Tutorial-GRHD_Equations-Cartesian.ipynb

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   "cell_type": "markdown",
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   "source": [
    "<script async src=\"https://www.googletagmanager.com/gtag/js?id=UA-59152712-8\"></script>\n",
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    "  window.dataLayer = window.dataLayer || [];\n",
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    "\n",
    "# Equations of General Relativistic Hydrodynamics (GRHD)\n",
    "\n",
    "## Authors: Zach Etienne & Patrick Nelson\n",
    "\n",
    "## This notebook documents and constructs a number of quantities useful for building symbolic (SymPy) expressions for the equations of general relativistic hydrodynamics (GRHD), using the same (Valencia) formalism as `IllinoisGRMHD`\n",
    "\n",
    "**Notebook Status:** <font color='orange'><b> Self-Validated </b></font>\n",
    "\n",
    "**Validation Notes:** This tutorial notebook has been confirmed to be self-consistent with its corresponding NRPy+ module, as documented [below](#code_validation). **Additional validation tests may have been performed, but are as yet, undocumented. (TODO)**\n",
    "\n",
    "## Introduction\n",
    "\n",
    "We write the equations of general relativistic hydrodynamics in conservative form as follows (adapted from Eqs. 41-44 of [Duez et al](https://arxiv.org/pdf/astro-ph/0503420.pdf):\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\ \\partial_t \\rho_* &+& \\partial_j \\left(\\rho_* v^j\\right) = 0 \\\\\n",
    "\\partial_t \\tilde{\\tau} &+& \\partial_j \\left(\\alpha^2 \\sqrt{\\gamma} T^{0j} - \\rho_* v^j \\right) = s \\\\\n",
    "\\partial_t \\tilde{S}_i &+& \\partial_j \\left(\\alpha \\sqrt{\\gamma} T^j{}_i \\right) = \\frac{1}{2} \\alpha\\sqrt{\\gamma} T^{\\mu\\nu} g_{\\mu\\nu,i},\n",
    "\\end{eqnarray}\n",
    "where we assume $T^{\\mu\\nu}$ is the stress-energy tensor of a perfect fluid:\n",
    "$$\n",
    "T^{\\mu\\nu} = \\rho_0 h u^{\\mu} u^{\\nu} + P g^{\\mu\\nu},\n",
    "$$\n",
    "the $s$ source term is given in terms of ADM quantities via\n",
    "$$\n",
    "s = \\alpha \\sqrt{\\gamma}\\left[\\left(T^{00}\\beta^i\\beta^j + 2 T^{0i}\\beta^j + T^{ij} \\right)K_{ij}\n",
    "- \\left(T^{00}\\beta^i + T^{0i} \\right)\\partial_i\\alpha \\right],\n",
    "$$\n",
    "\n",
    "and \n",
    "\\begin{align}\n",
    "v^j &= \\frac{u^j}{u^0} \\\\\n",
    "\\rho_* &= \\alpha\\sqrt{\\gamma} \\rho_0 u^0 \\\\\n",
    "h &= 1 + \\epsilon + \\frac{P}{\\rho_0}.\n",
    "\\end{align}\n",
    "\n",
    "Also we will write the 4-metric in terms of the ADM 3-metric, lapse, and shift using standard equations.\n",
    "\n",
    "Thus the full set of input variables include:\n",
    "* Spacetime quantities:\n",
    "    * ADM quantities $\\alpha$, $\\beta^i$, $\\gamma_{ij}$, $K_{ij}$\n",
    "* Hydrodynamical quantities:\n",
    "    * Rest-mass density $\\rho_0$\n",
    "    * Pressure $P$\n",
    "    * Internal energy $\\epsilon$\n",
    "    * 4-velocity $u^\\mu$\n",
    "\n",
    "For completeness, the rest of the conservative variables are given by\n",
    "\\begin{align}\n",
    "\\tilde{\\tau} &= \\alpha^2\\sqrt{\\gamma} T^{00} - \\rho_* \\\\\n",
    "\\tilde{S}_i  &= \\alpha \\sqrt{\\gamma} T^0{}_i\n",
    "\\end{align}\n",
    "\n",
    "### A Note on Notation\n",
    "\n",
    "As is standard in NRPy+, \n",
    "\n",
    "* Greek indices refer to four-dimensional quantities where the zeroth component indicates temporal (time) component.\n",
    "* Latin indices refer to three-dimensional quantities. This is somewhat counterintuitive since Python always indexes its lists starting from 0. As a result, the zeroth component of three-dimensional quantities will necessarily indicate the first *spatial* direction.\n",
    "\n",
    "For instance, in calculating the first term of $b^2 u^\\mu u^\\nu$, we use Greek indices:\n",
    "\n",
    "```python\n",
    "T4EMUU = ixp.zerorank2(DIM=4)\n",
    "for mu in range(4):\n",
    "    for nu in range(4):\n",
    "        # Term 1: b^2 u^{\\mu} u^{\\nu}\n",
    "        T4EMUU[mu][nu] = smallb2*u4U[mu]*u4U[nu]\n",
    "```\n",
    "\n",
    "When we calculate $\\beta_i = \\gamma_{ij} \\beta^j$, we use Latin indices:\n",
    "```python\n",
    "betaD = ixp.zerorank1(DIM=3)\n",
    "for i in range(3):\n",
    "    for j in range(3):\n",
    "        betaD[i] += gammaDD[i][j] * betaU[j]\n",
    "```\n",
    "\n",
    "As a corollary, any expressions involving mixed Greek and Latin indices will need to offset one set of indices by one: A Latin index in a four-vector will be incremented and a Greek index in a three-vector will be decremented (however, the latter case does not occur in this tutorial notebook). This can be seen when we handle $\\frac{1}{2} \\alpha \\sqrt{\\gamma} T^{\\mu \\nu}_{\\rm EM} \\partial_i g_{\\mu \\nu}$:\n",
    "```python\n",
    "# \\alpha \\sqrt{\\gamma} T^{\\mu \\nu}_{\\rm EM} \\partial_i g_{\\mu \\nu} / 2\n",
    "for i in range(3):\n",
    "    for mu in range(4):\n",
