===================
The Euler Equations
===================

The Euler equations consist of the mass conservation

.. math::
  :label: euler.mass

  \newcommand{\bvec}[1]{\mathbf{#1}}
  \newcommand{\defeq}{\buildrel{\text{def}}\over{=}}
  \newcommand{\dpd}[3][]{\mathinner{
  \dfrac{\partial{^{#1}}#2}{\partial{#3^{#1}}}
  }}

  \frac{\partial\rho}{\partial t} + \frac{\partial\rho v_j}{\partial x_j}
    = 0

momentum conservation

.. math::
  :label: euler.momentum

  \frac{\partial\rho v_i}{\partial t} 
    + \frac{\partial\rho v_iv_j}{\partial x_j}
    = \frac{\partial p}{\partial x_j} + \rho b_i

and energy conservation

.. math::
  :label: euler.energy

  \frac{\partial}{\partial t}
      \left[\rho\left( e + \frac{v_k^2}{2} \right)\right]
    + \frac{\partial}{\partial x_j}
        \left[\rho\left( e + \frac{v_k^2}{2} \right)v_j\right]
    = \rho \dot{q} - \frac{\partial pv_j}{\partial x_j} + \rho b_jv_j

Einstein's index summation convention was used.

Equations :eq:`euler.mass`, :eq:`euler.momentum`, and :eq:`euler.energy` aren't
closed even if we choose :math:`\bvec{b}` and :math:`\dot{q}` as given.  We
have 5 equtions but 6 unknowns (:math:`\rho`, :math:`\bvec{v}`, :math:`p`, and
:math:`e`).  To close the system of equations, I use the equation of state:

.. math::
  :label: euler:eos

  p = \rho RT

Internal energy is related to temperate:

.. math::
  :label: euler:internal

  e = c_vT = \frac{RT}{\gamma-1} = \frac{1}{\gamma-1}\frac{p}{\rho}

With the additional two equations (Eqs. :eq:`euler:eos` and
:eq:`euler:internal`) and one variable :math:`T`, the equations are closed.

Vector Flux Function
====================

Define the conservation variables:

.. math::
  :label: euler:unknown

  \bvec{u} \defeq \left(\begin{array}{c}
    u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5
  \end{array}\right) = \left(\begin{array}{c}
    \rho \\ \rho v_1 \\ \rho v_2 \\ \rho v_3 \\
    \rho\left(e+\frac{v_k^2}{2}\right)
  \end{array}\right)

Aided by writing the pressure with :math:`\bvec{u}`:

.. math::

  p = (\gamma-1)\left(u_5 - \frac{u_2^2+u_3^2+u_4^2}{2u_1}\right)

the conservation equations (Eqs. :eq:`euler.mass`, :eq:`euler.momentum`, and
:eq:`euler.energy`) can be cast to use only :math:`\bvec{u}`:

.. math::
  :label: euler:gov1

  \frac{\partial u_1}{\partial t}
    + \frac{\partial u_2}{\partial x_1}
    + \frac{\partial u_3}{\partial x_2}
    + \frac{\partial u_4}{\partial x_3} = 0

.. math::
  :label: euler:gov2

  \begin{aligned} &\frac{\partial u_2}{\partial t}
    + \frac{\partial}{\partial x_1}\left(\frac{u_2^2}{u_1}\right)
    + \frac{\partial}{\partial x_2}\left(\frac{u_2u_3}{u_1}\right)
    + \frac{\partial}{\partial x_3}\left(\frac{u_2u_4}{u_1}\right) = \\
    &\quad -\frac{\partial}{\partial x_1}\left[
        (\gamma-1)\left(u_5 - \frac{u_2^2+u_3^2+u_4^2}{2u_1}\right)
      \right] + b_1u_1
  \end{aligned}

.. math::
  :label: euler:gov3

  \begin{aligned} &\frac{\partial u_3}{\partial t}
    + \frac{\partial}{\partial x_1}\left(\frac{u_2u_3}{u_1}\right)
    + \frac{\partial}{\partial x_2}\left(\frac{u_3^2}{u_1}\right)
    + \frac{\partial}{\partial x_3}\left(\frac{u_3u_4}{u_1}\right) = \\
    &\quad -\frac{\partial}{\partial x_2}\left[
        (\gamma-1)\left(u_5 - \frac{u_2^2+u_3^2+u_4^2}{2u_1}\right)
      \right] + b_2u_1
  \end{aligned}

