for i in range(DIM):
PoynSU[i] = -alpha * (smallb2etk * uU[i] * u_0 +
sp.Rational(1, 2) * smallb2etk * g4UD[i + 1][0] -
smallb4U[i + 1] * smallb4D[0])
def GiRaFFE_Higher_Order_v2():
#Step 1.0: Set the spatial dimension parameter to 3.
par.set_parval_from_str("grid::DIM", 3)
DIM = par.parval_from_str("grid::DIM")
# Step 1.1: Set the finite differencing order to 4.
#par.set_parval_from_str("finite_difference::FD_CENTDERIVS_ORDER", 4)
thismodule = "GiRaFFE_NRPy"
# M_PI will allow the C code to substitute the correct value
M_PI = par.Cparameters("#define",thismodule,"M_PI","")
# ADMBase defines the 4-metric in terms of the 3+1 spacetime metric quantities gamma_{ij}, beta^i, and alpha
gammaDD = ixp.register_gridfunctions_for_single_rank2("AUX","gammaDD", "sym01",DIM=3)
betaU = ixp.register_gridfunctions_for_single_rank1("AUX","betaU",DIM=3)
alpha = gri.register_gridfunctions("AUX","alpha")
# GiRaFFE uses the Valencia 3-velocity and A_i, which are defined in the initial data module(GiRaFFEfood)
ValenciavU = ixp.register_gridfunctions_for_single_rank1("AUX","ValenciavU",DIM=3)
AD = ixp.register_gridfunctions_for_single_rank1("EVOL","AD",DIM=3)
# B^i must be computed at each timestep within GiRaFFE so that the Valencia 3-velocity can be evaluated
BU = ixp.register_gridfunctions_for_single_rank1("AUX","BU",DIM=3)
# <a id='step3'></a>
#
# ## Step 1.2: Build the four metric $g_{\mu\nu}$, its inverse $g^{\mu\nu}$ and spatial derivatives $g_{\mu\nu,i}$ from ADM 3+1 quantities $\gamma_{ij}$, $\beta^i$, and $\alpha$ \[Back to [top](#top)\]
#
# Notice that the time evolution equation for $\tilde{S}_i$
# $$
# \partial_t \tilde{S}_i = - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}
# $$
# contains $\partial_i g_{\mu \nu} = g_{\mu\nu,i}$. We will now focus on evaluating this term.
#
# The four-metric $g_{\mu\nu}$ is related to the three-metric $\gamma_{ij}$, index-lowered shift $\beta_i$, and lapse $\alpha$ by
# $$
# g_{\mu\nu} = \begin{pmatrix}
# -\alpha^2 + \beta^k \beta_k & \beta_j \\
# \beta_i & \gamma_{ij}
# \end{pmatrix}.
# $$
# This tensor and its inverse have already been built by the u0_smallb_Poynting__Cartesian.py module ([documented here](Tutorial-u0_smallb_Poynting-Cartesian.ipynb)), so we can simply load the module and import the variables.
# $$\label{step3}$$
# Step 1.2: import u0_smallb_Poynting__Cartesian.py to set
# the four metric and its inverse. This module also sets b^2 and u^0.
import u0_smallb_Poynting__Cartesian.u0_smallb_Poynting__Cartesian as u0b
u0b.compute_u0_smallb_Poynting__Cartesian(gammaDD,betaU,alpha,ValenciavU,BU)
betaD = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
betaD[i] += gammaDD[i][j] * betaU[j]
# Error check: fixed = to +=
# We will now pull in the four metric and its inverse.
import BSSN.ADMBSSN_tofrom_4metric as AB4m # NRPy+: ADM/BSSN <-> 4-metric conversions
AB4m.g4DD_ito_BSSN_or_ADM("ADM")
g4DD = AB4m.g4DD
AB4m.g4UU_ito_BSSN_or_ADM("ADM")
g4UU = AB4m.g4UU
# Next we compute spatial derivatives of the metric, $\partial_i g_{\mu\nu} = g_{\mu\nu,i}$, written in terms of the three-metric, shift, and lapse. Simply taking the derivative of the expression for $g_{\mu\nu}$ above, we find
