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Phase and Group Velocity

Because the stability of an explicit numerical method (such as those that we will examine in the rest of this thesis) which solves a system of hyperbolic equations is dependent on propagation velocities, it is worthwhile to spend a few moments here to define phase and group velocities [35,101] for a system such as (3.1).

Let us return to the unbounded domain problem with $ \mathcal{D} = \mathcal{R}^{n}$. Suppose that the matrices $ {\bf P}$, $ {\bf B}$ and $ {\bf A}_{k}$, $ k=1,\hdots,n$ which define system (3.1) are real constants; in particular, we assume that the driving term $ {\bf f}$ is zero, and that $ {\bf B}$ is anti-symmetric, so that system (3.1) is lossless. This is then a linear and shift-invariant system, and the solution can be written as a superposition of plane wave solutions of the form

$\displaystyle {\bf w}({\bf x}, t) = {\bf w}_{0}e^{j\omega t +\mbox{{\scriptsize\boldmath$\beta$}}\cdot {\bf x}}$    

where $ {\bf w}_{0}$ is a constant vector, $ \omega$ is a real frequency variable, and $ \beta$$ = [\beta_{1},\hdots,\beta_{n}]^{T}$ is the $ n$-component vector wavenumber defining the direction of propagation of the plane wave. Substituting this plane wave solution into the constant-coefficient system (3.1) gives

$\displaystyle \left(j\omega {\bf P}+\sum_{k=1}^{n}j\beta_{k}{\bf A}_{k} + {\bf B}\right) {\bf w} = {\bf0}$ (3.9)

Non-trivial solutions to (3.9) can only occur when

$\displaystyle \chi(\omega,$$\displaystyle \mbox{\boldmath$\beta$}$$\displaystyle ) \triangleq \det\left(j\omega {\bf P}+\sum_{k=1}^{n}j\beta_{k}{\bf A}_{k} + {\bf B}\right) = 0$ (3.10)

The $ n$ solutions to this equation,

$\displaystyle \omega_{k}($$\displaystyle \mbox{\boldmath$\beta$}$$\displaystyle ),\hspace{1.0in} k=1,\hdots,n$ (3.11)

(which are not necessarily distinct) define dispersion relations, from which we can derive much useful information.

All the linear systems to be examined in this thesis are isotropic; propagation characteristics are independent of direction (though not necessarily of location, or frequency). For LSI systems, this implies that the dispersion relations (3.11) can be written as functions of $ \Vert$$ \beta$$ \Vert _{2}$ alone, where $ \Vert$$ \beta$$ \Vert _{2}$ is simply the Euclidean norm of the vector $ \beta$. In this case, we may define the phase and group velocities for the $ k$th relation by

$\displaystyle \gamma_{k}^{p} \triangleq \frac{\omega_{k}}{\Vert\mbox{\boldmath$...
...k}^{g} \triangleq \frac{d \omega_{k}}{d \Vert\mbox{\boldmath$\beta$}\Vert _{2}}$ (3.12)

(For non-isotropic systems, we will need to resort to vector generalizations of these quantities [101,190].) Phase velocities define the speeds of single sinusoidal plane wave solutions, and the group velocities can be interpreted as the speeds of propagation of a wave packet; from the point of view of the stability of numerical methods, it is the group velocities which are of most importance, because they define the speeds of information or energy transfer [35]. It is interesting to note that if $ {\bf B}$ is non-zero, phase velocities may become unbounded in the limit as $ \beta$ becomes small--this occurs in several of the systems that will be discussed in Chapter 5, though for all these systems, the group velocities will be bounded. This is related to the fact that the system characteristics [74] are independent of $ {\bf B}$.

In the interest of extending these ideas to spatially inhomogeneous systems (of the form of (3.1) where $ {\bf P}$ and $ {\bf B}$ may exhibit a smooth functional dependence on $ {\bf x}\in\mathcal{D}$), we note that about any location $ {\bf x} = {\bf x}_{0}\in \mathcal{D}$, solutions to system (3.1) behave locally as solutions to the frozen-coefficient system [82] defined by $ {\bf P}({\bf x}_{0})$ and $ {\bf B}({\bf x}_{0})$. We may then define local group velocities $ \gamma_{k}^{g}(\Vert$$ \beta$$ \Vert _{2}, {\bf x}_{0})$, $ k=1,\hdots,n$ in the same way as in (3.12). A quantity which will appear frequently in our subsequent treatment of the stability of numerical methods for these systems will be the maximum global group velocity, defined as

$\displaystyle \gamma_{max}^{g} \triangleq \max_{\begin{minipage}[t]{1.0in}\vspa...
...d{minipage}}\gamma_{k}^{g}(\Vert\mbox{\boldmath$\beta$}\Vert _{2}, {\bf x}_{0})$    

which, more simply stated, is the maximum propagation velocity over all system modes, wavenumbers, and throughout the entire spatial problem domain.

next up previous
Next: Coordinate Changes and Grid Up: Symmetric Hyperbolic Systems Previous: Note on Boundary Conditions
Stefan Bilbao 2002-01-22