Here, [bar.K] is an [N.sub.1] x [N.sub.1] matrix with only [N.sub.1,[partial derivative]] nonzero rows, similar with [[partial derivative].sub.n][bar.g] introduced for

scalar wave. The (i,j) element [[[bar.K]].sub.i,j] associated with j-th primal edge and i-th dual face at surface, which only has one edge [L.sup.[partial derivative].sub.i] on the surface, is defined as:

Kallivokas, "Mixed unsplit-field perfectly matched layers for transient simulations of

scalar waves in heterogeneous domains," Computational Geosciences, vol.

As mentioned, to obtain a comparative analysis of the method and better understand these results, we carry out the same experiment using PML for the

scalar wave (13).

In other words, [g.sup.n.sub.i,j,k] is the solution of the second order central difference approximation of the

scalar wave equation with Kronecker delta excitation expressed as (it has been considered as i' = j' = k' = n' = 0 due to the shifting capability of the Green's functions):

The function f ([LAMBDA]) is the NtD map corresponding to the optimal PML for the

scalar wave equation.

This description above of in-and out-waves is almost identical to the quantum waves of the electron that can be obtained rigorously using a

scalar wave equation in Section H.

Vieira and Bezerra [122] study "resonant frequencies, Hawking radiation and scattering of

scalar waves ..." and find confluent Heun solutions.

Jiang, "The solution of the

scalar wave equation in the exterior of a sphere," Journal of Computational Physics, Vol.

We have shown that such a

scalar wave is purely electric and has no magnetic component.

Here we demonstrate the use of optimal grids for the two dimensional

scalar wave equation with domain decomposition.

The resulting optical field is denoted by a

scalar wave function,

Principle I is: Quantum matter waves exist in space and are solutions of a

scalar wave equation.