# $$
# g_{\mu\nu,l} = \begin{pmatrix}
# -2\alpha \alpha_{,l} + \beta^k_{\ ,l} \beta_k + \beta^k \beta_{k,l} & \beta_{j,l} \\
# \beta_{i,l} & \gamma_{ij,l}
# \end{pmatrix}.
# $$
#
# Notice the derivatives of the shift vector with its indexed lowered, $\beta_{i,j} = \partial_j \beta_i$. This can be easily computed in terms of the given ADMBase quantities $\beta^i$ and $\gamma_{ij}$ via:
# \begin{align}
# \beta_{i,j} &= \partial_j \beta_i \\
# &= \partial_j (\gamma_{ik} \beta^k) \\
# &= \gamma_{ik} \partial_j\beta^k + \beta^k \partial_j \gamma_{ik} \\
# \beta_{i,j} &= \gamma_{ik} \beta^k_{\ ,j} + \beta^k \gamma_{ik,j}.
# \end{align}
#
# Since this expression mixes Greek and Latin indices, we will declare this as a 4 dimensional quantity, but only set the three spatial components of its last index (that is, leaving $l=0$ unset).
#
# So, we will first set
# $$ g_{00,l} = \underbrace{-2\alpha \alpha_{,l}}_{\rm Term\ 1} + \underbrace{\beta^k_{\ ,l} \beta_k}_{\rm Term\ 2} + \underbrace{\beta^k \beta_{k,l}}_{\rm Term\ 3} $$
# Step 1.2, cont'd: Build spatial derivatives of the four metric
# Step 1.2.a: Declare derivatives of grid functions. These will be handled by FD_outputC
alpha_dD = ixp.declarerank1("alpha_dD")
betaU_dD = ixp.declarerank2("betaU_dD","nosym")
gammaDD_dD = ixp.declarerank3("gammaDD_dD","sym01")
# Step 1.2.b: These derivatives will be constructed analytically.
betaDdD = ixp.zerorank2()
g4DDdD = ixp.zerorank3(DIM=4)
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
# \gamma_{ik} \beta^k_{,j} + \beta^k \gamma_{ik,j}
betaDdD[i][j] += gammaDD[i][k] * betaU_dD[k][j] + betaU[k] * gammaDD_dD[i][k][j]
#Error check: changed = to += above
# Step 1.2.c: Set the 00 components
# Step 1.2.c.i: Term 1: -2\alpha \alpha_{,l}
for l in range(DIM):
g4DDdD[0][0][l+1] = -2*alpha*alpha_dD[l]
# Step 1.2.c.ii: Term 2: \beta^k_{\ ,l} \beta_k
for l in range(DIM):
for k in range(DIM):
g4DDdD[0][0][l+1] += betaU_dD[k][l] * betaD[k]
# Step 1.2.c.iii: Term 3: \beta^k \beta_{k,l}
for l in range(DIM):
for k in range(DIM):
g4DDdD[0][0][l+1] += betaU[k] * betaDdD[k][l]
# Now we will contruct the other components of $g_{\mu\nu,l}$. We will first construct
# $$ g_{i0,l} = g_{0i,l} = \beta_{i,l}, $$
# then
# $$ g_{ij,l} = \gamma_{ij,l} $$
# Step 1.2.d: Set the i0 and 0j components
for l in range(DIM):
for i in range(DIM):
# \beta_{i,l}
g4DDdD[i+1][0][l+1] = g4DDdD[0][i+1][l+1] = betaDdD[i][l]
#Step 1.2.e: Set the ij components
for l in range(DIM):
for i in range(DIM):
for j in range(DIM):
# \gamma_{ij,l}
g4DDdD[i+1][j+1][l+1] = gammaDD_dD[i][j][l]
# <a id='step4'></a>
#
# # $T^{\mu\nu}_{\rm EM}$ and its derivatives
#
# Now that the metric and its derivatives are out of the way, let's return to the evolution equation for $\tilde{S}_i$,
# $$
# \partial_t \tilde{S}_i = - \partial_j \left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right) + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}.
