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"\n",
"\n",
"\n",
"# BSSN Time-Evolution Equations for the Gauge Fields $\\alpha$ and $\\beta^i$\n",
"\n",
"## Authors: Zach Etienne & Terrence Pierre Jacques\n",
"### Formatting improvements courtesy Brandon Clark\n",
"\n",
"[comment]: <> (Abstract: TODO, or make the introduction an abstract and addiotnal notes section, and write a new Introduction)\n",
"\n",
"**Notebook Status:** Validated \n",
"\n",
"**Validation Notes:** All expressions generated in this module have been validated against a trusted code (the original NRPy+/SENR code, which itself was validated against [Baumgarte's code](https://arxiv.org/abs/1211.6632)).\n",
"\n",
"### NRPy+ Source Code for this module: [BSSN/BSSN_gauge_RHSs.py](../edit/BSSN/BSSN_gauge_RHSs.py)\n",
"\n",
"\n",
"## Introduction:\n",
"This tutorial notebook constructs SymPy expressions for the right-hand sides of the time-evolution equations for the gauge fields $\\alpha$ (the lapse, governing how much proper time elapses at each point between one timestep in a 3+1 solution to Einstein's equations and the next) and $\\beta^i$ (the shift, governing how much proper distance numerical grid points move from one timestep in a 3+1 solution to Einstein's equations and the next).\n",
"\n",
"Though we are completely free to choose gauge conditions (i.e., free to choose the form of the right-hand sides of the gauge time evolution equations), very few have been found robust in the presence of (puncture) black holes. So we focus here only on a few of the most stable choices.\n",
"\n"
]
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"source": [
"\n",
"\n",
"# Table of Contents\n",
"$$\\label{toc}$$\n",
"\n",
"This notebook is organized as follows\n",
"\n",
"1. [Step 1](#initializenrpy): Initialize needed Python/NRPy+ modules\n",
"1. [Step 2](#lapseconditions): Right-hand side of $\\partial_t \\alpha$\n",
" 1. [Step 2.a](#onepluslog): $1+\\log$ lapse\n",
" 1. [Step 2.b](#harmonicslicing): Harmonic slicing\n",
" 1. [Step 2.c](#frozen): Frozen lapse\n",
" 1. [Step 2.d](#statictrumpet_onepluslog): Alternative 1+log condition for Static Trumpet initial data\n",
"1. [Step 3](#shiftconditions): Right-hand side of $\\partial_t \\beta^i$: Second-order Gamma-driving shift conditions\n",
" 1. [Step 3.a](#origgammadriving): Original, non-covariant Gamma-driving shift condition\n",
" 1. [Step 3.b](#covgammadriving): [Brown](https://arxiv.org/abs/0902.3652)'s suggested covariant Gamma-driving shift condition\n",
" 1. [Step 3.b.i](#partial_beta): The right-hand side of the $\\partial_t \\beta^i$ equation\n",
" 1. [Step 3.b.ii](#partial_upper_b): The right-hand side of the $\\partial_t B^i$ equation\n",
" 1. [Step 3.c](#statictrumpet_nonadvecgammadriving): Non-advecting Gamma-driving shift condition (used for evolving \"Static Trumpet\" initial data)\n",
"1. [Step 4](#rescale): Rescale right-hand sides of BSSN gauge equations\n",
"1. [Step 5](#code_validation): Code Validation against `BSSN.BSSN_gauge_RHSs` NRPy+ module\n",
"1. [Step 6](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file"
]
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"\n",
"\n",
"# Step 1: Initialize needed Python/NRPy+ modules \\[Back to [top](#toc)\\]\n",
"$$\\label{initializenrpy}$$\n",
"\n",
"Let's start by importing all the needed modules from Python/NRPy+:"
]
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"# Step 1: Import all needed modules from NRPy+:\n",
