Editing Euler equations (fluid dynamics) (section)

===Characteristic equations===
The compressible Euler equations can be decoupled into a set of N+2 [[wave]] equations that describes [[sound]] in Eulerian continuum if they are expressed in [[method of characteristics|characteristic variables]] instead of conserved variables.

In fact the tensor '''A''' is always [[Diagonalizable matrix|diagonalizable]]. If the [[eigenvalue]]s (the case of Euler equations) are all real the system is defined ''hyperbolic'', and physically eigenvalues represent the speeds of propagation of information.{{sfn|Toro|1999|p= 44|loc=par 2.1 Quasi-linear Equations}} If they are all distinguished, the system is defined ''strictly hyperbolic'' (it will be proved to be the case of one-dimensional Euler equations). Furthermore, diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case.

If <math>\mathbf p_i</math> is the [[right eigenvector]] of the matrix <math>\mathbf A</math> corresponding to the [[eigenvalue]] <math>\lambda_i</math>, by building the [[projection matrix]]:
<math display="block">\mathbf{P} = \left[\mathbf{p}_1, \mathbf{p}_2, ..., \mathbf{p}_n\right].</math>

One can finally find the ''characteristic variables'' as:
<math display="block">\mathbf{w} = \mathbf{P}^{-1} \mathbf{y}.</math>

Since '''A''' is constant, multiplying the original 1-D equation in flux-Jacobian form with '''P'''<sup>−1</sup> yields the characteristic equations:{{sfn|Toro|1999|p= 52|loc= par 2.3 Linear Hyperbolic System}}
<math display="block">
\frac{\partial w_i}{\partial t} + \lambda_j \frac{\partial w_i}{\partial r_j} = 0_i.
</math>

The original equations have been [[Linear independence|decoupled]] into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds.  The variables ''w''<sub>''i''</sub> are called the ''characteristic variables'' and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is:
<math display="block">w_i(x, t) = w_i\left(x - \lambda_i t, 0\right).</math>

Then the solution in terms of the original conservative variables is obtained by transforming back:
<math display="block">\mathbf{y} = \mathbf{P} \mathbf{w},</math>
this computation can be explicited as the linear combination of the eigenvectors:
<math display="block">\mathbf{y}(x, t) = \sum_{i=1}^m w_i\left(x - \lambda_i t, 0\right) \mathbf p_i.</math>

Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each ''i''-th wave has shape ''w''<sub>''i''</sub>''p''<sub>''i''</sub> and speed of propagation ''λ''<sub>''i''</sub>. In the following we show a very simple example of this solution procedure.