# $$
# We turn our focus now to $T^j_{{\rm EM} i}$ and its derivatives. To this end, we start by computing $T^{\mu \nu}_{\rm EM}$ (from eq. 27 of [Paschalidis & Shapiro's paper on their GRFFE code](https://arxiv.org/pdf/1310.3274.pdf)):
#
# $$\boxed{T^{\mu \nu}_{\rm EM} = b^2 u^{\mu} u^{\nu} + \frac{b^2}{2} g^{\mu \nu} - b^{\mu} b^{\nu}.}$$
#
# Notice that $T^{\mu\nu}_{\rm EM}$ is written in terms of
#
# * $b^\mu$, the 4-component magnetic field vector, related to the comoving magnetic field vector $B^i_{(u)}$
# * $u^\mu$, the 4-velocity
# * $g^{\mu \nu}$, the inverse 4-metric
#
# However, $\texttt{GiRaFFE}$ has access to only the following quantities, requiring in the following sections that we write the above quantities in terms of the following ones:
#
# * $\gamma_{ij}$, the 3-metric
# * $\alpha$, the lapse
# * $\beta^i$, the shift
# * $A_i$, the vector potential
# * $B^i$, the magnetic field (we assume only in the grid interior, not the ghost zones)
# * $\left[\sqrt{\gamma}\Phi\right]$, the zero-component of the vector potential $A_\mu$, times the square root of the determinant of the 3-metric
# * $v_{(n)}^i$, the Valencia 3-velocity
# * $u^0$, the zero-component of the 4-velocity
#
# ## Step 2.0: $u^i$ and $b^i$ and related quantities \[Back to [top](#top)\]
#
# We begin by importing what we can from u0_smallb_Poynting__Cartesian.py. We will need the four-velocity $u^\mu$, which is related to the Valencia 3-velocity $v^i_{(n)}$ used directly by $\texttt{GiRaFFE}$ (see also [Duez, et al, eqs. 53 and 56](https://arxiv.org/pdf/astro-ph/0503420.pdf))
# \begin{align}
# u^i &= u^0 (\alpha v^i_{(n)} - \beta^i), \\
# u_j &= \alpha u^0 \gamma_{ij} v^i_{(n)},
# \end{align}
# where $v^i_{(n)}$ is the Valencia three-velocity. These have already been constructed in terms of the Valencia 3-velocity and other 3+1 ADM quantities by the u0_smallb_Poynting__Cartesian.py module, so we can simply import these variables:
# $$\label{step4}$$
# Step 2.0: u^i, b^i, and related quantities
# Step 2.0.a: import the four-velocity, as written in terms of the Valencia 3-velocity
uD = ixp.zerorank1()
uU = ixp.zerorank1()
u4upperZero = gri.register_gridfunctions("AUX","u4upperZero")
for i in range(DIM):
uD[i] = u0b.uD[i].subs(u0b.u0,u4upperZero)
uU[i] = u0b.uU[i].subs(u0b.u0,u4upperZero)
# We also need the magnetic field 4-vector $b^{\mu}$, which is related to the magnetic field by [eqs. 23, 24, and 31 in Duez, et al](https://arxiv.org/pdf/astro-ph/0503420.pdf):
# \begin{align}
# b^0 &= \frac{1}{\sqrt{4\pi}} B^0_{\rm (u)} = \frac{u_j B^j}{\sqrt{4\pi}\alpha}, \\
# b^i &= \frac{1}{\sqrt{4\pi}} B^i_{\rm (u)} = \frac{B^i + (u_j B^j) u^i}{\sqrt{4\pi}\alpha u^0}, \\
# \end{align}
# where $B^i$ is the variable tracked by the HydroBase thorn in the Einstein Toolkit. Again, these have already been built by the u0_smallb_Poynting__Cartesian.py module, so we can simply import the variables.
