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