"import sympy as sp # SymPy: The Python computer algebra package upon which NRPy+ depends\n",
"import NRPy_param_funcs as par # NRPy+: Parameter interface\n",
"import grid as gri # NRPy+: Functions having to do with numerical grids\n",
"import indexedexp as ixp # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support\n",
"import reference_metric as rfm # NRPy+: Reference metric support\n",
"import BSSN.BSSN_quantities as Bq # NRPy+: Computes useful BSSN quantities\n",
"import BSSN.BSSN_RHSs as Brhs # NRPy+: Constructs BSSN right-hand-side expressions\n",
"import sys # Standard Python modules for multiplatform OS-level functions\n",
"\n",
"# Step 1.c: Declare/initialize parameters for this module\n",
"thismodule = \"BSSN_gauge_RHSs\"\n",
"par.initialize_param(par.glb_param(\"char\", thismodule, \"LapseEvolutionOption\", \"OnePlusLog\"))\n",
"par.initialize_param(par.glb_param(\"char\", thismodule, \"ShiftEvolutionOption\", \"GammaDriving2ndOrder_Covariant\"))\n",
"\n",
"# Step 1.d: Set spatial dimension (must be 3 for BSSN, as BSSN is\n",
"# a 3+1-dimensional decomposition of the general\n",
"# relativistic field equations)\n",
"DIM = 3\n",
"\n",
"# Step 1.e: Given the chosen coordinate system, set up\n",
"# corresponding reference metric and needed\n",
"# reference metric quantities\n",
"# The following function call sets up the reference metric\n",
"# and related quantities, including rescaling matrices ReDD,\n",
"# ReU, and hatted quantities.\n",
"rfm.reference_metric()\n",
"\n",
"# Step 1.f: Define BSSN scalars & tensors (in terms of rescaled BSSN quantities)\n",
"import BSSN.BSSN_quantities as Bq\n",
"Bq.BSSN_basic_tensors()\n",
"Bq.betaU_derivs()\n",
"\n",
"import BSSN.BSSN_RHSs as Brhs\n",
"Brhs.BSSN_RHSs()"
]
},
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"metadata": {},
"source": [
"\n",
"\n",
"# Step 2: Right-hand side of $\\partial_t \\alpha$ \\[Back to [top](#toc)\\]\n",
"$$\\label{lapseconditions}$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"## Step 2.a: $1+\\log$ lapse \\[Back to [top](#toc)\\]\n",
"$$\\label{onepluslog}$$\n",
"\n",
"The [$1=\\log$ lapse condition](https://arxiv.org/abs/gr-qc/0206072) is a member of the [Bona-Masso family of lapse choices](https://arxiv.org/abs/gr-qc/9412071), which has the desirable property of singularity avoidance. As is common (e.g., see [Campanelli *et al* (2005)](https://arxiv.org/abs/gr-qc/0511048)), we make the replacement $\\partial_t \\to \\partial_0 = \\partial_t - \\beta^i \\partial_i$ to ensure lapse characteristics advect with the shift. The bracketed term in the $1+\\log$ lapse condition below encodes the shift advection term:\n",
"\n",
"\\begin{align}\n",
"\\partial_0 \\alpha &= -2 \\alpha K \\\\\n",
"\\implies \\partial_t \\alpha &= \\left[\\beta^i \\partial_i \\alpha\\right] - 2 \\alpha K\n",
"\\end{align}"
]
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"source": [
"# Step 2.a: The 1+log lapse condition:\n",
"# \\partial_t \\alpha = \\beta^i \\alpha_{,i} - 2*\\alpha*K\n",
"# First import expressions from BSSN_quantities\n",
"cf = Bq.cf\n",
"trK = Bq.trK\n",
"alpha = Bq.alpha\n",
"betaU = Bq.betaU\n",
"\n",
"# Implement the 1+log lapse condition\n",
"if par.parval_from_str(thismodule+\"::LapseEvolutionOption\") == \"OnePlusLog\":\n",
" alpha_rhs = -2*alpha*trK\n",
" alpha_dupD = ixp.declarerank1(\"alpha_dupD\")\n",
" for i in range(DIM):\n",
