Changed intro text to GRMHD tutorial

Tutorial-Derivation_of_MHD_Evolution_Equations.ipynb → Tutorial-Derivation_of_GRMHD_Evolution_Equations.ipynb

@@ -13,37 +13,30 @@
     "  gtag('config', 'UA-59152712-8');\n",
     "</script>\n",
     "\n",
-    "# IllinoisGRMHD: Basic equations and modules\n",
+    "# Derivation of the GRMHD Evolution Equations\n",
     "\n",
-    "## Authors: Leo Werneck & Zach Etienne\n",
+    "## Samuel Cupp\n",
     "\n",
-    "<font color='red'>**This module is currently under development**</font>\n",
-    "\n",
-    "## This module introduces the basic equations solved by IllinoisGRMHD and provides an introduction to the modules inside the code.\n",
+    "## This module derives the evolution equations for the conservative GRMHD variables from the conservation equations.\n",
     "\n",
     "## Introduction:\n",
     "\n",
-    "[`IllinoisGRMHD`](http://arxiv.org/abs/1501.07276) solves the equations of General Relativistic MagnetoHydroDynamics (GRMHD) using a high-resolution shock capturing scheme. It is a rewrite of the Illinois Numerical Relativity (ILNR) group's GRMHD code, and generates results that agree to roundoff error with that original code. Its feature set coincides with the features of the ILNR group's recent code (ca. 2009-2014), which was used in their modeling of the following systems:\n",
-    "\n",
-    "1. Magnetized circumbinary disk accretion onto binary black holes\n",
-    "2. Magnetized black hole-neutron star mergers\n",
-    "3. Magnetized Bondi flow, Bondi-Hoyle-Littleton accretion\n",
-    "4. White dwarf-neutron star mergers\n",
+    "When considering the evolution of magnetohydrodynamics in a general relativistic setting, there are generally two choices for evolving the system. One can evolve the primitive variables\n",
     "\n",
-    "`IllinoisGRMHD` is particularly good at modeling GRMHD flows into black holes without the need for excision. Its [HARM-based conservative-to-primitive solver](https://arxiv.org/abs/astro-ph/0512420) has also been modified to check the physicality of conservative variables prior to primitive inversion, and move them into the physical range if they become unphysical.\n",
+    "$$\n",
+    "\\mathbf{P} = \\left\\{\\rho_0,P,v^i,B^i\\right\\}\n",
+    "$$\n",
     "\n",
-    "Currently IllinoisGRMHD consists of\n",
+    "or the conservative variables\n",
     "\n",
-    "1. the Piecewise Parabolic Method (PPM) for reconstruction, \n",
-    "2. the Harten, Lax, van Leer (HLL/HLLE) approximate Riemann solver, and\n",
-    "3. a modified HARM Conservative-to-Primitive solver. \n",
+    "$$\n",
+    "\\mathbf{C} = \\left\\{\\rho_*,\\tilde{\\tau},\\tilde{S}_i,\\tilde{B}^i\\right\\}.\n",
+    "$$\n",
     "\n",
-    "`IllinoisGRMHD` evolves the vector potential $A_{\\mu}$ (on staggered grids) instead of the magnetic fields ($B^i$) directly, to guarantee that the magnetic fields will remain divergenceless even at AMR boundaries. On uniform resolution grids, this vector potential formulation produces results equivalent to those generated using the standard, staggered flux-CT scheme. This scheme is based on that of [Del Zanna *et al.* (2003)](https://arxiv.org/abs/astro-ph/0210618).\n",
+    "The details of the individual variables in each set are discussed more in [Step 2](#suitable_equations). Evolving the conservative variables has the benefit of automatically ensuring that the constraints are satisfied, but at the cost of requiring complicated reconstruction of the primitive variables. IllinoisGRMHD evolves $\\mathbf{C}$, and so I focus on the evolution equations for the conservative variables. The derivation in this document is based on the work by Duez _et al_ (citation below), and I have rederived their results.\n",
     "\n",
-    "### Required and recommended citations:\n",
-    "* **(Required)** Etienne, Z. B., Paschalidis, V., Haas R., Mösta P., and Shapiro, S. L. IllinoisGRMHD: an open-source, user-friendly GRMHD code for dynamical spacetimes. Class. Quantum Grav. 32 (2015) 175009. ([arxiv:1501.07276](http://arxiv.org/abs/1501.07276)).\n",
-    "* **(Required)** Noble, S. C., Gammie, C. F., McKinney, J. C., Del Zanna, L. Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics. Astrophysical Journal, 641, 626 (2006) ([astro-ph/0512420](https://arxiv.org/abs/astro-ph/0512420)).\n",
-    "* **(Recommended)** Del Zanna, L., Bucciantini N., Londrillo, P. An efficient shock-capturing central-type scheme for multidimensional relativistic flows - II. Magnetohydrodynamics. A&A 400 (2) 397-413 (2003). DOI: 10.1051/0004-6361:20021641 ([astro-ph/0210618](https://arxiv.org/abs/astro-ph/0210618))."
+    "### Original Source\n",
+    "* M. D. Duez, Y. T. Liu, S. L. Shapiro, and B. C. Stephens. Relativistic magnetohydrodynamics in dynamical spacetimes: Numerical methods and tests. Phys. Rev. D 72, 024028 (2005). ([arxiv:0802.3210](https://arxiv.org/abs/0802.3210))."
    ]
   },
   {
@@ -76,8 +69,7 @@
    "source": [
     "<a id='EM_tensor'></a>\n",
     "\n",
-    "# Step 1: Deriving $T_{EM}^{\\mu\\nu}$ \\[Back to [top](#toc)\\]\n",
-    "$$\\label{EM_tensor}$$\n",
+    "C$$\\label{EM_tensor}$$\n",
     "\n",
     "In order to find evolution equations for the conserved quantities for the system, we start by finding a convenient form of the electromagnetic energy-momentum tensor $T_{EM}^{\\mu\\nu}$, which is defined as\n",
     "\n",
@@ -149,10 +141,10 @@
     "  + 2B^{\\mu} B_{\\mu}\n",
     "\\end{align}\n",
     "\n",
-    "I now consider two specific choices of $\\xi^\\mu$: the normal observer $n^\\mu$ and the observer co-moving with the fluid $u^\\mu$. For the normal observer, $n_\\mu=(\\alpha,0,0,0)$. This simplifies the second term because only the zero component can survive for the indices contracted with $n_\\mu$. However, $\\epsilon^{\\sigma\\alpha\\beta\\mu}$ cannot repeat indices. Therefore, \n",
+    "I now consider two specific choices of $\\xi^\\mu$: the normal observer $n^\\mu$ and the observer co-moving with the fluid $u^\\mu$. For the normal observer, $n_\\mu=(-\\alpha,0,0,0)$. This simplifies the second term because only the zero component can survive for the indices contracted with $n_\\mu$. However, $\\epsilon^{\\sigma\\alpha\\beta\\mu}$ cannot repeat indices. Therefore, \n",
     "\n",
     "\\begin{align}\n",
-    "n_\\alpha n_\\sigma \\epsilon^{\\sigma\\alpha\\beta\\mu} &= \\alpha n_\\sigma \\epsilon^{\\sigma 0\\beta\\mu} \\\\\n",
+    "n_\\alpha n_\\sigma \\epsilon^{\\sigma\\alpha\\beta\\mu} &= -\\alpha n_\\sigma \\epsilon^{\\sigma 0\\beta\\mu} \\\\\n",
     "&= \\alpha^2 \\epsilon^{0 0\\beta\\mu} \\\\\n",
     "&= 0\n",
     "\\end{align}\n",