.. math::
  :label: euler:gov4

  \begin{aligned} &\frac{\partial u_4}{\partial t}
    + \frac{\partial}{\partial x_1}\left(\frac{u_2u_4}{u_1}\right)
    + \frac{\partial}{\partial x_2}\left(\frac{u_3u_4}{u_1}\right)
    + \frac{\partial}{\partial x_3}\left(\frac{u_4^2}{u_1}\right) = \\
    &\quad -\frac{\partial}{\partial x_3}\left[
        (\gamma-1)\left(u_5 - \frac{u_2^2+u_3^2+u_4^2}{2u_1}\right)
      \right] + b_3u_1
  \end{aligned}

.. math::
  :label: euler:gov5

  \begin{aligned} &\frac{\partial u_5}{\partial t}
    + \frac{\partial}{\partial x_1}\left(\frac{u_2u_5}{u_1}\right)
    + \frac{\partial}{\partial x_2}\left(\frac{u_3u_5}{u_1}\right)
    + \frac{\partial}{\partial x_3}\left(\frac{u_4u_5}{u_1}\right) = \\
    &\quad - \frac{\partial}{\partial x_1}\left[
        (\gamma-1)\left(u_5 - \frac{u_2^2+u_3^2+u_4^2}{2u_1}\right)
        \frac{u_2}{u_1}
      \right] \\
    &\quad - \frac{\partial}{\partial x_2}\left[
        (\gamma-1)\left(u_5 - \frac{u_2^2+u_3^2+u_4^2}{2u_1}\right)
        \frac{u_3}{u_1}
      \right] \\
    &\quad - \frac{\partial}{\partial x_3}\left[
        (\gamma-1)\left(u_5 - \frac{u_2^2+u_3^2+u_4^2}{2u_1}\right)
        \frac{u_4}{u_1}
      \right]
    + \rho\dot{q} + b_1u_2 + b_2u_3 + b_3u_4
  \end{aligned}

Then organize Eqs. :eq:`euler:gov1` -- :eq:`euler:gov5` into a vector form:

.. math::
  :label: euler:vec

  \frac{\partial\bvec{u}}{\partial t}
    + \sum_{\mu=1}^3 \frac{\partial\bvec{f}^{(\mu)}}{\partial x_{\mu}}
    = \bvec{s}

The flux functions are defined as:

.. math::
  :label: euler:flux1

  \bvec{f}^{(1)} &= \left(\begin{array}{c}
    f^{(1)}_1 \\ f^{(1)}_2 \\ f^{(1)}_3 \\ f^{(1)}_4 \\ f^{(1)}_5
  \end{array}\right) \defeq \left(\begin{array}{l}
    u_2 \\
    (\gamma-1)u_5 - \frac{\gamma-3}{2}\frac{u_2^2}{u_1}
      - \frac{\gamma-1}{2}\frac{u_3^2}{u_1}
      - \frac{\gamma-1}{2}\frac{u_4^2}{u_1} \\
    \frac{u_2u_3}{u_1} \\
    \frac{u_2u_4}{u_1} \\
    \gamma\frac{u_2u_5}{u_1}
      - \frac{\gamma-1}{2}\frac{u_2^2+u_3^2+u_4^2}{u_1}\frac{u_2}{u_1}
  \end{array}\right)
  
.. math::
  :label: euler:flux2

  \bvec{f}^{(2)} &= \left(\begin{array}{c}
    f^{(2)}_1 \\ f^{(2)}_2 \\ f^{(2)}_3 \\ f^{(2)}_4 \\ f^{(2)}_5
  \end{array}\right) \defeq \left(\begin{array}{l}
    u_3 \\
    \frac{u_2u_3}{u_1} \\
    (\gamma-1)u_5 - \frac{\gamma-1}{2}\frac{u_2^2}{u_1}
      - \frac{\gamma-3}{2}\frac{u_3^2}{u_1}
      - \frac{\gamma-1}{2}\frac{u_4^2}{u_1} \\
    \frac{u_3u_4}{u_1} \\
    \gamma\frac{u_3u_5}{u_1}
      - \frac{\gamma-1}{2}\frac{u_2^2+u_3^2+u_4^2}{u_1}\frac{u_3}{u_1}
  \end{array}\right)