    "        for nu in range(4):\n",
    "            S_tilde_rhsD[i] += alpsqrtgam * T4EMUU[mu][nu] * g4DD_zerotimederiv_dD[mu][nu][i+1] / 2\n",
    "```"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='toc'></a>\n",
    "\n",
    "# Table of Contents\n",
    "$$\\label{toc}$$\n",
    "\n",
    "Each family of quantities is constructed within a given function (**boldfaced** below). This notebook is organized as follows\n",
    "\n",
    "\n",
    "1. [Step 1](#importmodules): Import needed NRPy+ & Python modules\n",
    "1. [Step 2](#stressenergy): Define the stress-energy tensor $T^{\\mu\\nu}$ and $T^\\mu{}_\\nu$:\n",
    "    * **compute_enthalpy()**, **compute_T4UU()**, **compute_T4UD()**: \n",
    "1. [Step 3](#primtoconserv): Writing the conservative variables in terms of the primitive variables:\n",
    "    * **compute_sqrtgammaDET()**, **compute_rho_star()**, **compute_tau_tilde()**, **compute_S_tildeD()**\n",
    "1. [Step 4](#grhdfluxes): Define the fluxes for the GRHD equations\n",
    "    1. [Step 4.a](#rhostarfluxterm): Define $\\rho_*$ flux term for GRHD equations:\n",
    "        * **compute_vU_from_u4U__no_speed_limit()**, **compute_rho_star_fluxU()**:\n",
    "    1. [Step 4.b](#taustildesourceterms) Define $\\tilde{\\tau}$ and $\\tilde{S}_i$ flux terms for GRHD equations:\n",
    "        * **compute_tau_tilde_fluxU()**, **compute_S_tilde_fluxUD()**\n",
    "1. [Step 5](#grhdsourceterms): Define source terms on RHSs of GRHD equations\n",
    "    1. [Step 5.a](#ssourceterm): Define $s$ source term on RHS of $\\tilde{\\tau}$ equation:\n",
    "        * **compute_s_source_term()**\n",
    "    1. [Step 5.b](#stildeisourceterm): Define source term on RHS of $\\tilde{S}_i$ equation\n",
    "        1. [Step 5.b.i](#fourmetricderivs): Compute $g_{\\mu\\nu,i}$ in terms of ADM quantities and their derivatives:\n",
    "            * **compute_g4DD_zerotimederiv_dD()**\n",
    "        1. [Step 5.b.ii](#stildeisource): Compute source term of the $\\tilde{S}_i$ equation: $\\frac{1}{2} \\alpha\\sqrt{\\gamma} T^{\\mu\\nu} g_{\\mu\\nu,i}$:\n",
    "            * **compute_S_tilde_source_termD()**\n",
    "1. [Step 6](#convertvtou): Conversion of $v^i$ to $u^\\mu$ (Courtesy Patrick Nelson):\n",
    "    * **u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit()**, **u4U_in_terms_of_vU__rescale_vU_by_applying_speed_limit()**\n",
    "1. [Step 7](#declarevarsconstructgrhdeqs): Declare ADM and hydrodynamical input variables, and construct GRHD equations\n",
    "1. [Step 8](#code_validation): Code Validation against `GRHD.equations` NRPy+ module\n",
    "1. [Step 9](#latex_pdf_output): Output this notebook to $\\LaTeX$-formatted PDF file"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='importmodules'></a>\n",
    "\n",
    "# Step 1: Import needed NRPy+ & Python modules \\[Back to [top](#toc)\\]\n",
    "$$\\label{importmodules}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 1: Import needed core NRPy+ modules\n",
    "from outputC import nrpyAbs      # NRPy+: Core C code output module\n",
    "import NRPy_param_funcs as par   # NRPy+: parameter interface\n",
    "import sympy as sp               # SymPy: The Python computer algebra package upon which NRPy+ depends\n",
    "import indexedexp as ixp         # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='stressenergy'></a>\n",
    "\n",
    "# Step 2: Define the stress-energy tensor $T^{\\mu\\nu}$ and $T^\\mu{}_\\nu$ \\[Back to [top](#toc)\\]\n",
    "$$\\label{stressenergy}$$\n",
    "\n",
    "Recall from above that\n",
    "\n",
    "$$\n",
    "T^{\\mu\\nu} = \\rho_0 h u^{\\mu} u^{\\nu} + P g^{\\mu\\nu},\n",
    "$$\n",
    "where $h = 1 + \\epsilon + \\frac{P}{\\rho_0}$. Also \n",
    "\n",
    "$$\n",
    "T^\\mu{}_{\\nu} = T^{\\mu\\delta} g_{\\delta \\nu}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 2.a: First define h, the enthalpy:\n",
    "def compute_enthalpy(rho_b,P,epsilon):\n",
    "    global h\n",
    "    h = 1 + epsilon + P/rho_b\n",
    "\n",
    "# Step 2.b: Define T^{mu nu} (a 4-dimensional tensor)\n",
    "def compute_T4UU(gammaDD,betaU,alpha, rho_b,P,epsilon,u4U):\n",
    "    global T4UU\n",
    "\n",
    "    compute_enthalpy(rho_b,P,epsilon)\n",
    "    # Then define g^{mu nu} in terms of the ADM quantities:\n",
    "    import BSSN.ADMBSSN_tofrom_4metric as AB4m\n",
    "    AB4m.g4UU_ito_BSSN_or_ADM(\"ADM\",gammaDD,betaU,alpha)\n",
    "\n",
    "    # Finally compute T^{mu nu}\n",
    "    T4UU = ixp.zerorank2(DIM=4)\n",
    "    for mu in range(4):\n",
    "        for nu in range(4):\n",
    "            T4UU[mu][nu] = rho_b * h * u4U[mu]*u4U[nu] + P*AB4m.g4UU[mu][nu]\n",
    "\n",
    "# Step 2.c: Define T^{mu}_{nu} (a 4-dimensional tensor)\n",
    "def compute_T4UD(gammaDD,betaU,alpha, T4UU):\n",
    "    global T4UD\n",
    "    # Next compute T^mu_nu = T^{mu delta} g_{delta nu}, needed for S_tilde flux.\n",
    "    # First we'll need g_{alpha nu} in terms of ADM quantities:\n",
    "    import BSSN.ADMBSSN_tofrom_4metric as AB4m\n",