# Step 2.0.b: import the small b terms
smallb4U = ixp.zerorank1(DIM=4)
smallb4D = ixp.zerorank1(DIM=4)
for mu in range(4):
smallb4U[mu] = u0b.smallb4U[mu].subs(u0b.u0,u4upperZero)
smallb4D[mu] = u0b.smallb4D[mu].subs(u0b.u0,u4upperZero)
smallb2 = u0b.smallb2etk.subs(u0b.u0,u4upperZero)
# <a id='step5'></a>
#
# ## Step 2.1: Construct all components of the electromagnetic stress-energy tensor $T^{\mu \nu}_{\rm EM}$ \[Back to [top](#top)\]
#
# We now have all the pieces to calculate the stress-energy tensor,
# $$T^{\mu \nu}_{\rm EM} = \underbrace{b^2 u^{\mu} u^{\nu}}_{\rm Term\ 1} +
# \underbrace{\frac{b^2}{2} g^{\mu \nu}}_{\rm Term\ 2}
# - \underbrace{b^{\mu} b^{\nu}}_{\rm Term\ 3}.$$
# Because $u^0$ is a separate variable, we could build the $00$ component separately, then the $\mu0$ and $0\nu$ components, and finally the $\mu\nu$ components. Alternatively, for clarity, we could create a temporary variable $u^\mu=\left( u^0, u^i \right)$
# $$\label{step5}$$
# Step 2.1: Construct the electromagnetic stress-energy tensor
# Step 2.1.a: Set up the four-velocity vector
u4U = ixp.zerorank1(DIM=4)
u4U[0] = u4upperZero
for i in range(DIM):
u4U[i+1] = uU[i]
# Step 2.1.b: Build T4EMUU itself
T4EMUU = ixp.zerorank2(DIM=4)
for mu in range(4):
for nu in range(4):
# Term 1: b^2 u^{\mu} u^{\nu}
T4EMUU[mu][nu] = smallb2*u4U[mu]*u4U[nu]
for mu in range(4):
for nu in range(4):
# Term 2: b^2 / 2 g^{\mu \nu}
T4EMUU[mu][nu] += smallb2*g4UU[mu][nu]/2
for mu in range(4):
for nu in range(4):
# Term 3: -b^{\mu} b^{\nu}
T4EMUU[mu][nu] += -smallb4U[mu]*smallb4U[nu]
#
# # Evolution equation for $\tilde{S}_i$
# <a id='step7'></a>
#
# ## Step 3.0: Construct the evolution equation for $\tilde{S}_i$ \[Back to [top](#top)\]
#
# Finally, we will return our attention to the time evolution equation (from eq. 13 of the [original paper](https://arxiv.org/pdf/1704.00599.pdf)),
# \begin{align}
# \partial_t \tilde{S}_i &= - \partial_j \underbrace{\left( \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \right)}_{\rm SevolParenUD} + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} \\
# &= - \partial_j{\rm SevolParenUD[i][j]} + \frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} .
# \end{align}
# We will first take construct ${\rm SevolParenUD}$, then use its derivatives to construct the evolution equation. Note that
# \begin{align}
# {\rm SevolParenUD[i][j]} &= \alpha \sqrt{\gamma} T^j_{{\rm EM} i} \\
# &= \alpha \sqrt{\gamma} g_{\mu i} T^{\mu j}_{\rm EM}.
# \end{align}
# $$\label{step7}$$
# Step 3.0: Construct the evolution equation for \tilde{S}_i
# Here, we set up the necessary machinery to take FD derivatives of alpha * sqrt(gamma)
global gammaUU,gammadet
gammaUU = ixp.register_gridfunctions_for_single_rank2("AUX","gammaUU","sym01")
gammadet = gri.register_gridfunctions("AUX","gammadet")
gammaUU, gammadet = ixp.symm_matrix_inverter3x3(gammaDD)
alpsqrtgam = alpha*sp.sqrt(gammadet)
global SevolParenUD
SevolParenUD = ixp.register_gridfunctions_for_single_rank2("AUX","SevolParenUD","nosym")
SevolParenUD = ixp.zerorank2()
for i in range(DIM):
for j in range (DIM):
for mu in range(4):
SevolParenUD[j][i] += alpsqrtgam * g4DD[mu][i+1] * T4EMUU[mu][j+1]
SevolParenUD_dD = ixp.declarerank3("SevolParenUD_dD","nosym")
global Stilde_rhsD
Stilde_rhsD = ixp.zerorank1()
# The first term: \alpha \sqrt{\gamma} \partial_j T^j_{{\rm EM} i}
for i in range(DIM):
for j in range(DIM):
Stilde_rhsD[i] += -SevolParenUD_dD[j][i][j]
# The second term: \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} / 2
for i in range(DIM):
for mu in range(4):
for nu in range(4):
Stilde_rhsD[i] += alpsqrtgam * T4EMUU[mu][nu] * g4DDdD[mu][nu][i+1] / 2
# # Evolution equations for $A_i$ and $\Phi$
# <a id='step8'></a>
#
# ## Step 4.0: Construct the evolution equations for $A_i$ and $[\sqrt{\gamma}\Phi]$ \[Back to [top](#top)\]
#
# We will also need to evolve the vector potential $A_i$. This evolution is given as eq. 17 in the [$\texttt{GiRaFFE}$](https://arxiv.org/pdf/1704.00599.pdf) paper:
# $$\boxed{\partial_t A_i = \epsilon_{ijk} v^j B^k - \partial_i (\underbrace{\alpha \Phi - \beta^j A_j}_{\rm AevolParen}),}$$
# where $\epsilon_{ijk} = [ijk] \sqrt{\gamma}$ is the antisymmetric Levi-Civita tensor, the drift velocity $v^i = u^i/u^0$, $\gamma$ is the determinant of the three metric, $B^k$ is the magnetic field, $\alpha$ is the lapse, and $\beta$ is the shift.
