Finite Difference Schemes for the (2+1)D Wave Equation

Waveguide meshes of rectilinear [10], interpolated rectilinear [4], triangular [4,11] and hexagonal [11] forms have all been applied to solve the (2+1)D wave equation. Though they have often been written as scattering forms, we showed in Chapter 4 of [2] that such meshes can also be written as finite difference schemes. There are quite a few computational issues that arise which serve to distinguish between these difference schemes. Among them are the density of grid points, the possibility of decomposing a given scheme into more computationally efficient subschemes, the operation count, spectral characteristics, the ease with which boundary conditions can be implemented, as well as the maximum allowable time step. The stability issue discussed in §2.2 may also be a concern, and thus favor a waveguide mesh implementation instead of a straightforward difference scheme. It is, of course, impossible to say which is best, without knowing problem specifics. The following is intended partly as a catalogue, as well as an indication of certain features which probably deserve more attention, in particular the distinction between passivity and stability which becomes apparent in the cases of the triangular and interpolated meshes.

It is worthwhile introducing two new quantities at this point. In addition to $\Delta$, the ``nearest-neighbor'' grid spacing, or inter-junction spacing, $T$ the time step, $v_{0}$, which will always be equal to $\Delta/T$, and $\lambda = \gamma/v_{0}$, we also define $\rho_{S}$, the computational density of a particular scheme $S$ to be number of grid points at which the the difference scheme is operative, per unit volume and per unit time. Thus if the $N$-dimensional volume of the spatial domain $\mathcal{D}$ of a particular problem is $\vert\mathcal{D}\vert$ and the total time over which it operates is $\mathcal{T}$, then the total number of grid point calculations which will need to be made will be $\mathcal{\vert D\vert T}\rho_{S}$. Similarly, we can define the add density $\sigma_{S}$ to be $A_{S}\rho_{S}$ if scheme $S$ requires $A_{S}$ adds in order to update at any given grid point. A multiply density could be defined similarly, though we will not, for reasons of space, do so here.