.. math::
  :label: euler:flux3

  \bvec{f}^{(3)} &= \left(\begin{array}{c}
    f^{(3)}_1 \\ f^{(3)}_2 \\ f^{(3)}_3 \\ f^{(3)}_4 \\ f^{(3)}_5
  \end{array}\right) \defeq \left(\begin{array}{l}
    u_4 \\
    \frac{u_2u_4}{u_1} \\
    \frac{u_3u_4}{u_1} \\
    (\gamma-1)u_5 - \frac{\gamma-1}{2}\frac{u_2^2}{u_1}
      - \frac{\gamma-1}{2}\frac{u_3^2}{u_1}
      - \frac{\gamma-3}{2}\frac{u_4^2}{u_1} \\
    \gamma\frac{u_4u_5}{u_1}
      - \frac{\gamma-1}{2}\frac{u_2^2+u_3^2+u_4^2}{u_1}\frac{u_4}{u_1}
  \end{array}\right)

At the right-hand side, the source term is

.. math::
  :label: euler:sterm

  \bvec{s} = \left(\begin{array}{c}
    s_1 \\ s_2 \\ s_3 \\ s_4 \\ s_5
  \end{array}\right) \defeq \left(\begin{array}{l}
    0 \\ b_1u_1 \\ b_2u_1 \\ b_3u_3 \\ \dot{q}u_1 + b_1u_2 + b_2u_3 + b_3u_4
  \end{array}\right)

Quasi-linear System Equation
============================

Expand Eq. :eq:`euler:vec` to an index form:

.. math::
  :label: euler:idx

  \frac{\partial u_m}{\partial t}
    + \sum_{\mu=1}^3 \frac{\partial f^{(\mu)}_m}{\partial x_{\mu}}
    = s_m, \quad m = 1, \ldots, 5

Because we want to construct an inviscid baseline solver, later we will drop
the source term from Eq. :eq:`euler:idx`.

Define

.. math::

  u_{mt} &\defeq \dpd{u_m}{t}, \\
  u_{mx_{\mu}} &\defeq \dpd{u_m}{x_{\mu}}, \\
  f^{(\mu)}_{m,l} &\defeq \dpd{f^{(\mu)}_m}{u_l}
  
where :math:`\mu = 1, 2, 3,` and :math:`m, l = 1, 2, \ldots, 5`.

Aided by the above definition, we rewrite the equation to a matrix-vector form:

.. math::
  :label: euler:qlinear

  \dpd{\bvec{u}}{t} + \sum_{\mu=1}^3
                      \mathrm{A}^{(\mu)} \dpd{\bvec{u}}{x_{\mu}} = 0

where :math:`\mathrm{A}^{(1)}`, :math:`\mathrm{A}^{(2)}`, and
:math:`\mathrm{A}^{(3)}` are the Jacobian matrices, of which the components are
defined as

.. math::

  \left[\mathrm{A}^{(\mu)}\right]_{ml} \defeq f^{(\mu)}_{m,l}, 
  \quad m, l = 1, \ldots, 5

and tabulated in what follows.

Component group 1:

.. math::
  :label: euler:jaco0

  f^{(1)}_{1,1} &= f^{(1)}_{1,3} = f^{(1)}_{1,4} = f^{(1)}_{1,5} = \\
  f^{(2)}_{1,1} &= f^{(2)}_{1,2} = f^{(2)}_{1,4} = f^{(2)}_{1,5} = \\
  f^{(3)}_{1,1} &= f^{(3)}_{1,2} = f^{(3)}_{1,3} = f^{(3)}_{1,5} = 0, \\
  f^{(1)}_{1,2} &= f^{(2)}_{1,3} = f^{(3)}_{1,4} = 1

Component group 2 (for :math:`A^{(1)}`):