    "    AB4m.g4DD_ito_BSSN_or_ADM(\"ADM\",gammaDD,betaU,alpha)\n",
    "    T4UD = ixp.zerorank2(DIM=4)\n",
    "    for mu in range(4):\n",
    "        for nu in range(4):\n",
    "            for delta in range(4):\n",
    "                T4UD[mu][nu] += T4UU[mu][delta]*AB4m.g4DD[delta][nu]"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='primtoconserv'></a>\n",
    "\n",
    "# Step 3: Writing the conservative variables in terms of the primitive variables \\[Back to [top](#toc)\\]\n",
    "$$\\label{primtoconserv}$$\n",
    "\n",
    "Recall from above that the conservative variables may be written as\n",
    "\\begin{align}\n",
    "\\rho_* &= \\alpha\\sqrt{\\gamma} \\rho_0 u^0 \\\\\n",
    "\\tilde{\\tau} &= \\alpha^2\\sqrt{\\gamma} T^{00} - \\rho_* \\\\\n",
    "\\tilde{S}_i  &= \\alpha \\sqrt{\\gamma} T^0{}_i\n",
    "\\end{align}\n",
    "\n",
    "$T^{\\mu\\nu}$ and $T^\\mu{}_\\nu$ have already been defined $-$ all in terms of primitive variables. Thus we'll just need $\\sqrt{\\gamma}=$`gammaDET`, and all conservatives can then be written in terms of other defined quantities, which themselves are written in terms of primitive variables and the ADM metric."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 3: Writing the conservative variables in terms of the primitive variables\n",
    "def compute_sqrtgammaDET(gammaDD):\n",
    "    global sqrtgammaDET\n",
    "    gammaUU, gammaDET = ixp.symm_matrix_inverter3x3(gammaDD)\n",
    "    sqrtgammaDET = sp.sqrt(gammaDET)\n",
    "\n",
    "def compute_rho_star(alpha, sqrtgammaDET, rho_b,u4U):\n",
    "    global rho_star\n",
    "    # Compute rho_star:\n",
    "    rho_star = alpha*sqrtgammaDET*rho_b*u4U[0]\n",
    "\n",
    "def compute_tau_tilde(alpha, sqrtgammaDET, T4UU,rho_star):\n",
    "    global tau_tilde\n",
    "    tau_tilde = alpha**2*sqrtgammaDET*T4UU[0][0] - rho_star\n",
    "\n",
    "def compute_S_tildeD(alpha, sqrtgammaDET, T4UD):\n",
    "    global S_tildeD\n",
    "    S_tildeD = ixp.zerorank1(DIM=3)\n",
    "    for i in range(3):\n",
    "        S_tildeD[i] = alpha*sqrtgammaDET*T4UD[0][i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='grhdfluxes'></a>\n",
    "\n",
    "# Step 4: Define the fluxes for the GRHD equations \\[Back to [top](#toc)\\]\n",
    "$$\\label{grhdfluxes}$$\n",
    "\n",
    "<a id='rhostarfluxterm'></a>\n",
    "\n",
    "## Step 4.a: Define $\\rho_*$ flux term for GRHD equations \\[Back to [top](#toc)\\]\n",
    "$$\\label{rhostarfluxterm}$$\n",
    "\n",
    "Recall from above that\n",
    "\\begin{array}\n",
    "\\ \\partial_t \\rho_* &+ \\partial_j \\left(\\rho_* v^j\\right) = 0.\n",
    "\\end{array}\n",
    "\n",
    "Here we will define the $\\rho_* v^j$ that constitutes the flux of $\\rho_*$, first defining $v^j=u^j/u^0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 4: Define the fluxes for the GRHD equations\n",
    "# Step 4.a: vU from u4U may be needed for computing rho_star_flux from u4U\n",
    "def compute_vU_from_u4U__no_speed_limit(u4U):\n",
    "    global vU\n",
    "    # Now compute v^i = u^i/u^0:\n",
    "    vU = ixp.zerorank1(DIM=3)\n",
    "    for j in range(3):\n",
    "        vU[j] = u4U[j+1]/u4U[0]\n",
    "\n",
    "# Step 4.b: rho_star flux\n",
    "def compute_rho_star_fluxU(vU, rho_star):\n",
    "    global rho_star_fluxU\n",
    "    rho_star_fluxU = ixp.zerorank1(DIM=3)\n",
    "    for j in range(3):\n",
    "        rho_star_fluxU[j] = rho_star*vU[j]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='taustildesourceterms'></a>\n",
    "\n",
    "## Step 4.b: Define $\\tilde{\\tau}$ and $\\tilde{S}_i$ flux terms for GRHD equations \\[Back to [top](#toc)\\]\n",
    "$$\\label{taustildesourceterms}$$\n",
    "\n",
    "Recall from above that\n",
    "\\begin{array}\n",
    "\\ \\partial_t \\tilde{\\tau} &+ \\partial_j \\underbrace{\\left(\\alpha^2 \\sqrt{\\gamma} T^{0j} - \\rho_* v^j \\right)} &= s \\\\\n",
    "\\partial_t \\tilde{S}_i &+ \\partial_j \\underbrace{\\left(\\alpha \\sqrt{\\gamma} T^j{}_i \\right)} &= \\frac{1}{2} \\alpha\\sqrt{\\gamma} T^{\\mu\\nu} g_{\\mu\\nu,i}.\n",
    "\\end{array}\n",
    "\n",
    "Here we will define all terms that go inside the $\\partial_j$'s on the left-hand side of the above equations (i.e., the underbraced expressions):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 4.c: tau_tilde flux\n",
    "def compute_tau_tilde_fluxU(alpha, sqrtgammaDET, vU,T4UU, rho_star):\n",
    "    global tau_tilde_fluxU\n",
    "    tau_tilde_fluxU = ixp.zerorank1(DIM=3)\n",
    "    for j in range(3):\n",
    "        tau_tilde_fluxU[j] = alpha**2*sqrtgammaDET*T4UU[0][j+1] - rho_star*vU[j]\n",
    "\n",
    "# Step 4.d: S_tilde flux\n",
    "def compute_S_tilde_fluxUD(alpha, sqrtgammaDET, T4UD):\n",
    "    global S_tilde_fluxUD\n",
    "    S_tilde_fluxUD = ixp.zerorank2(DIM=3)\n",
    "    for j in range(3):\n",
    "        for i in range(3):\n",
    "            S_tilde_fluxUD[j][i] = alpha*sqrtgammaDET*T4UD[j+1][i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='grhdsourceterms'></a>\n",
    "\n",