" alpha_rhs += betaU[i]*alpha_dupD[i]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"## Step 2.b: Harmonic slicing \\[Back to [top](#toc)\\]\n",
"$$\\label{harmonicslicing}$$\n",
"\n",
"As defined on Pg 2 of https://arxiv.org/pdf/gr-qc/9902024.pdf , this is given by \n",
"\n",
"$$\n",
"\\partial_t \\alpha = \\partial_t e^{6 \\phi} = 6 e^{6 \\phi} \\partial_t \\phi\n",
"$$\n",
"\n",
"If \n",
"\n",
"$$\\text{cf} = W = e^{-2 \\phi},$$ \n",
"\n",
"then\n",
"\n",
"$$\n",
"6 e^{6 \\phi} \\partial_t \\phi = 6 W^{-3} \\partial_t \\phi.\n",
"$$\n",
"\n",
"However,\n",
"$$\n",
"\\partial_t \\phi = -\\partial_t \\text{cf} / (2 \\text{cf})$$\n",
"\n",
"(as described above), so if `cf`$=W$, then\n",
"\\begin{align}\n",
"\\partial_t \\alpha &= 6 e^{6 \\phi} \\partial_t \\phi \\\\\n",
"&= 6 W^{-3} \\left(-\\frac{\\partial_t W}{2 W}\\right) \\\\\n",
"&= -3 \\text{cf}^{-4} \\text{cf}\\_\\text{rhs}\n",
"\\end{align}\n",
"\n",
"**Exercise to students: Implement Harmonic slicing for `cf`$=\\chi$** "
]
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"source": [
"# Step 2.b: Implement the harmonic slicing lapse condition\n",
"if par.parval_from_str(thismodule+\"::LapseEvolutionOption\") == \"HarmonicSlicing\":\n",
" if par.parval_from_str(\"BSSN.BSSN_quantities::EvolvedConformalFactor_cf\") == \"W\":\n",
" alpha_rhs = -3*cf**(-4)*Brhs.cf_rhs\n",
" elif par.parval_from_str(\"BSSN.BSSN_quantities::EvolvedConformalFactor_cf\") == \"phi\":\n",
" alpha_rhs = 6*sp.exp(6*cf)*Brhs.cf_rhs\n",
" else:\n",
" print(\"Error LapseEvolutionOption==HarmonicSlicing unsupported for EvolvedConformalFactor_cf!=(W or phi)\")\n",
" sys.exit(1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"## Step 2.c: Frozen lapse \\[Back to [top](#toc)\\]\n",
"$$\\label{frozen}$$\n",
"\n",
"This slicing condition is given by\n",
"$$\\partial_t \\alpha = 0,$$\n",
"\n",
"which is rarely a stable lapse condition."
]
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"source": [
"# Step 2.c: Frozen lapse\n",
"# \\partial_t \\alpha = 0\n",
"if par.parval_from_str(thismodule+\"::LapseEvolutionOption\") == \"Frozen\":\n",
" alpha_rhs = sp.sympify(0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"## Step 2.d: Alternative $1+\\log$ lapse for Static Trumpet initial data \\[Back to [top](#toc)\\]\n",
"$$\\label{statictrumpet_onepluslog}$$\n",
"\n",
"An alternative to the standard 1+log condition to be used with Static Trumpet initial data, the lapse is evolved according to a\n",
"condition consistent with staticity, given by equation 67 in [Ruchlin, Etienne, & Baumgarte (2018)](https://arxiv.org/pdf/1712.07658.pdf)\n",
"\n",
"\\begin{align}\n",
"\\partial_0 \\alpha &= -\\alpha(1 - \\alpha) K \\\\\n",
"\\implies \\partial_t \\alpha &= \\left[\\beta^i \\partial_i \\alpha\\right] -\\alpha(1 - \\alpha) K\n",
"\\end{align}"
]
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"source": [
"# Step 2.d: Alternative 1+log lapse condition:\n",
"# \\partial_t \\alpha = \\beta^i \\alpha_{,i} -\\alpha*(1 - \\alpha)*K\n",
"\n",
"# Implement the alternative 1+log lapse condition\n",
"if par.parval_from_str(thismodule+\"::LapseEvolutionOption\") == \"OnePlusLogAlt\":\n",
" alpha_rhs = -alpha*(1 - alpha)*trK\n",
" alpha_dupD = ixp.declarerank1(\"alpha_dupD\")\n",
" for i in range(DIM):\n",
" alpha_rhs += betaU[i]*alpha_dupD[i]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Step 3: Right-hand side of $\\partial_t \\beta^i$: Second-order Gamma-driving shift conditions \\[Back to [top](#toc)\\]\n",
