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,

$$\frac{\partial {\bf u}}{\partial t} + \frac{\partial {\bf f}({\bf u})}{\partial x}= {\bf0}$$ (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,

$$\frac{\partial \mathcal{U}}{\partial t} + \frac{\partial \mathcal{F}}{\partial x}= 0$$ (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

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

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

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

so that we have

$${\bf P}_{\mathcal{U}}\frac{\partial {\bf u}}{\partial t} + {\bf A}_{\mathcal{U}}\frac{\partial {\bf u}}{\partial x} = {\bf0}$$ (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

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

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

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

if $\mathcal{U}$ is chosen as

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

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