Editing Euler equations (fluid dynamics) (section)

===Conservation form===
{{See also|Conservation equation|}}
The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for [[computational fluid dynamics]] simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods
called conservative methods.{{sfn|Toro|1999|p= 24}}

The '''free Euler equations are conservative''', in the sense they are equivalent to a conservation equation:
<math display="block">
\frac{\partial \mathbf y}{\partial t}+ \nabla \cdot \mathbf F ={\mathbf 0},
</math>
or simply in Einstein notation:
<math display="block">
\frac{\partial y_j}{\partial t}+
\frac{\partial f_{ij}}{\partial r_i}= 0_i,
</math>
where the conservation quantity <math>\mathbf y</math> in this case is a vector, and <math>\mathbf F</math> is a [[flux]] matrix. This can be simply proved.

{{hidden
|Demonstration of the conservation form
|First, the following identities hold:
<math display="block">\nabla \cdot (w \mathbf I) = \mathbf I \cdot \nabla w + w \nabla \cdot \mathbf I = \nabla w </math>
<math display="block">\mathbf u \cdot \nabla \cdot \mathbf u = \nabla \cdot (\mathbf u \otimes \mathbf u)</math>
where <math>\otimes</math> denotes the [[outer product]]. The same identities expressed in [[Einstein notation]] are:
<math display="block">\partial_i\left(w \delta_{ij}\right) = \delta_{ij} \partial_i w + w \partial_i \delta_{ij} = \delta_{ij} \partial_i w =  \partial_j w</math>
<math display="block">u_j \partial_i u_i = \partial_i \left(u_i u_j\right)</math>
where {{mvar|I}} is the [[identity matrix]] with dimension {{mvar|N}} and {{mvar|δ<sub>ij</sub>}} its general element, the Kroenecker delta.

Thanks to these vector identities, the incompressible Euler equations with constant and uniform density and without external field can be put in the so-called ''conservation'' (or Eulerian) differential form, with vector notation:
<math display="block">\left\{\begin{align} 
  {\partial\mathbf{u} \over \partial t} + \nabla \cdot \left(\mathbf{u} \otimes \mathbf{u} + w\mathbf{I}\right) &= \mathbf{0} \\
  {\partial 0 \over \partial t} + \nabla \cdot \mathbf{u} &= 0,
\end{align}\right.</math>
or with Einstein notation:
<math display="block">\left\{\begin{align} 
  \partial_t u_j + \partial_i \left(u_i u_j + w \delta_{ij}\right) &= 0 \\
                                     \partial_t 0 + \partial_j u_j &= 0,
\end{align}\right.</math>

Then '''incompressible''' Euler equations with uniform density have conservation variables:
<math display="block">
\mathbf y = \begin{pmatrix}\mathbf u \\ 0 \end{pmatrix}; \qquad
\mathbf F = \begin{pmatrix}\mathbf u \otimes \mathbf u + w \mathbf I \\ \mathbf u \end{pmatrix}.
</math>

Note that in the second component u is by itself a vector, with length N, so y has length N+1 and F has size N(N+1). In 3D for example y has length 4, I has size 3×3 and F has size 4×3, so the explicit forms are:
<math display="block">
{\mathbf y}=\begin{pmatrix}  u_1 \\ u_2  \\ u_3 \\0 \end{pmatrix}; \quad
{\mathbf F}=\begin{pmatrix}
u_1^2 + w &  u_1u_2 &  u_1u_3
\\  u_2 u_1 & u_2^2 + w &  u_2 u_3
\\ u_3 u_1 &  u_3 u_2 &  u_3^2 + w 
\\ u_1 & u_2 & u_3 \end{pmatrix}.
</math>

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At last Euler equations can be recast into the particular equation:

{{Equation box 1
|indent=:
|title='''Incompressible Euler equation(s) with constant and uniform density'''<br/>(''conservation or Eulerian form'')
|equation=<math>
\frac {\partial}{\partial t}\begin{pmatrix} \mathbf u \\ 0 \end{pmatrix} + \nabla \cdot \begin{pmatrix}\mathbf u \otimes \mathbf u + w \mathbf I \\ \mathbf u \end{pmatrix} = \begin{pmatrix}\mathbf g \\ 0\end{pmatrix}
</math>
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|border colour = #0073CF
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}}