"$$\\label{shiftconditions}$$\n",
"\n",
"The motivation behind Gamma-driving shift conditions are well documented in the book [*Numerical Relativity* by Baumgarte & Shapiro](https://www.amazon.com/Numerical-Relativity-Einsteins-Equations-Computer/dp/052151407X/).\n",
"\n",
"\n",
"\n",
"## Step 3.a: Original, non-covariant Gamma-driving shift condition \\[Back to [top](#toc)\\]\n",
"$$\\label{origgammadriving}$$\n",
"\n",
"**Option 1: Non-Covariant, Second-Order Shift**\n",
"\n",
"We adopt the [*shifting (i.e., advecting) shift*](https://arxiv.org/abs/gr-qc/0605030) non-covariant, second-order shift condition:\n",
"\\begin{align}\n",
"\\partial_0 \\beta^i &= B^{i} \\\\\n",
"\\partial_0 B^i &= \\frac{3}{4} \\partial_{0} \\bar{\\Lambda}^{i} - \\eta B^{i} \\\\\n",
"\\implies \\partial_t \\beta^i &= \\left[\\beta^j \\partial_j \\beta^i\\right] + B^{i} \\\\\n",
"\\partial_t B^i &= \\left[\\beta^j \\partial_j B^i\\right] + \\frac{3}{4} \\partial_{0} \\bar{\\Lambda}^{i} - \\eta B^{i},\n",
"\\end{align}\n",
"where $\\eta$ is the shift damping parameter, and $\\partial_{0} \\bar{\\Lambda}^{i}$ in the right-hand side of the $\\partial_{0} B^{i}$ equation is computed by adding $\\beta^j \\partial_j \\bar{\\Lambda}^i$ to the right-hand side expression given for $\\partial_t \\bar{\\Lambda}^i$ in the BSSN time-evolution equations as listed [here](Tutorial-BSSN_formulation.ipynb), so no explicit time dependence occurs in the right-hand sides of the BSSN evolution equations and the Method of Lines can be applied directly."
]
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"# Step 3.a: Set \\partial_t \\beta^i\n",
"# First import expressions from BSSN_quantities\n",
"BU = Bq.BU\n",
"betU = Bq.betU\n",
"betaU_dupD = Bq.betaU_dupD\n",
"# Define needed quantities\n",
"beta_rhsU = ixp.zerorank1()\n",
"B_rhsU = ixp.zerorank1()\n",
"if par.parval_from_str(thismodule+\"::ShiftEvolutionOption\") == \"GammaDriving2ndOrder_NoCovariant\":\n",
" # Step 3.a.i: Compute right-hand side of beta^i\n",
" # * \\partial_t \\beta^i = \\beta^j \\beta^i_{,j} + B^i\n",
" for i in range(DIM):\n",
" beta_rhsU[i] += BU[i]\n",
" for j in range(DIM):\n",
" beta_rhsU[i] += betaU[j]*betaU_dupD[i][j]\n",
" # Compute right-hand side of B^i:\n",
" eta = par.Cparameters(\"REAL\", thismodule, [\"eta\"],2.0)\n",
"\n",
" # Step 3.a.ii: Compute right-hand side of B^i\n",
" # * \\partial_t B^i = \\beta^j B^i_{,j} + 3/4 * \\partial_0 \\Lambda^i - eta B^i\n",
" # Step 3.a.iii: Define BU_dupD, in terms of derivative of rescaled variable \\bet^i\n",
" BU_dupD = ixp.zerorank2()\n",
" betU_dupD = ixp.declarerank2(\"betU_dupD\",\"nosym\")\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" BU_dupD[i][j] = betU_dupD[i][j]*rfm.ReU[i] + betU[i]*rfm.ReUdD[i][j]\n",
"\n",
" # Step 3.a.iv: Compute \\partial_0 \\bar{\\Lambda}^i = (\\partial_t - \\beta^i \\partial_i) \\bar{\\Lambda}^j\n",
" Lambdabar_partial0 = ixp.zerorank1()\n",
" for i in range(DIM):\n",
" Lambdabar_partial0[i] = Brhs.Lambdabar_rhsU[i]\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" Lambdabar_partial0[j] += -betaU[i]*Brhs.LambdabarU_dupD[j][i]\n",
"\n",
" # Step 3.a.v: Evaluate RHS of B^i:\n",
" for i in range(DIM):\n",
" B_rhsU[i] += sp.Rational(3,4)*Lambdabar_partial0[i] - eta*BU[i]\n",
" for j in range(DIM):\n",