# The scalar electric potential $\Phi$ is also evolved by eq. 19:
# $$\boxed{\partial_t [\sqrt{\gamma} \Phi] = -\partial_j (\underbrace{\alpha\sqrt{\gamma} \gamma^{ij} A_i - \beta^j [\sqrt{\gamma} \Phi]}_{\rm PevolParenU[j]}) - \xi \alpha [\sqrt{\gamma} \Phi],}$$
# with $\xi$ chosen as a damping factor.
#
# ### Step 4.0.a: Construct some useful auxiliary gridfunctions for the other evolution equations
#
# After declaring a some needed quantities, we will also define the parenthetical terms (underbrace above) that we need to take derivatives of. That way, we can take finite-difference derivatives easily. Note that the above equations incorporate the fact that $\gamma^{ij}$ is the appropriate raising operator for $A_i$: $A^j = \gamma^{ij} A_i$. This is correct since $n_\mu A^\mu = 0$, where $n_\mu$ is a normal to the hypersurface, so $A^0=0$ (according to Sec. II, subsection C of [the "Improved EM gauge condition" paper of Etienne *et al*](https://arxiv.org/pdf/1110.4633.pdf)).
# $$\label{step8}$$
# Step 4.0: Construct the evolution equations for A_i and sqrt(gamma)Phi
# Step 4.0.a: Construct some useful auxiliary gridfunctions for the other evolution equations
xi = par.Cparameters("REAL",thismodule,"xi", 0.1) # The (dimensionful) Lorenz damping factor. Recommendation: set to ~1.5/max(delta t).
# Define sqrt(gamma)Phi as psi6Phi
psi6Phi = gri.register_gridfunctions("EVOL","psi6Phi")
Phi = psi6Phi / sp.sqrt(gammadet)
# We'll define a few extra gridfunctions to avoid complicated derivatives
global AevolParen,PevolParenU
AevolParen = gri.register_gridfunctions("AUX","AevolParen")
PevolParenU = ixp.register_gridfunctions_for_single_rank1("AUX","PevolParenU")
# {\rm AevolParen} = \alpha \Phi - \beta^j A_j
AevolParen = alpha*Phi
for j in range(DIM):
AevolParen += -betaU[j] * AD[j]
# {\rm PevolParenU[j]} = \alpha\sqrt{\gamma} \gamma^{ij} A_i - \beta^j [\sqrt{\gamma} \Phi]
for j in range(DIM):
PevolParenU[j] = -betaU[j] * psi6Phi
for i in range(DIM):
PevolParenU[j] += alpsqrtgam * gammaUU[i][j] * AD[i]
AevolParen_dD = ixp.declarerank1("AevolParen_dD")
PevolParenU_dD = ixp.declarerank2("PevolParenU_dD","nosym")
# ### Step 4.0.b: Complete the construction of the evolution equations for $A_i$ and $[\sqrt{\gamma}\Phi]$
#
# Now to set the evolution equations ([eqs. 17 and 19](https://arxiv.org/pdf/1704.00599.pdf)), recalling that the drift velocity $v^i = u^i/u^0$:
# \begin{align}
# \partial_t A_i &= \epsilon_{ijk} v^j B^k - \partial_i (\alpha \Phi - \beta^j A_j) \\
# &= \epsilon_{ijk} \frac{u^j}{u^0} B^k - {\rm AevolParen\_dD[i]} \\
# \partial_t [\sqrt{\gamma} \Phi] &= -\partial_j \left(\left(\alpha\sqrt{\gamma}\right)A^j - \beta^j [\sqrt{\gamma} \Phi]\right) - \xi \alpha [\sqrt{\gamma} \Phi] \\
# &= -{\rm PevolParenU\_dD[j][j]} - \xi \alpha [\sqrt{\gamma} \Phi]. \\
# \end{align}
# Step 4.0.b: Construct the actual evolution equations for A_i and sqrt(gamma)Phi
global A_rhsD,psi6Phi_rhs
A_rhsD = ixp.zerorank1()
psi6Phi_rhs = sp.sympify(0)
# We already have a handy function to define the Levi-Civita symbol in WeylScalars
import WeylScal4NRPy.WeylScalars_Cartesian as weyl
# Initialize the Levi-Civita tensor by setting it equal to the Levi-Civita symbol
LeviCivitaSymbolDDD = weyl.define_LeviCivitaSymbol_rank3()
LeviCivitaTensorDDD = ixp.zerorank3()
#LeviCivitaTensorUUU = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
LeviCivitaTensorDDD[i][j][k] = LeviCivitaSymbolDDD[i][j][k] * sp.sqrt(gammadet)
#LeviCivitaTensorUUU[i][j][k] = LeviCivitaSymbolDDD[i][j][k] / sp.sqrt(gammadet)
for i in range(DIM):
A_rhsD[i] = -AevolParen_dD[i]
for j in range(DIM):
for k in range(DIM):
A_rhsD[i] += LeviCivitaTensorDDD[i][j][k]*(uU[j]/u4upperZero)*BU[k]
psi6Phi_rhs = -xi*alpha*psi6Phi
for j in range(DIM):
psi6Phi_rhs += -PevolParenU_dD[j][j]
def generate_memory_access_code(gammaDD, betaU, alpha):