    "# Step 5: Define source terms on RHSs of GRHD equations \\[Back to [top](#toc)\\]\n",
    "$$\\label{grhdsourceterms}$$\n",
    "\n",
    "<a id='ssourceterm'></a>\n",
    "\n",
    "## Step 5.a: Define $s$ source term on RHS of $\\tilde{\\tau}$ equation \\[Back to [top](#toc)\\]\n",
    "$$\\label{ssourceterm}$$\n",
    "\n",
    "\n",
    "Recall again from above the $s$ source term on the right-hand side of the $\\tilde{\\tau}$ evolution equation is given in terms of ADM quantities and the stress-energy tensor via\n",
    "$$\n",
    "s = \\underbrace{\\alpha \\sqrt{\\gamma}}_{\\text{Term 3}}\\left[\\underbrace{\\left(T^{00}\\beta^i\\beta^j + 2 T^{0i}\\beta^j + T^{ij} \\right)K_{ij}}_{\\text{Term 1}}\n",
    "\\underbrace{- \\left(T^{00}\\beta^i + T^{0i} \\right)\\partial_i\\alpha}_{\\text{Term 2}} \\right],\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_s_source_term(KDD,betaU,alpha, sqrtgammaDET,alpha_dD, T4UU):\n",
    "    global s_source_term\n",
    "    s_source_term = sp.sympify(0)\n",
    "    # Term 1:\n",
    "    for i in range(3):\n",
    "        for j in range(3):\n",
    "            s_source_term += (T4UU[0][0]*betaU[i]*betaU[j] + 2*T4UU[0][i+1]*betaU[j] + T4UU[i+1][j+1])*KDD[i][j]\n",
    "    # Term 2:\n",
    "    for i in range(3):\n",
    "        s_source_term += -(T4UU[0][0]*betaU[i] + T4UU[0][i+1])*alpha_dD[i]\n",
    "    # Term 3:\n",
    "    s_source_term *= alpha*sqrtgammaDET"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='stildeisourceterm'></a>\n",
    "\n",
    "## Step 5.b: Define source term on RHS of $\\tilde{S}_i$ equation \\[Back to [top](#toc)\\]\n",
    "$$\\label{stildeisourceterm}$$\n",
    "\n",
    "Recall from above\n",
    "$$\n",
    "\\partial_t \\tilde{S}_i + \\partial_j \\left(\\alpha \\sqrt{\\gamma} T^j{}_i \\right) = \\frac{1}{2} \\alpha\\sqrt{\\gamma} T^{\\mu\\nu} g_{\\mu\\nu,i}.\n",
    "$$\n",
    "Our goal here will be to compute\n",
    "$$\n",
    "\\frac{1}{2} \\alpha\\sqrt{\\gamma} T^{\\mu\\nu} g_{\\mu\\nu,i}.\n",
    "$$\n",
    "\n",
    "<a id='fourmetricderivs'></a>\n",
    "\n",
    "### Step 5.b.i: Compute $g_{\\mu\\nu,i}$ in terms of ADM quantities and their derivatives \\[Back to [top](#toc)\\]\n",
    "$$\\label{fourmetricderivs}$$\n",
    "\n",
    "\n",
    "To compute $g_{\\mu\\nu,i}$ we need to evaluate the first derivative of $g_{\\mu\\nu}$ in terms of ADM variables.\n",
    "\n",
    "We are given $\\gamma_{ij}$, $\\alpha$, and $\\beta^i$, and the 4-metric is given in terms of these quantities via\n",
    "$$\n",
    "g_{\\mu\\nu} = \\begin{pmatrix} \n",
    "-\\alpha^2 + \\beta^k \\beta_k & \\beta_i \\\\\n",
    "\\beta_j & \\gamma_{ij}\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "Thus \n",
    "$$\n",
    "g_{\\mu\\nu,k} = \\begin{pmatrix} \n",
    "-2 \\alpha\\alpha_{,i} + \\beta^j_{,k} \\beta_j + \\beta^j \\beta_{j,k} & \\beta_{i,k} \\\\\n",
    "\\beta_{j,k} & \\gamma_{ij,k}\n",
    "\\end{pmatrix},\n",
    "$$\n",
    "where $\\beta_{i} = \\gamma_{ij} \\beta^j$, so\n",
    "$$\n",
    "\\beta_{i,k} = \\gamma_{ij,k} \\beta^j + \\gamma_{ij} \\beta^j_{,k}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def compute_g4DD_zerotimederiv_dD(gammaDD,betaU,alpha, gammaDD_dD,betaU_dD,alpha_dD):\n",
    "    global g4DD_zerotimederiv_dD\n",
    "    # Eq. 2.121 in B&S\n",
    "    betaD = ixp.zerorank1(DIM=3)\n",
    "    for i in range(3):\n",
    "        for j in range(3):\n",
    "            betaD[i] += gammaDD[i][j]*betaU[j]\n",
    "\n",
    "    betaDdD = ixp.zerorank2(DIM=3)\n",
    "    for i in range(3):\n",
    "        for j in range(3):\n",
    "            for k in range(3):\n",
    "                # Recall that betaD[i] = gammaDD[i][j]*betaU[j] (Eq. 2.121 in B&S)\n",
    "                betaDdD[i][k] += gammaDD_dD[i][j][k]*betaU[j] + gammaDD[i][j]*betaU_dD[j][k]\n",
    "\n",
    "    # Eq. 2.122 in B&S\n",
    "    g4DD_zerotimederiv_dD = ixp.zerorank3(DIM=4)\n",
    "    for k in range(3):\n",
    "        # Recall that g4DD[0][0] = -alpha^2 + betaU[j]*betaD[j]\n",
    "        g4DD_zerotimederiv_dD[0][0][k+1] += -2*alpha*alpha_dD[k]\n",
    "        for j in range(3):\n",
    "            g4DD_zerotimederiv_dD[0][0][k+1] += betaU_dD[j][k]*betaD[j] + betaU[j]*betaDdD[j][k]\n",
    "\n",
    "    for i in range(3):\n",
    "        for k in range(3):\n",
    "            # Recall that g4DD[i][0] = g4DD[0][i] = betaD[i]\n",
    "            g4DD_zerotimederiv_dD[i+1][0][k+1] = g4DD_zerotimederiv_dD[0][i+1][k+1] = betaDdD[i][k]\n",
    "    for i in range(3):\n",
    "        for j in range(3):\n",
    "            for k in range(3):\n",
    "                # Recall that g4DD[i][j] = gammaDD[i][j]\n",
    "                g4DD_zerotimederiv_dD[i+1][j+1][k+1] = gammaDD_dD[i][j][k]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='stildeisource'></a>\n",
    "\n",
    "### Step 5.b.ii: Compute source term of the $\\tilde{S}_i$ equation: $\\frac{1}{2} \\alpha\\sqrt{\\gamma} T^{\\mu\\nu} g_{\\mu\\nu,i}$ \\[Back to [top](#toc)\\]\n",
    "$$\\label{stildeisource}$$\n",
    "\n",