" B_rhsU[i] += betaU[j]*BU_dupD[i][j]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"## Step 3.b: [Brown](https://arxiv.org/abs/0902.3652)'s suggested covariant Gamma-driving shift condition \\[Back to [top](#toc)\\]\n",
"$$\\label{covgammadriving}$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"### Step 3.b.i: The right-hand side of the $\\partial_t \\beta^i$ equation \\[Back to [top](#toc)\\]\n",
"$$\\label{partial_beta}$$\n",
"\n",
"This is [Brown's](https://arxiv.org/abs/0902.3652) suggested formulation (Eq. 20b; note that Eq. 20a is the same as our lapse condition, as $\\bar{D}_j \\alpha = \\partial_j \\alpha$ for scalar $\\alpha$):\n",
"$$\\partial_t \\beta^i = \\left[\\beta^j \\bar{D}_j \\beta^i\\right] + B^{i}$$\n",
"Based on the definition of covariant derivative, we have\n",
"$$\n",
"\\bar{D}_{j} \\beta^{i} = \\beta^i_{,j} + \\bar{\\Gamma}^i_{mj} \\beta^m,\n",
"$$\n",
"so the above equation becomes\n",
"\\begin{align}\n",
"\\partial_t \\beta^i &= \\left[\\beta^j \\left(\\beta^i_{,j} + \\bar{\\Gamma}^i_{mj} \\beta^m\\right)\\right] + B^{i}\\\\\n",
"&= {\\underbrace {\\textstyle \\beta^j \\beta^i_{,j}}_{\\text{Term 1}}} + \n",
"{\\underbrace {\\textstyle \\beta^j \\bar{\\Gamma}^i_{mj} \\beta^m}_{\\text{Term 2}}} + \n",
"{\\underbrace {\\textstyle B^i}_{\\text{Term 3}}} \n",
"\\end{align}"
]
},
{
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"source": [
"# Step 3.b: The right-hand side of the \\partial_t \\beta^i equation\n",
"if par.parval_from_str(thismodule+\"::ShiftEvolutionOption\") == \"GammaDriving2ndOrder_Covariant\":\n",
" # Step 3.b Option 2: \\partial_t \\beta^i = \\left[\\beta^j \\bar{D}_j \\beta^i\\right] + B^{i}\n",
" # First we need GammabarUDD, defined in Bq.gammabar__inverse_and_derivs()\n",
" Bq.gammabar__inverse_and_derivs()\n",
" GammabarUDD = Bq.GammabarUDD\n",
" # Then compute right-hand side:\n",
" # Term 1: \\beta^j \\beta^i_{,j}\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" beta_rhsU[i] += betaU[j]*betaU_dupD[i][j]\n",
"\n",
" # Term 2: \\beta^j \\bar{\\Gamma}^i_{mj} \\beta^m\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" for m in range(DIM):\n",
" beta_rhsU[i] += betaU[j]*GammabarUDD[i][m][j]*betaU[m]\n",
" # Term 3: B^i\n",
" for i in range(DIM):\n",
" beta_rhsU[i] += BU[i]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"### Step 3.b.ii: The right-hand side of the $\\partial_t B^i$ equation \\[Back to [top](#toc)\\]\n",
"$$\\label{partial_upper_b}$$\n",
"\n",
"$$\\partial_t B^i = \\left[\\beta^j \\bar{D}_j B^i\\right] + \\frac{3}{4}\\left( \\partial_t \\bar{\\Lambda}^{i} - \\beta^j \\bar{D}_j \\bar{\\Lambda}^{i} \\right) - \\eta B^{i}$$\n",
"\n",
"Based on the definition of covariant derivative, we have for vector $V^i$\n",
"$$\n",
"\\bar{D}_{j} V^{i} = V^i_{,j} + \\bar{\\Gamma}^i_{mj} V^m,\n",
"$$\n",
"so the above equation becomes\n",
"\\begin{align}\n",
"\\partial_t B^i &= \\left[\\beta^j \\left(B^i_{,j} + \\bar{\\Gamma}^i_{mj} B^m\\right)\\right] + \\frac{3}{4}\\left[ \\partial_t \\bar{\\Lambda}^{i} - \\beta^j \\left(\\bar{\\Lambda}^i_{,j} + \\bar{\\Gamma}^i_{mj} \\bar{\\Lambda}^m\\right) \\right] - \\eta B^{i} \\\\\n",
"&= {\\underbrace {\\textstyle \\beta^j B^i_{,j}}_{\\text{Term 1}}} + \n",
"{\\underbrace {\\textstyle \\beta^j \\bar{\\Gamma}^i_{mj} B^m}_{\\text{Term 2}}} + \n",
"{\\underbrace {\\textstyle \\frac{3}{4}\\partial_t \\bar{\\Lambda}^{i}}_{\\text{Term 3}}} -\n",
"{\\underbrace {\\textstyle \\frac{3}{4}\\beta^j \\bar{\\Lambda}^i_{,j}}_{\\text{Term 4}}} -\n",
