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) + ";"