    "Now that we've computed `g4DD_zerotimederiv_dD`$=g_{\\mu\\nu,i}$, the $\\tilde{S}_i$ evolution equation source term may be quickly constructed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 5.b.ii: Compute S_tilde source term\n",
    "def compute_S_tilde_source_termD(alpha, sqrtgammaDET,g4DD_zerotimederiv_dD, T4UU):\n",
    "    global S_tilde_source_termD\n",
    "    S_tilde_source_termD = ixp.zerorank1(DIM=3)\n",
    "    for i in range(3):\n",
    "        for mu in range(4):\n",
    "            for nu in range(4):\n",
    "                S_tilde_source_termD[i] += sp.Rational(1,2)*alpha*sqrtgammaDET*T4UU[mu][nu]*g4DD_zerotimederiv_dD[mu][nu][i+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='convertvtou'></a>\n",
    "\n",
    "# Step 6: Conversion of $v^i$ to $u^\\mu$ (Courtesy Patrick Nelson) \\[Back to [top](#toc)\\]\n",
    "$$\\label{convertvtou}$$\n",
    "\n",
    "According to Eqs. 9-11 of [the IllinoisGRMHD paper](https://arxiv.org/pdf/1501.07276.pdf), the Valencia 3-velocity $v^i_{(n)}$ is related to the 4-velocity $u^\\mu$ via\n",
    "\n",
    "\\begin{align}\n",
    "\\alpha v^i_{(n)} &= \\frac{u^i}{u^0} + \\beta^i \\\\\n",
    "\\implies u^i &= u^0 \\left(\\alpha v^i_{(n)} - \\beta^i\\right)\n",
    "\\end{align}\n",
    "\n",
    "Defining $v^i = \\frac{u^i}{u^0}$, we get\n",
    "\n",
    "$$v^i = \\alpha v^i_{(n)} - \\beta^i,$$\n",
    "\n",
    "and in terms of this variable we get\n",
    "\n",
    "\\begin{align}\n",
    "g_{00} \\left(u^0\\right)^2 + 2 g_{0i} u^0 u^i + g_{ij} u^i u^j &= \\left(u^0\\right)^2 \\left(g_{00} + 2 g_{0i} v^i + g_{ij} v^i v^j\\right)\\\\\n",
    "\\implies u^0 &= \\pm \\sqrt{\\frac{-1}{g_{00} + 2 g_{0i} v^i + g_{ij} v^i v^j}} \\\\\n",
    "&= \\pm \\sqrt{\\frac{-1}{(-\\alpha^2 + \\beta^2) + 2 \\beta_i v^i + \\gamma_{ij} v^i v^j}} \\\\\n",
    "&= \\pm \\sqrt{\\frac{1}{\\alpha^2 - \\gamma_{ij}\\left(\\beta^i + v^i\\right)\\left(\\beta^j + v^j\\right)}}\\\\\n",
    "&= \\pm \\sqrt{\\frac{1}{\\alpha^2 - \\alpha^2 \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}\\\\\n",
    "&= \\pm \\frac{1}{\\alpha}\\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}\n",
    "\\end{align}\n",
    "\n",
    "Generally speaking, numerical errors will occasionally drive expressions under the radical to either negative values or potentially enormous values (corresponding to enormous Lorentz factors). Thus a reliable approach for computing $u^0$ requires that we first rewrite the above expression in terms of the Lorentz factor squared: $\\Gamma^2=\\left(\\alpha u^0\\right)^2$:\n",
    "\\begin{align}\n",
    "u^0 &= \\pm \\frac{1}{\\alpha}\\sqrt{\\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}}}\\\\\n",
    "\\implies \\left(\\alpha u^0\\right)^2 &= \\frac{1}{1 - \\gamma_{ij}v^i_{(n)}v^j_{(n)}} \\\\\n",
    "\\implies \\gamma_{ij}v^i_{(n)}v^j_{(n)} &= 1 - \\frac{1}{\\left(\\alpha u^0\\right)^2} \\\\\n",
    "&= 1 - \\frac{1}{\\Gamma^2}\n",
    "\\end{align}\n",
    "\n",
    "In order for the bottom expression to hold true, the left-hand side must be between 0 and 1. Again, this is not guaranteed due to the appearance of numerical errors. In fact, a robust algorithm will not allow $\\Gamma^2$ to become too large (which might contribute greatly to the stress-energy of a given gridpoint), so let's define the largest allowed Lorentz factor as $\\Gamma_{\\rm max}$.\n",
    "\n",
    "Then our algorithm for computing $u^0$ is as follows:\n",
    "\n",
    "If\n",
    "$$R=\\gamma_{ij}v^i_{(n)}v^j_{(n)}>1 - \\frac{1}{\\Gamma_{\\rm max}^2},$$ \n",
    "then adjust the 3-velocity $v^i$ as follows:\n",
    "\n",
    "$$v^i_{(n)} \\to \\sqrt{\\frac{1 - \\frac{1}{\\Gamma_{\\rm max}^2}}{R}}v^i_{(n)}.$$\n",
    "\n",
    "After this rescaling, we are then guaranteed that if $R$ is recomputed, it will be set to its ceiling value $R=R_{\\rm max} = 1 - \\frac{1}{\\Gamma_{\\rm max}^2}$.\n",
    "\n",
    "Then, regardless of whether the ceiling on $R$ was applied, $u^0$ can be safely computed via\n",
    "\n",
    "$$\n",
    "u^0 = \\frac{1}{\\alpha \\sqrt{1-R}},\n",
    "$$\n",
    "and the remaining components $u^i$ via\n",
    "$$\n",
    "u^i = u^0 v^i.\n",
    "$$\n",
    "\n",
    "In summary our algorithm for computing $u^{\\mu}$ from $v^i = \\frac{u^i}{u^0}$ is as follows:\n",
    "\n",
    "1. Choose a maximum Lorentz factor $\\Gamma_{\\rm max}$=`GAMMA_SPEED_LIMIT`, and define $v^i_{(n)} = \\frac{1}{\\alpha}\\left( \\frac{u^i}{u^0} + \\beta^i\\right)$.\n",
    "1. Compute $R=\\gamma_{ij}v^i_{(n)}v^j_{(n)}=1 - \\frac{1}{\\Gamma^2}$\n",
    "1. If $R \\le 1 - \\frac{1}{\\Gamma_{\\rm max}^2}$, then skip the next step.\n",
    "1. Otherwise if $R > 1 - \\frac{1}{\\Gamma_{\\rm max}^2}$ then adjust $v^i_{(n)}\\to \\sqrt{\\frac{1 - \\frac{1}{\\Gamma_{\\rm max}^2}}{R}}v^i_{(n)}$, which will force $R=R_{\\rm max}$.\n",
    "1. Given the $R$ computed in the above step, $u^0 = \\frac{1}{\\alpha \\sqrt{1-R}}$, and $u^i=u^0 v^i$.\n",
    "\n",