"{\\underbrace {\\textstyle \\frac{3}{4}\\beta^j \\bar{\\Gamma}^i_{mj} \\bar{\\Lambda}^m}_{\\text{Term 5}}} -\n",
"{\\underbrace {\\textstyle \\eta B^i}_{\\text{Term 6}}}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:21:58.678281Z",
"iopub.status.busy": "2021-03-07T17:21:58.642565Z",
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"shell.execute_reply": "2021-03-07T17:21:58.700720Z"
}
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"outputs": [],
"source": [
"if par.parval_from_str(thismodule+\"::ShiftEvolutionOption\") == \"GammaDriving2ndOrder_Covariant\":\n",
" # Step 3.c: Covariant option:\n",
" # \\partial_t B^i = \\beta^j \\bar{D}_j B^i\n",
" # + \\frac{3}{4} ( \\partial_t \\bar{\\Lambda}^{i} - \\beta^j \\bar{D}_j \\bar{\\Lambda}^{i} )\n",
" # - \\eta B^{i}\n",
" # = \\beta^j B^i_{,j} + \\beta^j \\bar{\\Gamma}^i_{mj} B^m\n",
" # + \\frac{3}{4}[ \\partial_t \\bar{\\Lambda}^{i}\n",
" # - \\beta^j (\\bar{\\Lambda}^i_{,j} + \\bar{\\Gamma}^i_{mj} \\bar{\\Lambda}^m)]\n",
" # - \\eta B^{i}\n",
" # Term 1, part a: First compute B^i_{,j} using upwinded derivative\n",
" BU_dupD = ixp.zerorank2()\n",
" betU_dupD = ixp.declarerank2(\"betU_dupD\",\"nosym\")\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" BU_dupD[i][j] = betU_dupD[i][j]*rfm.ReU[i] + betU[i]*rfm.ReUdD[i][j]\n",
" # Term 1: \\beta^j B^i_{,j}\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" B_rhsU[i] += betaU[j]*BU_dupD[i][j]\n",
" # Term 2: \\beta^j \\bar{\\Gamma}^i_{mj} B^m\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" for m in range(DIM):\n",
" B_rhsU[i] += betaU[j]*GammabarUDD[i][m][j]*BU[m]\n",
" # Term 3: \\frac{3}{4}\\partial_t \\bar{\\Lambda}^{i}\n",
" for i in range(DIM):\n",
" B_rhsU[i] += sp.Rational(3,4)*Brhs.Lambdabar_rhsU[i]\n",
" # Term 4: -\\frac{3}{4}\\beta^j \\bar{\\Lambda}^i_{,j}\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" B_rhsU[i] += -sp.Rational(3,4)*betaU[j]*Brhs.LambdabarU_dupD[i][j]\n",
" # Term 5: -\\frac{3}{4}\\beta^j \\bar{\\Gamma}^i_{mj} \\bar{\\Lambda}^m\n",
" for i in range(DIM):\n",
" for j in range(DIM):\n",
" for m in range(DIM):\n",
" B_rhsU[i] += -sp.Rational(3,4)*betaU[j]*GammabarUDD[i][m][j]*Bq.LambdabarU[m]\n",
" # Term 6: - \\eta B^i\n",
" # eta is a free parameter; we declare it here:\n",
" eta = par.Cparameters(\"REAL\", thismodule, [\"eta\"],2.0)\n",
" for i in range(DIM):\n",
" B_rhsU[i] += -eta*BU[i]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"## Step 3.c: Non-advecting Gamma-driving shift condition (used for evolving \"Static Trumpet\" initial data) \\[Back to [top](#toc)\\]\n",
"$$\\label{statictrumpet_nonadvecgammadriving}$$\n",
"\n",
"\n",
"For the shift vector evolution equation, we desire only that the right-hand sides vanish analytically (although numerical error is expected to result in specious evolution). To this end, we adopt the nonadvecting Gamma-driver condition, given by equations 68a and 68b in [Ruchlin, Etienne, & Baumgarte (2018)](https://arxiv.org/pdf/1712.07658.pdf)\n",
"\n",
"\\begin{align}\n",
"\\partial_t \\beta^i &= B^{i} \\\\\n",
"\\partial_t B^i &= \\frac{3}{4} \\partial_{t} \\bar{\\Lambda}^{i} - \\eta B^{i},\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:21:58.707000Z",
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"shell.execute_reply": "2021-03-07T17:21:58.709124Z"
}
},
"outputs": [],
"source": [
"# Step 3.c: Set \\partial_t \\beta^i\n",
"\n",