# There are several pieces of C code that we will write ourselves because we need to do things
# a little bit outside of what NRPy+ is built for.
# First, we will write general memory access. We will read in values from memory at a given point
# for each quantity we care about.
global general_access
general_access = ""
for var in [
"GAMMADD00", "GAMMADD01", "GAMMADD02", "GAMMADD11", "GAMMADD12",
"GAMMADD22", "BETAU0", "BETAU1", "BETAU2", "ALPHA", "BU0", "BU1",
"BU2", "VALENCIAVU0", "VALENCIAVU1", "VALENCIAVU2"
]:
lhsvar = var.lower().replace("dd", "DD").replace("u", "U").replace(
"bU", "BU").replace("valencia", "Valencia")
# e.g.,
# const REAL gammaDD00dD0 = auxevol_gfs[IDX4S(GAMMA_FACEDD00GF,i0,i1,i2)];
general_access += "const REAL " + lhsvar + " = auxevol_gfs[IDX4S(" + var + "GF,i0,i1,i2)];\n"
# This quick function returns a nearby point for memory access. We need this because derivatives are not local operations.
def idxp1(dirn):
if dirn == 0:
return "i0+1,i1,i2"
if dirn == 1:
return "i0,i1+1,i2"
if dirn == 2:
return "i0,i1,i2+1"
# Next we evaluate needed derivatives of the metric, based on their values at cell faces
global metric_deriv_access
metric_deriv_access = []
# for dirn in range(3):
# metric_deriv_access.append("")
# for var in ["GAMMA_FACEDDdD00", "GAMMA_FACEDDdD01", "GAMMA_FACEDDdD02",
# "GAMMA_FACEDDdD11", "GAMMA_FACEDDdD12", "GAMMA_FACEDDdD22",
# "BETA_FACEUdD0", "BETA_FACEUdD1", "BETA_FACEUdD2","ALPHA_FACEdD"]:
# lhsvar = var.lower().replace("dddd","DDdD").replace("udd","UdD").replace("dd","dD").replace("u","U").replace("_face","")
# rhsvar = var.replace("dD","")
# # e.g.,
# # const REAL gammaDDdD000 = (auxevol_gfs[IDX4S(GAMMA_FACEDD00GF,i0+1,i1,i2)]-auxevol_gfs[IDX4S(GAMMA_FACEDD00GF,i0,i1,i2)])/dxx0;
# metric_deriv_access[dirn] += "const REAL "+lhsvar+str(dirn)+" = (auxevol_gfs[IDX4S("+rhsvar+"GF,"+idxp1(dirn)+")]-auxevol_gfs[IDX4S("+rhsvar+"GF,i0,i1,i2)])/dxx"+str(dirn)+";\n"
# metric_deriv_access[dirn] += "REAL Stilde_rhsD"+str(dirn)+";\n"
# For this workaround, instead of taking the derivative of the metric components and then building the
# four-metric, we build the four-metric and then take derivatives. Do this at i and i+1
for dirn in range(3):
metric_deriv_access.append("")
for var in [
"GAMMA_FACEDD00", "GAMMA_FACEDD01", "GAMMA_FACEDD02",
"GAMMA_FACEDD11", "GAMMA_FACEDD12", "GAMMA_FACEDD22",
"BETA_FACEU0", "BETA_FACEU1", "BETA_FACEU2", "ALPHA_FACE"
]:
lhsvar = var.lower().replace("dd", "DD").replace("u", "U")
rhsvar = var
# e.g.,
# const REAL gammaDD00 = auxevol_gfs[IDX4S(GAMMA_FACEDD00GF,i0,i1,i2)];
metric_deriv_access[
dirn] += "const REAL " + lhsvar + " = auxevol_gfs[IDX4S(" + rhsvar + "GF,i0,i1,i2)];\n"