    "While the above algorithm is quite robust, its `if()` statement in the fourth step is not very friendly to NRPy+ or an optimizing C compiler, as it would require NRPy+ to generate separate C kernels for each branch of the `if()`. Let's instead try the following trick, which Roland Haas taught us. \n",
    "\n",
    "Define $R^*$ as\n",
    "\n",
    "$$\n",
    "R^* = \\frac{1}{2} \\left(R_{\\rm max} + R - |R_{\\rm max} - R| \\right).\n",
    "$$\n",
    "\n",
    "If $R>R_{\\rm max}$, then $|R_{\\rm max} - R|=R - R_{\\rm max}$, and we get:\n",
    "\n",
    "$$\n",
    "R^* = \\frac{1}{2} \\left(R_{\\rm max} + R - (R - R_{\\rm max}) \\right) = \\frac{1}{2} \\left(2 R_{\\rm max}\\right) = R_{\\rm max}\n",
    "$$\n",
    "\n",
    "If $R\\le R_{\\rm max}$, then $|R_{\\rm max} - R|=R_{\\rm max} - R$, and we get:\n",
    "\n",
    "$$\n",
    "R^* = \\frac{1}{2} \\left(R_{\\rm max} + R - (R_{\\rm max} - R) \\right) = \\frac{1}{2} \\left(2 R\\right) = R\n",
    "$$\n",
    "\n",
    "Then we can rescale *all* $v^i_{(n)}$ via\n",
    "\n",
    "$$\n",
    "v^i_{(n)} \\to v^i_{(n)} \\sqrt{\\frac{R^*}{R}},\n",
    "$$\n",
    "\n",
    "though we must be very careful to carefully handle the case in which $R=0$. To avoid any problems in this case, we simply adjust the above rescaling by adding a tiny number [`TINYDOUBLE`](https://en.wikipedia.org/wiki/Tiny_Bubbles) to $R$ in the denominator, typically `1e-100`:\n",
    "\n",
    "$$\n",
    "v^i_{(n)} \\to v^i_{(n)} \\sqrt{\\frac{R^*}{R + {\\rm TINYDOUBLE}}}.\n",
    "$$\n",
    "\n",
    "Finally, $u^0$ can be immediately and safely computed, via:\n",
    "$$\n",
    "u^0 = \\frac{1}{\\alpha \\sqrt{1-R^*}},\n",
    "$$\n",
    "and $u^i$ via \n",
    "$$\n",
    "u^i = u^0 v^i = u^0 \\left(\\alpha v^i_{(n)} - \\beta^i\\right).\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 6.a: Convert Valencia 3-velocity v_{(n)}^i into u^\\mu, and apply a speed limiter\n",
    "#           Speed-limited ValenciavU is output to rescaledValenciavU global.\n",
    "def u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(alpha,betaU,gammaDD, ValenciavU):\n",
    "    # Inputs:  Metric lapse alpha, shift betaU, 3-metric gammaDD, Valencia 3-velocity ValenciavU\n",
    "    # Outputs (as globals): u4U_ito_ValenciavU, rescaledValenciavU\n",
    "\n",
    "    # R = gamma_{ij} v^i v^j\n",
    "    R = sp.sympify(0)\n",
    "    for i in range(3):\n",
    "        for j in range(3):\n",
    "            R += gammaDD[i][j]*ValenciavU[i]*ValenciavU[j]\n",
    "\n",
    "    thismodule = \"GRHD\"\n",
    "    # The default value isn't terribly important here, since we can overwrite in the main C code\n",
    "    GAMMA_SPEED_LIMIT = par.Cparameters(\"REAL\", thismodule, \"GAMMA_SPEED_LIMIT\", 10.0)  # Default value based on\n",
    "                                                                                        # IllinoisGRMHD.\n",
    "    # GiRaFFE default = 2000.0\n",
    "    Rmax = 1 - 1 / (GAMMA_SPEED_LIMIT * GAMMA_SPEED_LIMIT)\n",
    "    # Now, we set Rstar = min(Rmax,R):\n",
    "    # If R <  Rmax, then Rstar = 0.5*(Rmax+R-Rmax+R) = R\n",
    "    # If R >= Rmax, then Rstar = 0.5*(Rmax+R+Rmax-R) = Rmax\n",
    "    Rstar = sp.Rational(1, 2) * (Rmax + R - nrpyAbs(Rmax - R))\n",
    "\n",
    "    # We add TINYDOUBLE to R below to avoid a 0/0, which occurs when\n",
    "    #    ValenciavU == 0 for all Valencia 3-velocity components.\n",
    "    # \"Those tiny *doubles* make me warm all over\n",
    "    #  with a feeling that I'm gonna love you till the end of time.\"\n",
    "    #    - Adapted from Connie Francis' \"Tiny Bubbles\"\n",
    "    TINYDOUBLE = par.Cparameters(\"#define\",thismodule,\"TINYDOUBLE\",1e-100)\n",
    "\n",
    "    # The rescaled (speed-limited) Valencia 3-velocity\n",
    "    #     is given by, v_{(n)}^i = sqrt{Rstar/R} v^i\n",
    "    global rescaledValenciavU\n",
    "    rescaledValenciavU = ixp.zerorank1(DIM=3)\n",
    "    for i in range(3):\n",
    "        # If R == 0, then Rstar == 0, so sqrt( Rstar/(R+TINYDOUBLE) )=sqrt(0/1e-100) = 0\n",
    "        #   If your velocities are of order 1e-100 and this is physically\n",
    "        #   meaningful, there must be something wrong with your unit conversion.\n",
    "        rescaledValenciavU[i] = ValenciavU[i]*sp.sqrt(Rstar/(R + TINYDOUBLE))\n",
    "\n",
    "    # Finally compute u^mu in terms of Valenciav^i\n",
    "    # u^0 = 1/(alpha-sqrt(1-R^*))\n",
    "    global u4U_ito_ValenciavU\n",
    "    u4U_ito_ValenciavU = ixp.zerorank1(DIM=4)\n",
    "    u4U_ito_ValenciavU[0] = 1/(alpha*sp.sqrt(1-Rstar))\n",
    "    # u^i = u^0 ( alpha v^i_{(n)} - beta^i ), where v^i_{(n)} is the Valencia 3-velocity\n",
    "    for i in range(3):\n",
    "        u4U_ito_ValenciavU[i+1] = u4U_ito_ValenciavU[0] * (alpha * rescaledValenciavU[i] - betaU[i])\n",
    "\n",
    "# Step 6.b: Convert v^i into u^\\mu, and apply a speed limiter.\n",
    "#           Speed-limited vU is output to rescaledvU global.\n",
    "def u4U_in_terms_of_vU__rescale_vU_by_applying_speed_limit(alpha,betaU,gammaDD, vU):\n",
    "    ValenciavU = ixp.zerorank1(DIM=3)\n",