.. math::
  :label: euler:jaco1

  f^{(1)}_{2,1} &= \frac{\gamma-3}{2}\frac{u_2^2}{u_1^2}
    + \frac{\gamma-1}{2}\frac{u_3^2}{u_1^2}
    + \frac{\gamma-1}{2}\frac{u_4^2}{u_1^2}, \\
  f^{(1)}_{2,2} &= -(\gamma-3)\frac{u_2}{u_1}, \quad
  f^{(1)}_{2,3} = -(\gamma-1)\frac{u_3}{u_1}, \quad
  f^{(1)}_{2,4} = -(\gamma-1)\frac{u_4}{u_1}, \quad
  f^{(1)}_{2,5} = \gamma-1, \\
  f^{(1)}_{3,1} &= -\frac{u_2u_3}{u_1^2}, \quad
  f^{(1)}_{3,2} = \frac{u_3}{u_1}, \quad
  f^{(1)}_{3,3} = \frac{u_2}{u_1}, \quad
  f^{(1)}_{3,4} = f^{(1)}_{3,5} = 0, \\
  f^{(1)}_{4,1} &= -\frac{u_2u_4}{u_1^2}, \quad
  f^{(1)}_{4,2} = \frac{u_4}{u_1}, \quad
  f^{(1)}_{4,4} = \frac{u_2}{u_1}, \quad
  f^{(1)}_{4,3} = f^{(1)}_{4,5} = 0, \\
  f^{(1)}_{5,1} &= -\gamma\frac{u_2u_5}{u_1^2}
    + (\gamma-1)\frac{u_2^2+u_3^2+u_4^2}{u_1^2}\frac{u_2}{u_1}, \quad
  f^{(1)}_{5,2} = \gamma\frac{u_5}{u_1}
    - \frac{\gamma-1}{2}\frac{3u_2^2 + u_3^2 + u_4^2}{u_1^2}, \\
  f^{(1)}_{5,3} &= -(\gamma-1)\frac{u_2u_3}{u_1^2}, \quad
  f^{(1)}_{5,4} = -(\gamma-1)\frac{u_2u_4}{u_1^2}, \quad
  f^{(1)}_{5,5} = \gamma\frac{u_2}{u_1}

Component group 3 (for :math:`A^{(2)}`):

.. math::
  :label: euler:jaco2

  f^{(2)}_{2,1} &= -\frac{u_2u_3}{u_1^2}, \quad
  f^{(2)}_{2,2} = \frac{u_3}{u_1}, \quad
  f^{(2)}_{2,3} = \frac{u_2}{u_1}, \quad
  f^{(2)}_{2,4} = f^{(2)}_{2,5} = 0, \\
  f^{(2)}_{3,1} &= \frac{\gamma-1}{2}\frac{u_2^2}{u_1^2}
    + \frac{\gamma-3}{2}\frac{u_3^2}{u_1^2}
    + \frac{\gamma-1}{2}\frac{u_4^2}{u_1^2}, \\
  f^{(2)}_{3,2} &= -(\gamma-1)\frac{u_2}{u_1}, \quad
  f^{(2)}_{3,3} = -(\gamma-3)\frac{u_3}{u_1}, \quad
  f^{(2)}_{3,4} = -(\gamma-1)\frac{u_4}{u_1}, \quad
  f^{(2)}_{3,5} = \gamma-1, \\
  f^{(2)}_{4,1} &= -\frac{u_3u_4}{u_1^2}, \quad
  f^{(2)}_{4,3} = \frac{u_4}{u_1}, \quad
  f^{(2)}_{4,4} = \frac{u_3}{u_1}, \quad
  f^{(2)}_{4,2} = f^{(2)}_{4,5} = 0, \\
  f^{(2)}_{5,1} &= -\gamma\frac{u_3u_5}{u_1^2}
    + (\gamma-1)\frac{u_2^2+u_3^2+u_4^2}{u_1^2}\frac{u_3}{u_1}, \quad
  f^{(2)}_{5,3} = \gamma\frac{u_5}{u_1}
    - \frac{\gamma-1}{2}\frac{u_2^2 + 3u_3^2 + u_4^2}{u_1^2}, \\
  f^{(2)}_{5,2} &= -(\gamma-1)\frac{u_2u_3}{u_1^2}, \quad
  f^{(2)}_{5,4} = -(\gamma-1)\frac{u_3u_4}{u_1^2}, \quad
  f^{(2)}_{5,5} = \gamma\frac{u_3}{u_1}

