Symmetric Hyperbolic Systems

In the previous chapter, we examined the discretization of lumped analog circuits; by lumped, we mean that the voltages and currents in these circuits are functions of only one independent variable: time. Although the described procedure was originally intended as a means of developing robust digital filtering structures, an equivalent point of view is that such structures in fact numerically integrate the set of ordinary differential equations describing the time evolution of these currents and voltages.

In a distributed problem, the dependent variables are functions not only of time $t$, but also of location within an $n$-dimensional spatial domain $\mathcal{D}$, with coordinates ${\bf x} = [x_{1},\hdots,x_{n}]^{T}$. Such a problem is referred to as an $(n+1)$D problem in the WDF literature [131]. Problems without spatial dependence will be called lumped problems. If the equations which define the problem include differential operators, we are faced with solving a set of partial differential equations (PDEs).

A particularly important family of PDE systems are the symmetric hyperbolic [74,82] systems of the form

$${\bf P}\frac{\partial {\bf w}}{\partial t} +\sum_{k=1}^{n}{\bf A}_{k}\frac{\partial {\bf w}}{\partial x_{k}} + {\bf Bw} + {\bf f} = {\bf0}$$ (3.1)

Here, ${\bf w}$, the state, is a $q$-element column vector defined over coordinates ${\bf x}\in\mathcal{D}\subset \mathcal{R}^{n}$ and $t\geq 0$. ${\bf P}$ and ${\bf A}_{k}$, $k=1,\hdots,n$, are real symmetric $q\times q$ matrices; in particular, ${\bf P}$ is assumed to be positive definite. ${\bf B}$ is a real $q\times q$ matrix (not necessarily symmetric) whose symmetric part models energy loss or growth, and the $q$-element real column vector ${\bf f}$ is a forcing function or excitation. For all the systems to be discussed in this thesis (except the fluid dynamic systems of Appendix B), the matrices ${\bf A}_{k}$ are assumed to be constant, though ${\bf P}$ and ${\bf B}$ are allowed to depend on ${\bf x}$. These systems are thus linear and time-invariant, but not generally shift-invariant, so that we cannot apply spatial Fourier transforms directly to analyze them. System (3.1) must be complemented by initial and boundary conditions [82], in order for the solution to exist and be unique.

Though it is possible to extend this definition to include cases where the matrices ${\bf A}_{k}$ may depend on ${\bf x}$, $t$ or even ${\bf w}$ (in which case system (3.1) is nonlinear), this simpler form describes a wide variety of physical systems, from electromagnetics to string, membrane, beam, plate, shell, and elastic solid dynamics, to transmission line systems, to linear acoustics, etc. Symmetric hyperbolic systems are important because they form a subclass of strongly hyperbolic systems, for which the initial-value problem is well-posed [176]. Roughly speaking, to say that a system is well-posed is to say that the growth of its solution is bounded in a well-defined way; growth in an $L_{2}$ norm cannot be faster than exponential. This concept is elaborated in detail in [82,176]. We can examine this growth in the present case as follows.

First, assume that the problem is defined over an unbounded spatial domain $\mathcal{D} = \mathbb{R}^{n}$, so that we can drop any consideration of boundary conditions, and also that the forcing function ${\bf f} = {\bf0}$. We now take the inner product of ${\bf w}^{T}$ (the transpose of ${\bf w}$) with (3.1) to get

$${\bf w}^{T}{\bf P}\frac{\partial {\bf w}}{\partial t} +\sum_{k=1}... ...bf w}}{\partial x_{k}} + \frac{1}{2}{\bf w}^{T}({\bf B}+{\bf B}^{T}){\bf w} = 0$$ (3.2)

where we have replaced ${\bf B}$ by its symmetric part $\frac{1}{2}\left({\bf B}+{\bf B}^{T}\right)$. Due to the symmetry of ${\bf P}$ and the ${\bf A}_{k}$, we can then write

$$\frac{1}{2}\frac{\partial}{\partial t}\left({\bf w}^{T}{\bf Pw}\r... ...f w}\right) + \frac{1}{2}{\bf w}^{T}\left({\bf B}+{\bf B}^{T}\right){\bf w} = 0$$ (3.3)

Now, integrate (3.3) over $\mathbb{R}^{n}$, to get

$$\frac{d}{d t}\int_{\mathbb{R}^{n}}\frac{1}{2}\left({\bf w}^{T}{\b... ...+ \frac{1}{2}\int_{\mathbb{R}^{n}}{\bf w}^{T}({\bf B}+{\bf B}^{T}){\bf w}dV = 0$$ (3.4)

where $dV = dx_{1}dx_{2}\hdots dx_{n}$ is the $n$D differential volume element. The expression $\sum_{k=1}^{n}\frac{\partial}{\partial x_{k}}\left({\bf w}^{T}{\bf A}_{k}{\bf w}\right)$ is easily seen to be the divergence of a vector field, and by the Divergence Theorem [174], the integral of this quantity can be replaced by a surface integral over the problem boundary--because we have assumed no boundary, this integral vanishes, and we are left with

$$\frac{d}{d t}\int_{\mathbb{R}^{n}}\frac{1}{2}\left({\bf w}^{T}{\b... ...+ \frac{1}{2}\int_{\mathbb{R}^{n}}{\bf w}^{T}({\bf B}+{\bf B}^{T}){\bf w}dV = 0$$ (3.5)

The quantity

$$E(t) \triangleq \int_{\mathbb{R}^{n}}\frac{1}{2}\left({\bf w}^{T}{\bf Pw}\right)dV$$ (3.6)

can be interpreted as the total energy of system (3.1) at time $t$. Note that due to the positivity requirement on ${\bf P}$, it is a positive definite function of the state, ${\bf w}$. If ${\bf B}+{\bf B}^{T}$ is positive semi-definite, then we must have, from (3.5), that

$$\frac{d}{dt}E \leq 0$$

which implies that

$$E(t_{2})\leq E(t_{1}) \hspace{0.3in}t_{2}\geq t_{1}$$ (3.7)

In other words, the energy of the system must decrease as time progresses.

In the MD circuit models that we will discuss, what we will be doing, in essence, is dividing this energy up among various reactive MD circuit elements. We will elaborate on this in the sections on the (1+1)D transmission line and (2+1)D parallel-plate system. The passivity condition is essentially equivalent to (3.7). Also, the symmetric nature of the systems will be reflected, in the circuit models, by the use of mainly reciprocal [12] circuit elements, though non-reciprocal elements (gyrators) will come into play if ${\bf B}$ is not symmetric (it is not required to be, and note that system (3.1) is well-posed regardless of the form of ${\bf B}$ [82]). We have not explored the application of passive circuit methods to systems which are more generally strongly hyperbolic, for which energy estimates such as (3.7) can also be derived [82]. This would appear to be a worthy direction of future research.