    "    for i in range(3):\n",
    "        ValenciavU[i] = (vU[i] + betaU[i])/alpha\n",
    "    u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(alpha,betaU,gammaDD, ValenciavU)\n",
    "    # Since ValenciavU is written in terms of vU,\n",
    "    #   u4U_ito_ValenciavU is actually u4U_ito_vU\n",
    "    global u4U_ito_vU\n",
    "    u4U_ito_vU = ixp.zerorank1(DIM=4)\n",
    "    for mu in range(4):\n",
    "        u4U_ito_vU[mu] = u4U_ito_ValenciavU[mu]\n",
    "    # Finally compute the rescaled (speed-limited) vU\n",
    "    global rescaledvU\n",
    "    rescaledvU = ixp.zerorank1(DIM=3)\n",
    "    for i in range(3):\n",
    "        rescaledvU[i] = alpha * rescaledValenciavU[i] - betaU[i]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='declarevarsconstructgrhdeqs'></a>\n",
    "\n",
    "# Step 7: Declare ADM and hydrodynamical input variables, and construct GRHD equations \\[Back to [top](#toc)\\]\n",
    "$$\\label{declarevarsconstructgrhdeqs}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# First define hydrodynamical quantities\n",
    "u4U = ixp.declarerank1(\"u4U\", DIM=4)\n",
    "rho_b,P,epsilon = sp.symbols('rho_b P epsilon',real=True)\n",
    "\n",
    "# Then ADM quantities\n",
    "gammaDD = ixp.declarerank2(\"gammaDD\",\"sym01\",DIM=3)\n",
    "KDD     = ixp.declarerank2(\"KDD\"    ,\"sym01\",DIM=3)\n",
    "betaU   = ixp.declarerank1(\"betaU\", DIM=3)\n",
    "alpha   = sp.symbols('alpha', real=True)\n",
    "\n",
    "# First compute stress-energy tensor T4UU and T4UD:\n",
    "compute_T4UU(gammaDD,betaU,alpha, rho_b,P,epsilon,u4U)\n",
    "compute_T4UD(gammaDD,betaU,alpha, T4UU)\n",
    "\n",
    "# Next sqrt(gamma)\n",
    "compute_sqrtgammaDET(gammaDD)\n",
    "\n",
    "# Compute conservative variables in terms of primitive variables\n",
    "compute_rho_star( alpha, sqrtgammaDET, rho_b,u4U)\n",
    "compute_tau_tilde(alpha, sqrtgammaDET, T4UU,rho_star)\n",
    "compute_S_tildeD( alpha, sqrtgammaDET, T4UD)\n",
    "\n",
    "# Then compute v^i from u^mu\n",
    "compute_vU_from_u4U__no_speed_limit(u4U)\n",
    "\n",
    "# Next compute fluxes of conservative variables\n",
    "compute_rho_star_fluxU(                                   vU,      rho_star)\n",
    "compute_tau_tilde_fluxU(alpha,              sqrtgammaDET, vU,T4UU, rho_star)\n",
    "compute_S_tilde_fluxUD( alpha,              sqrtgammaDET,    T4UD)\n",
    "\n",
    "# Then declare derivatives & compute g4DD_zerotimederiv_dD\n",
    "gammaDD_dD = ixp.declarerank3(\"gammaDD_dD\",\"sym01\",DIM=3)\n",
    "betaU_dD   = ixp.declarerank2(\"betaU_dD\"  ,\"nosym\",DIM=3)\n",
    "alpha_dD   = ixp.declarerank1(\"alpha_dD\"          ,DIM=3)\n",
    "compute_g4DD_zerotimederiv_dD(gammaDD,betaU,alpha, gammaDD_dD,betaU_dD,alpha_dD)\n",
    "\n",
    "# Then compute source terms on tau_tilde and S_tilde equations\n",
    "compute_s_source_term(KDD,betaU,alpha, sqrtgammaDET,alpha_dD, T4UU)\n",
    "compute_S_tilde_source_termD(    alpha, sqrtgammaDET,g4DD_zerotimederiv_dD,   T4UU)\n",
    "\n",
    "# Then compute the 4-velocities in terms of an input Valencia 3-velocity testValenciavU[i]\n",
    "testValenciavU = ixp.declarerank1(\"testValenciavU\",DIM=3)\n",
    "u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(alpha,betaU,gammaDD, testValenciavU)\n",
    "\n",
    "# Finally compute the 4-velocities in terms of an input 3-velocity testvU[i] = u^i/u^0\n",
    "testvU = ixp.declarerank1(\"testvU\",DIM=3)\n",
    "u4U_in_terms_of_vU__rescale_vU_by_applying_speed_limit(alpha,betaU,gammaDD, testvU)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='code_validation'></a>\n",
    "\n",
    "# Step 8: Code Validation against `GRHD.equations` NRPy+ module \\[Back to [top](#toc)\\]\n",
    "$$\\label{code_validation}$$\n",
    "\n",
    "As a code validation check, we verify agreement in the SymPy expressions for the GRHD equations generated in\n",
    "1. this tutorial versus\n",
    "2. the NRPy+ [GRHD.equations](../edit/GRHD/equations.py) module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "import GRHD.equations as Ge\n",
    "\n",
    "# First compute stress-energy tensor T4UU and T4UD:\n",
    "Ge.compute_T4UU(gammaDD,betaU,alpha, rho_b,P,epsilon,u4U)\n",
    "Ge.compute_T4UD(gammaDD,betaU,alpha, Ge.T4UU)\n",
    "\n",
    "# Next sqrt(gamma)\n",
    "Ge.compute_sqrtgammaDET(gammaDD)\n",
    "\n",
    "# Compute conservative variables in terms of primitive variables\n",
    "Ge.compute_rho_star( alpha, Ge.sqrtgammaDET, rho_b,u4U)\n",
    "Ge.compute_tau_tilde(alpha, Ge.sqrtgammaDET, Ge.T4UU,Ge.rho_star)\n",
    "Ge.compute_S_tildeD( alpha, Ge.sqrtgammaDET, Ge.T4UD)\n",
    "\n",
    "# Then compute v^i from u^mu\n",
    "Ge.compute_vU_from_u4U__no_speed_limit(u4U)\n",
    "\n",
    "# Next compute fluxes of conservative variables\n",
    "Ge.compute_rho_star_fluxU (                                     Ge.vU,         Ge.rho_star)\n",
    "Ge.compute_tau_tilde_fluxU(alpha,              Ge.sqrtgammaDET, Ge.vU,Ge.T4UU, Ge.rho_star)\n",