"if par.parval_from_str(thismodule+\"::ShiftEvolutionOption\") == \"NonAdvectingGammaDriving\":\n",
" # Step 3.c.i: Compute right-hand side of beta^i\n",
" # * \\partial_t \\beta^i = B^i\n",
" for i in range(DIM):\n",
" beta_rhsU[i] += BU[i]\n",
"\n",
" # Compute right-hand side of B^i:\n",
" eta = par.Cparameters(\"REAL\", thismodule, [\"eta\"],2.0)\n",
"\n",
" # Step 3.c.ii: Compute right-hand side of B^i\n",
" # * \\partial_t B^i = 3/4 * \\partial_t \\Lambda^i - eta B^i\n",
" # Step 3.c.iii: Evaluate RHS of B^i:\n",
" for i in range(DIM):\n",
" B_rhsU[i] += sp.Rational(3,4)*Brhs.Lambdabar_rhsU[i] - eta*BU[i]\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Step 4: Rescale right-hand sides of BSSN gauge equations \\[Back to [top](#toc)\\]\n",
"$$\\label{rescale}$$\n",
"\n",
"Next we rescale the right-hand sides of the BSSN equations so that the evolved variables are $\\left\\{h_{i j},a_{i j},\\text{cf}, K, \\lambda^{i}, \\alpha, \\mathcal{V}^i, \\mathcal{B}^i\\right\\}$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:21:58.762715Z",
"iopub.status.busy": "2021-03-07T17:21:58.747634Z",
"iopub.status.idle": "2021-03-07T17:21:58.765074Z",
"shell.execute_reply": "2021-03-07T17:21:58.765648Z"
},
"scrolled": true
},
"outputs": [],
"source": [
"# Step 4: Rescale the BSSN gauge RHS quantities so that the evolved\n",
"# variables may remain smooth across coord singularities\n",
"vet_rhsU = ixp.zerorank1()\n",
"bet_rhsU = ixp.zerorank1()\n",
"for i in range(DIM):\n",
" vet_rhsU[i] = beta_rhsU[i] / rfm.ReU[i]\n",
" bet_rhsU[i] = B_rhsU[i] / rfm.ReU[i]\n",
"#print(str(Abar_rhsDD[2][2]).replace(\"**\",\"^\").replace(\"_\",\"\").replace(\"xx\",\"x\").replace(\"sin(x2)\",\"Sin[x2]\").replace(\"sin(2*x2)\",\"Sin[2*x2]\").replace(\"cos(x2)\",\"Cos[x2]\").replace(\"detgbaroverdetghat\",\"detg\"))\n",
"#print(str(Dbarbetacontraction).replace(\"**\",\"^\").replace(\"_\",\"\").replace(\"xx\",\"x\").replace(\"sin(x2)\",\"Sin[x2]\").replace(\"detgbaroverdetghat\",\"detg\"))\n",
"#print(betaU_dD)\n",
"#print(str(trK_rhs).replace(\"xx2\",\"xx3\").replace(\"xx1\",\"xx2\").replace(\"xx0\",\"xx1\").replace(\"**\",\"^\").replace(\"_\",\"\").replace(\"sin(xx2)\",\"Sinx2\").replace(\"xx\",\"x\").replace(\"sin(2*x2)\",\"Sin2x2\").replace(\"cos(x2)\",\"Cosx2\").replace(\"detgbaroverdetghat\",\"detg\"))\n",
"#print(str(bet_rhsU[0]).replace(\"xx2\",\"xx3\").replace(\"xx1\",\"xx2\").replace(\"xx0\",\"xx1\").replace(\"**\",\"^\").replace(\"_\",\"\").replace(\"sin(xx2)\",\"Sinx2\").replace(\"xx\",\"x\").replace(\"sin(2*x2)\",\"Sin2x2\").replace(\"cos(x2)\",\"Cosx2\").replace(\"detgbaroverdetghat\",\"detg\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Step 5: Code Validation against `BSSN.BSSN_gauge_RHSs` NRPy+ module \\[Back to [top](#toc)\\]\n",
"$$\\label{code_validation}$$\n",
"\n",
"Here, as a code validation check, we verify agreement in the SymPy expressions for the RHSs of the BSSN gauge equations between\n",
"\n",
"1. this tutorial and \n",
"2. the NRPy+ [BSSN.BSSN_gauge_RHSs](../edit/BSSN/BSSN_gauge_RHSs.py) module.\n",
"\n",
"By default, we analyze the RHSs in Spherical coordinates and with the covariant Gamma-driving second-order shift condition, though other coordinate systems & gauge conditions may be chosen."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:21:58.772900Z",
"iopub.status.busy": "2021-03-07T17:21:58.772178Z",
"iopub.status.idle": "2021-03-07T17:21:59.150639Z",
