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Structure of Coordinate Changes

These same authors provide some more detailed guidelines as to what types of coordinate changes are of interest [62]. In particular, they describe transformations of the form:

$\displaystyle \begin{eqnarray}{\bf u} &=& {\bf V}^{-1}{\bf Ht}\\ {\bf t} &=& {\bf H}^{-1}{\bf Vu} \end{eqnarray}$ (3.14a)

where $ \mathbf{t} = [t_{1},\hdots, t_{n+1}]^{T}$ are the new coordinates and $ \mathbf{u}=[x_{1},\hdots,x_{n},t]^{T}$ are the old. $ \mathbf{V}$ is prescribed to be diag $ (1,1,\hdots,1,v_{0})$ and can be thought of as a simple scaling of the original coordinates $ {\bf u}$ to a non-dimensional (or rather, ``all-spatial'') form. $ v_{0}$ thus plays an important role, as we shall see later in a discrete setting, as the space step/time step ratio on a numerical grid. Its magnitude will be governed by a stability bound [176], sometimes called the Courant-Friedrichs-Lewy (CFL) criterion, as in conventional explicit finite difference methods (although the manifestation of the condition in the networks we will derive is of a quite different character). The invertible matrix $ \mathbf{H}$ is usually chosen to be orthogonal [62].

Here, we can see that the requirement (3.14a) will be satisfied if the elements in the rightmost column of $ {\bf H}^{-1}$ are positive; if $ {\bf H}$ is orthogonal, we have $ {\bf H}^{-1} = {\bf H}^{T}$. The bottom row of $ {\bf H}$ then consists of positive elements (often chosen equal, so as to give equal contributions from all components $ t_{j}$ to $ t$), in order to satisfy requirement (3.14b).

The differential operators $ \nabla_{\mathbf{t}}=[\frac{\partial}{\partial t_{1}}, \hdots ,\frac{\partial}{\partial t_{k}}]^{T}$ and $ \nabla_{\mathbf{u}}=[\frac{\partial}{\partial x_{1}}, \hdots ,\frac{\partial}{\partial x_{n}},\frac{\partial}{\partial t}]^{T}$ are related by:

$\displaystyle \nabla_{\mathbf{t}}=\mathbf{H}^{T}{\bf V^{-1}}\nabla_{\mathbf{u}}\hspace{0.5in}\nabla_{\mathbf{u}}=\mathbf{VH}^{-T}\nabla_{\mathbf{t}}$ (3.15)

Also, we introduce the scaled time variable

$\displaystyle t' = v_{0}t$ (3.16)

which will necessitate a special treatment in the circuit models to follow. See §3.5.1 for more details.


next up previous
Next: Coordinate Changes in (1+1)D Up: Coordinate Changes and Grid Previous: Coordinate Changes and Grid
Stefan Bilbao 2002-01-22