    "Ge.compute_S_tilde_fluxUD (alpha,              Ge.sqrtgammaDET,       Ge.T4UD)\n",
    "\n",
    "# Then declare derivatives & compute g4DD_zerotimederiv_dD\n",
    "# gammaDD_dD = ixp.declarerank3(\"gammaDD_dD\",\"sym01\",DIM=3)\n",
    "# betaU_dD   = ixp.declarerank2(\"betaU_dD\"  ,\"nosym\",DIM=3)\n",
    "# alpha_dD   = ixp.declarerank1(\"alpha_dD\"          ,DIM=3)\n",
    "Ge.compute_g4DD_zerotimederiv_dD(gammaDD,betaU,alpha, gammaDD_dD,betaU_dD,alpha_dD)\n",
    "\n",
    "# Finally compute source terms on tau_tilde and S_tilde equations\n",
    "Ge.compute_s_source_term(KDD,betaU,alpha, Ge.sqrtgammaDET,alpha_dD, Ge.T4UU)\n",
    "Ge.compute_S_tilde_source_termD(   alpha, Ge.sqrtgammaDET,Ge.g4DD_zerotimederiv_dD,Ge.T4UU)\n",
    "\n",
    "GetestValenciavU = ixp.declarerank1(\"testValenciavU\")\n",
    "Ge.u4U_in_terms_of_ValenciavU__rescale_ValenciavU_by_applying_speed_limit(alpha, betaU, gammaDD, GetestValenciavU)\n",
    "\n",
    "GetestvU = ixp.declarerank1(\"testvU\")\n",
    "Ge.u4U_in_terms_of_vU__rescale_vU_by_applying_speed_limit(                alpha, betaU, gammaDD, GetestvU)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ALL TESTS PASSED!\n"
     ]
    }
   ],
   "source": [
    "all_passed=True\n",
    "def comp_func(expr1,expr2,basename,prefixname2=\"Ge.\"):\n",
    "    if str(expr1-expr2)!=\"0\":\n",
    "        print(basename+\" - \"+prefixname2+basename+\" = \"+ str(expr1-expr2))\n",
    "        all_passed=False\n",
    "\n",
    "def gfnm(basename,idx1,idx2=None,idx3=None):\n",
    "    if idx2 is None:\n",
    "        return basename+\"[\"+str(idx1)+\"]\"\n",
    "    if idx3 is None:\n",
    "        return basename+\"[\"+str(idx1)+\"][\"+str(idx2)+\"]\"\n",
    "    return basename+\"[\"+str(idx1)+\"][\"+str(idx2)+\"][\"+str(idx3)+\"]\"\n",
    "\n",
    "expr_list = []\n",
    "exprcheck_list = []\n",
    "namecheck_list = []\n",
    "\n",
    "namecheck_list.extend([\"sqrtgammaDET\",\"rho_star\",\"tau_tilde\",\"s_source_term\"])\n",
    "exprcheck_list.extend([Ge.sqrtgammaDET,Ge.rho_star,Ge.tau_tilde,Ge.s_source_term])\n",
    "expr_list.extend([sqrtgammaDET,rho_star,tau_tilde,s_source_term])\n",
    "for mu in range(4):\n",
    "    namecheck_list.extend([gfnm(\"u4_ito_ValenciavU\",mu),gfnm(\"u4U_ito_vU\",mu)])\n",
    "    exprcheck_list.extend([   Ge.u4U_ito_ValenciavU[mu],   Ge.u4U_ito_vU[mu]])\n",
    "    expr_list.extend(        [   u4U_ito_ValenciavU[mu],      u4U_ito_vU[mu]])\n",
    "    for nu in range(4):\n",
    "        namecheck_list.extend([gfnm(\"T4UU\",mu,nu),gfnm(\"T4UD\",mu,nu)])\n",
    "        exprcheck_list.extend([Ge.T4UU[mu][nu],Ge.T4UD[mu][nu]])\n",
    "        expr_list.extend([T4UU[mu][nu],T4UD[mu][nu]])\n",
    "        for delta in range(4):\n",
    "            namecheck_list.extend([gfnm(\"g4DD_zerotimederiv_dD\",mu,nu,delta)])\n",
    "            exprcheck_list.extend([Ge.g4DD_zerotimederiv_dD[mu][nu][delta]])\n",
    "            expr_list.extend([g4DD_zerotimederiv_dD[mu][nu][delta]])\n",
    "\n",
    "\n",
    "for i in range(3):\n",
    "    namecheck_list.extend([gfnm(\"S_tildeD\",i),gfnm(\"vU\",i),gfnm(\"rho_star_fluxU\",i),\n",
    "                           gfnm(\"tau_tilde_fluxU\",i),gfnm(\"S_tilde_source_termD\",i),\n",
    "                           gfnm(\"rescaledValenciavU\",i), gfnm(\"rescaledvU\",i)])\n",
    "    exprcheck_list.extend([Ge.S_tildeD[i],Ge.vU[i],Ge.rho_star_fluxU[i],\n",
    "                           Ge.tau_tilde_fluxU[i],Ge.S_tilde_source_termD[i],\n",
    "                           Ge.rescaledValenciavU[i],Ge.rescaledvU[i]])\n",
    "    expr_list.extend([S_tildeD[i],vU[i],rho_star_fluxU[i],\n",
    "                      tau_tilde_fluxU[i],S_tilde_source_termD[i],\n",
    "                      rescaledValenciavU[i],rescaledvU[i]])\n",
    "    for j in range(3):\n",
    "        namecheck_list.extend([gfnm(\"S_tilde_fluxUD\",i,j)])\n",
    "        exprcheck_list.extend([Ge.S_tilde_fluxUD[i][j]])\n",
    "        expr_list.extend([S_tilde_fluxUD[i][j]])\n",
    "\n",
    "for i in range(len(expr_list)):\n",
    "    comp_func(expr_list[i],exprcheck_list[i],namecheck_list[i])\n",
    "\n",
    "import sys\n",
    "if all_passed:\n",
    "    print(\"ALL TESTS PASSED!\")\n",
    "else:\n",
    "    print(\"ERROR: AT LEAST ONE TEST DID NOT PASS\")\n",
    "    sys.exit(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='latex_pdf_output'></a>\n",
    "\n",
    "# Step 9: 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-GRHD_Equations-Cartesian.pdf](Tutorial-GRHD_Equations-Cartesian.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": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Created Tutorial-GRHD_Equations-Cartesian.tex, and compiled LaTeX file to\n",
      "    PDF file Tutorial-GRHD_Equations-Cartesian.pdf\n"
     ]
    }
   ],
   "source": [
    "import cmdline_helper as cmd    # NRPy+: Multi-platform Python command-line interface\n",
    "cmd.output_Jupyter_notebook_to_LaTeXed_PDF(\"Tutorial-GRHD_Equations-Cartesian\")"
   ]
  }
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