"shell.execute_reply": "2021-03-07T17:21:59.151117Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Consistency check between BSSN.BSSN_gauge_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.\n",
"alpha_rhs - bssnrhs.alpha_rhs = 0\n",
"vet_rhsU[0] - bssnrhs.vet_rhsU[0] = 0\n",
"bet_rhsU[0] - bssnrhs.bet_rhsU[0] = 0\n",
"vet_rhsU[1] - bssnrhs.vet_rhsU[1] = 0\n",
"bet_rhsU[1] - bssnrhs.bet_rhsU[1] = 0\n",
"vet_rhsU[2] - bssnrhs.vet_rhsU[2] = 0\n",
"bet_rhsU[2] - bssnrhs.bet_rhsU[2] = 0\n"
]
}
],
"source": [
"# Step 5: We already have SymPy expressions for BSSN gauge RHS expressions\n",
"# in terms of other SymPy variables. Even if we reset the\n",
"# list of NRPy+ gridfunctions, these *SymPy* expressions for\n",
"# BSSN RHS variables *will remain unaffected*.\n",
"#\n",
"# Here, we will use the above-defined BSSN gauge RHS expressions\n",
"# to validate against the same expressions in the\n",
"# BSSN/BSSN_gauge_RHSs.py file, to ensure consistency between\n",
"# this tutorial and the module itself.\n",
"#\n",
"# Reset the list of gridfunctions, as registering a gridfunction\n",
"# twice will spawn an error.\n",
"gri.glb_gridfcs_list = []\n",
"\n",
"\n",
"# Step 5.a: Call the BSSN_gauge_RHSs() function from within the\n",
"# BSSN/BSSN_gauge_RHSs.py module,\n",
"# which should generate exactly the same expressions as above.\n",
"import BSSN.BSSN_gauge_RHSs as Bgrhs\n",
"par.set_parval_from_str(\"BSSN.BSSN_gauge_RHSs::ShiftEvolutionOption\",\"GammaDriving2ndOrder_Covariant\")\n",
"Bgrhs.BSSN_gauge_RHSs()\n",
"\n",
"print(\"Consistency check between BSSN.BSSN_gauge_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.\")\n",
"\n",
"print(\"alpha_rhs - bssnrhs.alpha_rhs = \" + str(alpha_rhs - Bgrhs.alpha_rhs))\n",
"\n",
"for i in range(DIM):\n",
" print(\"vet_rhsU[\"+str(i)+\"] - bssnrhs.vet_rhsU[\"+str(i)+\"] = \" + str(vet_rhsU[i] - Bgrhs.vet_rhsU[i]))\n",
" print(\"bet_rhsU[\"+str(i)+\"] - bssnrhs.bet_rhsU[\"+str(i)+\"] = \" + str(bet_rhsU[i] - Bgrhs.bet_rhsU[i]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Step 6: Output this notebook to $\\LaTeX$-formatted PDF file \\[Back to [top](#toc)\\]\n",
"$$\\label{latex_pdf_output}$$\n",
"\n",
"The following code cell converts this Jupyter notebook into a proper, clickable $\\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename\n",
"[Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.pdf](Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.pdf) (Note that clicking on this link may not work; you may need to open the PDF file through another means.)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"execution": {
"iopub.execute_input": "2021-03-07T17:21:59.155623Z",
"iopub.status.busy": "2021-03-07T17:21:59.154923Z",
"iopub.status.idle": "2021-03-07T17:22:03.242672Z",
"shell.execute_reply": "2021-03-07T17:22:03.243409Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Created Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.tex, and compiled\n",
" LaTeX file to PDF file Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.pdf\n"
]
}
],
"source": [
"import cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface\n",
"cmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs\")"
]
}
],
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"kernelspec": {
"display_name": "Python 3",
"language": "python",
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"file_extension": ".py",
"mimetype": "text/x-python",
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