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Entropy Variables

In §B.3, we showed how Fettweis et al. have effectively employed a skew-selfadjoint form of the gas dynamics system in order to generate a circuit model. Such forms have been the subject of a great deal of research, especially in the last few years [181]. One particular form, which makes use of so-called entropy variables would appear to be of fundamental importance, because it arises from a change of variables (and not a simple scaling of the system, as was the case for the system arrived at in §B.3). We recap the results from [181] here.

Consider a system of conservation laws,

$\displaystyle \frac{\partial {\bf u}}{\partial t} + \frac{\partial {\bf f}({\bf u})}{\partial x}= {\bf0}$ (B.18)

where the $ {\bf f}({\bf u})$ are smooth, possibly nonlinear mappings. The gas dynamics system (B.9), with $ {\bf u} = [\rho, \rho v, \rho e]^{T}$ and $ {\bf f}({\bf u}) = [\rho v, \rho v^{2} +p, \rho v e +pv]^{T}$, again complemented by the constitutive relation (B.10) is of this form. It is noted in [68,116,182] that (B.18) implies a further conservation law,

$\displaystyle \frac{\partial \mathcal{U}}{\partial t} + \frac{\partial \mathcal{F}}{\partial x}= 0$ (B.19)

for some smooth convex scalar function $ \mathcal{U}({\bf u})$, and a scalar flux $ \mathcal{F}({\bf u})$, over any time interval over which solutions to (B.18) remain smooth. If discontinuities (shocks) develop, then (B.19) becomes an inequality ($ \leq$). $ \mathcal{U}$ and $ \mathcal{F}$ are related by

$\displaystyle \left(\frac{d \mathcal{U}}{d {\bf u}}\right)^{T}\frac{d {\bf f}}{d{\bf u}} = \left(\frac{d \mathcal{F}}{d {\bf u}}\right)^{T}$    

It was shown in [183] that system (B.18) is symmetrized through left-multiplication by the Hessian $ {\bf P}_{\mathcal{U}}$ of $ \mathcal{U}$,

$\displaystyle {\bf P}_{\mathcal{U}} = \frac{d^{2}\mathcal{U}}{d{\bf u}^{2}}$    

so that we have

$\displaystyle {\bf P}_{\mathcal{U}}\frac{\partial {\bf u}}{\partial t} + {\bf A}_{\mathcal{U}}\frac{\partial {\bf u}}{\partial x} = {\bf0}$ (B.20)

where $ {\bf A}_{\mathcal{U}}$ and $ {\bf P}_{\mathcal{U}}$ are symmetric, and in addition $ {\bf P}_{\mathcal{U}}$ is positive definite (a result of the convexity requirement on $ \mathcal{U}$). This nonlinear system is of the same form as the (linear) symmetric hyperbolic system (3.1) discussed in §3.2, and possesses many similar properties; this form, however, can not be easily approached through MD circuit methods. Furthermore, weak solutions (i.e. solutions involving discontinuities) will not be preserved under such a scaling [181]. This is the same defect as that of Fettweis's MDKC for the Euler system, as discussed in §B.3.

It was later shown that (B.18) can also symmetrized with respect to a new variable $ {\bf z}$, defined by

$\displaystyle {\bf z} = \frac{d \mathcal{U}}{d{\bf u}}$    

In this case, symmetrization is carried out through a variable change and not a scaling, so weak solutions are indeed preserved. If, furthermore, the flux $ {\bf f}$ is homogeneous [181], it can be shown that there is also a skew-selfadjoint form of (B.18). The gas dynamics system (B.9) can be written in skew-selfadjoint form as

$\displaystyle {\bf P}_{\mathcal{U}}\frac{\partial{\bf z}}{\partial t}+ \frac{\p...
...}{\partial x}+\frac{\partial {\bf A}_{\mathcal{U}}{\bf z}}{\partial x} = {\bf0}$ (B.21)

if $ \mathcal{U}$ is chosen as

$\displaystyle \mathcal{U} = -(p\rho^{\alpha})^{1/\alpha + \gamma}\hspace{1.0in}\alpha>0$    

which is closely related to the physical entropy of the system [181]. The new variables $ {\bf z}$ are referred to as entropy variables.

next up previous
Next: An open problem Up: The Gas Dynamics Equations Previous: An Alternate MDKC and
Stefan Bilbao 2002-01-22