Component group 4 (for :math:`A^{(3)}`):

.. math::
  :label: euler:jaco3

  f^{(3)}_{2,1} &= -\frac{u_2u_4}{u_1^2}, \quad
  f^{(3)}_{2,2} = \frac{u_4}{u_1}, \quad
  f^{(3)}_{2,4} = \frac{u_2}{u_1}, \quad
  f^{(3)}_{2,3} = f^{(3)}_{2,5} = 0, \\
  f^{(3)}_{3,1} &= -\frac{u_3u_4}{u_1^2}, \quad
  f^{(3)}_{3,3} = \frac{u_4}{u_1}, \quad
  f^{(3)}_{3,4} = \frac{u_3}{u_1}, \quad
  f^{(3)}_{3,2} = f^{(3)}_{3,5} = 0, \\
  f^{(3)}_{4,1} &= \frac{\gamma-1}{2}\frac{u_2^2}{u_1^2}
    + \frac{\gamma-1}{2}\frac{u_3^2}{u_1^2}
    + \frac{\gamma-3}{2}\frac{u_4^2}{u_1^2}, \\
  f^{(3)}_{4,2} &= -(\gamma-1)\frac{u_2}{u_1}, \quad
  f^{(3)}_{4,3} = -(\gamma-1)\frac{u_3}{u_1}, \quad
  f^{(3)}_{4,4} = -(\gamma-3)\frac{u_4}{u_1}, \quad
  f^{(3)}_{4,5} = \gamma-1, \\
  f^{(3)}_{5,1} &= -\gamma\frac{u_4u_5}{u_1^2}
    + (\gamma-1)\frac{u_2^2+u_3^2+u_4^2}{u_1^2}\frac{u_4}{u_1}, \quad
  f^{(3)}_{5,4} = \gamma\frac{u_5}{u_1}
    - \frac{\gamma-1}{2}\frac{u_2^2 + u_3^2 + 3u_4^2}{u_1^2}, \\
  f^{(3)}_{5,2} &= -(\gamma-1)\frac{u_2u_4}{u_1^2}, \quad
  f^{(3)}_{5,3} = -(\gamma-1)\frac{u_3u_4}{u_1^2}, \quad
  f^{(3)}_{5,5} = \gamma\frac{u_4}{u_1}

Nomenclature
============

:math:`\bvec{x} \defeq (x_1, x_2, x_3)^t`
  Space vector.

:math:`t`
  Time.

:math:`\rho`
  Mass density.

:math:`\bvec{v} \defeq (v_1, v_2, v_3)^t`
  Flow velocity vector.

:math:`p`
  Pressure.

:math:`\bvec{b} \defeq (b_1, b_2, b_3)^t`
  Body force vector.

:math:`e`
  Internal energy density per unit mass.

:math:`\dot{q}`
  Heat generation rate per unit volume.

:math:`R`
  Universal gas constant.

:math:`T`
  Temperature.

:math:`c_v`
  Specific heat at constant volume.

:math:`c_p`
  Specific heat at constant pressure.

:math:`\gamma \defeq c_p/c_v`
  Ratio of specific heat.

:math:`\bvec{u} \defeq (u_1, u_2, u_3, u_4, u_5)^t`
  Conservation variables.

:math:`\bvec{f}^{(1)}, \bvec{f}^{(2)}, \bvec{f}^{(3)}`
  Vector flux functions.  :math:`\bvec{f}^{(\mu)} \defeq (f^{(\mu)}_1,
  f^{(\mu)}_2, f^{(\mu)}_3, f^{(\mu)}_4, f^{(\mu)}_5)^t` where :math:`\mu = 1,
  2, 3`.

:math:`\bvec{s} \defeq (s_1, s_2, s_3, s_4, s_5)^t`
  Source term.

:math:`\mathrm{A}^{(1)}, \mathrm{A}^{(2)}, \mathrm{A}^{(3)}`
  Jacobian matrices.