2.5.2 Characteristic fields

Let us introduce into system (122) the Jacobian matrices ${\bf\cal A}^{\alpha}({\bf w})$ associated to the five-vectors ${\bf F}^{\alpha}({\bf w})$

$${\bf\cal A}^{\alpha} = \frac{\partial{\bf F}^{\alpha}({\bf w})} {\partial {\bf w}}$$ (54)

then, system (122) can be written as a quasi-linear system of 5 first order partial differential equations for the unknown field $\bf w$

$${\bf\cal A}^{\mu}({\bf w}) \frac {\partial {\bf w}}{\partial x^{\mu}} = 0$$ (55)

Finally, we introduce the vectors

$${\bf u} = {\bf F}^{0}({\bf w})$$ (56)

$${\bf f}^{i} = {\bf F}^{i} \circ ({\bf F}^{0})^{-1}$$ (57)

where $\circ$ means composition of functions and $(.)^{-1}$ stands for the inverse function.

With the above definitions, system (122) reads as a system of conservation laws for the new vector of unknowns u

$$\frac{\partial {\bf u}}{\partial x^{0}} + \frac{\partial {\bf f}^{i}({\bf u})}{\partial x^{i}} = 0$$ (58)

In the above system (59) we can define three $5\times 5$-Jacobian matrices ${\bf\cal B}^{i}$(u), the Jacobian matrices associated to the vector ${\bf f}^{i}$(u), the flux in the $i$-direction of the system (59), as:

$${\bf\cal B}^{i} = \frac{\partial{\bf f}^{i}({\bf u})} {\partial\bf u \rm }$$ (59)

Following a procedure similar to the one described in Font et al.(1994) we have succeeded in obtaining analytical expressions for the spectral decomposition of the matrices ${\bf\cal B}^{i}$(u). For the sake of clarity, let us restrict to a 2D case, i.e., to problems involving a dependence on two spatial coordinates (that we will denote with the scripts i = x, y instead of i = 1,2)

The eigenvalues and the right and left eigenvectors of ${\bf\cal B}^{x}$ (similar for i=y) are going to be displayed explicitly.

The eigenvalues of matrix ${\bf\cal B}^x$(u) are:

$$\lambda_{\pm} = \frac{1}{1- v^2 c_s^2} \left\{v^{x}(1-c_{s}^... ...{(1-{\rm v}^{2}) [1-v^x v^x- (v^2-v^x v^x)c_{s}^{2}]} \right\}$$ (60)

$$\lambda_0 = v^x \mbox{ (double)}$$ (61)

Let us make several comments to the above expressions for the eigenvalues:

i) In the case ($v^x=v, v^y=0$), expression (61) gives the corresponding one-dimensional eigenvalues

$$\lambda_{\pm} = \frac{v \pm c_{s}}{1\pm v c_{s}}$$ (62)

ii) In the limit $\vert v^x\vert \rightarrow 1$, the genuinely nonlinear characteristic fields $\lambda_{\pm}$ become linearly degenerate.

For the sake of concisness and before showing the complete set of right and left-eigenvectors, let us define the following quantities:

$${\cal K} = {\displaystyle{\frac{\tilde{\kappa}} {\tilde{\kappa}-c_s^2}}}$$ (63)

$${\cal A}_{\pm} = {\displaystyle{\frac{1 - v^x v^x} {1 - v^x {\lambda}_{\pm}}}}$$ (64)

$$\Delta = h^2 W ({\cal K} - 1) (1 - v^x v^x) ({\cal A}_{+} {\lambda}_{+} - {\cal A}_{-} {\lambda}_{-})$$ (65)

being $\Delta$ the determinant of the matrix of right-eigenvectors. As it can be proved, ${\cal K}$ is always greater than one, and $\Delta$ different from zero.

A complete set of right-eigenvectors is,

$${\bf r}_{0,1} = \left( {\displaystyle{\frac{{\cal K}}{h W}}}, v^x, v^y, 1 - {\displaystyle{\frac{{\cal K}}{h W}}} \right)$$ (66)

$${\bf r}_{0,2} = \left( W v^y , 2 h W^2 v^x v^y, h(1+2 W^2 v^y v^y), 2 h W^2 v^y - W v^y\right)$$ (67)

$${\bf r}_{\pm} = (1, h W {\cal A}_{\pm} {\lambda}_{\pm}, h W v^y, h W {\cal A}_{\pm} - 1)$$ (68)

The corresponding complete set of left-eigenvectors is

$$\begin{eqnarray*} {\bf l}_{0,1} &=& {\displaystyle{\frac{W}{{\cal K} - 1}}} (h - W, W v^x, W v^y, -W) \end{eqnarray*}$$

$$\begin{eqnarray*} {\bf l}_{0,2} &=& {\displaystyle{\frac{1}{h (1 - v^x v^x)}}} (- v^y, v^x v^y, 1 - v^x v^x, -v^y) \end{eqnarray*}$$

$${\bf l}_{\mp} = ({\pm} 1){\displaystyle{\frac{h}{\Delta}}} ... ... {\cal K} {\cal A}_{\pm} {\lambda}_{\pm} \end{array} \right]$$

Symmetry relations allow to extend the above spectral decomposition to the other spatial direction $y$ and also to the general 3D case.

As we have emphasized above, in order to apply HRSC methods to solve the equations of relativistic hydrodynamics as written in (59), the knowledge of the spectral decomposition of the Jacobian matrices ${\bf\cal B}^i$ is crucial.