# Read in at the next grid point
for var in [
"GAMMA_FACEDD00", "GAMMA_FACEDD01", "GAMMA_FACEDD02",
"GAMMA_FACEDD11", "GAMMA_FACEDD12", "GAMMA_FACEDD22",
"BETA_FACEU0", "BETA_FACEU1", "BETA_FACEU2", "ALPHA_FACE"
]:
lhsvar = var.lower().replace("dd", "DD").replace("u", "U").replace(
"_face", "_facep1")
rhsvar = var
# e.g.,
# const REAL gammaDD00 = auxevol_gfs[IDX4S(GAMMA_FACEDD00GF,i0+1,i1,i2)];
metric_deriv_access[
dirn] += "const REAL " + lhsvar + " = auxevol_gfs[IDX4S(" + rhsvar + "GF," + idxp1(
dirn) + ")];\n"
metric_deriv_access[dirn] += "REAL Stilde_rhsD" + str(dirn) + ";\n"
import BSSN.ADMBSSN_tofrom_4metric as AB4m
AB4m.g4DD_ito_BSSN_or_ADM("ADM", gammaDD, betaU, alpha)
four_metric_vars = [
AB4m.g4DD[0][0], AB4m.g4DD[0][1], AB4m.g4DD[0][2], AB4m.g4DD[0][3],
AB4m.g4DD[1][1], AB4m.g4DD[1][2], AB4m.g4DD[1][3], AB4m.g4DD[2][2],
AB4m.g4DD[2][3], AB4m.g4DD[3][3]
]
four_metric_names = [
"g4DD00", "g4DD01", "g4DD02", "g4DD03", "g4DD11", "g4DD12", "g4DD13",
"g4DD22", "g4DD23", "g4DD33"
]
global four_metric_C, four_metric_Cp1
four_metric_C = outputC(four_metric_vars,
four_metric_names,
"returnstring",
params="outCverbose=False,CSE_sorting=none")
for ii in range(len(four_metric_names)):
four_metric_names[ii] += "p1"
four_metric_Cp1 = outputC(four_metric_vars,
four_metric_names,
"returnstring",
params="outCverbose=False,CSE_sorting=none")
four_metric_C = four_metric_C.replace("gamma", "gamma_face").replace(
"beta",
"beta_face").replace("alpha", "alpha_face").replace("{", "").replace(
"}", "").replace("g4",
"const REAL g4").replace("tmp_", "tmp_deriv")
four_metric_Cp1 = four_metric_Cp1.replace("gamma", "gamma_facep1").replace(
"beta", "beta_facep1").replace("alpha", "alpha_facep1").replace(
"{", "").replace("}", "").replace("g4", "const REAL g4").replace(
"tmp_", "tmp_derivp")
global four_metric_deriv
four_metric_deriv = []
for dirn in range(3):
four_metric_deriv.append("")
for var in [
"g4DDdD00", "g4DDdD01", "g4DDdD02", "g4DDdD03", "g4DDdD11",
"g4DDdD12", "g4DDdD13", "g4DDdD22", "g4DDdD23", "g4DDdD33"
]:
lhsvar = var + str(dirn + 1)
rhsvar = var.replace("dD", "")
rhsvarp1 = rhsvar + "p1"
# e.g.,
# const REAL g44DDdD000 = (g4DD00p1 - g4DD00)/dxx0;
four_metric_deriv[
dirn] += "const REAL " + lhsvar + " = (" + rhsvarp1 + " - " + rhsvar + ")/dxx" + str(
dirn) + ";\n"
# This creates the C code that writes to the Stilde_rhs direction specified.
global write_final_quantity
write_final_quantity = []
for dirn in range(3):
write_final_quantity.append("")
write_final_quantity[dirn] += "rhs_gfs[IDX4S(STILDED" + str(
dirn) + "GF,i0,i1,i2)] += Stilde_rhsD" + str(dirn) + ";"