Accuracy as a function of collocation cube size

Le us consider a time-harmonic field with wave number k radiating from a point within a cubic cell ci of side length H. On two opposite faces D1 and D2 of ci, an appropriate number of locations for the equivalent sources nS = 2×_S×_{S is selected (compare with Figure 3.1). We note that in [12, 13, 14],} it is proposed to place the equivalent sources on the points of the extended planes ofD1 andD2 which lie within the union of two circular domains concentric with (and containing) the faces ofci. The radius of these domains is chosen to be equal to (or slightly larger than) half the length of the diagonals of the faces. However, our finding is that in the context of the present work there is no disadvantage in terms of accuracy and computing time if we choose to place equivalent sources directly on the Cartesian grids τS(1) ∪τ

(2)

S , and thus this shall be our standard choice in this thesis. We refer to Appendix B.2 for further discussion on this issue. We recall that in order to evaluate the equivalent sources, the field needs to be specified at the nC = 6×_C×_C−₁₂×_{C+ 8 collocation points of the surface} S_{C. The} number nC results as sources are placed on each one of the six faces on the cube: on the first of the three pairs of opposite faces, we place 2×_C×_{C Cartesian points; on the second pair of opposite faces,} only 2×_C×_C−₄×_{C new positions can be located; and finally, on the last opposite pair, only the} 2×_C×_C−₈×_{C+8 interior points of the Cartesian grid can be selected. We assume that the wave values} at the collocation points along with the parametersk, H, S, C are known. The purpose of this subsection is to determine the dimensions of a suitable collocation cube which is characterized by the edge length HC. To this end, we select the specific values k= 10, S = 7, andC = 7 (which means thatnS = 98 and nC = 218).Numerical results suggest it is best to choosenC at least 2nS.Numerical experiments further indicate that under these constraints, any other choice of parameters for k, S, C lead quantitatively to the same conclusions. We consider the five different values 2H,2.5H,3H,4H, and 5H forHC. Once the collocation cube is known, Algorithm B.2.1 can be used to obtain an approximation of the wave at any point outside of the collocation cube At the fixed point in space P = [0,1.25,1.26]t

, we compute the numerical error EP(HC) to the exact solution in absolute norm. The results are displayed in Figure 3.2 for the four different values 0.00025,0.0025,0.025, and 0.25 for H. We note that the point P lies outside of the collocation cube for all five choices ofHC. Figure 3.2 leads to the following observation: for any fixedH,the equivalent source computation yields a more accurate solution atP as the distance

3.1 Parameter value identification 16 10−4 10−3 10−2 10−1 100 101 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 H_C E P H=0.00025 H=0.0025 H=0.025 H=0.25

Figure 3.2: The errorEP(HC) as a function ofHC at P = [0,1.25,1.26]t

for different values of H

from the two facesD₁∪_{D2 to the collocation points} increases. _{Clearly, in this experiment the increased} distance is realized by increasing the size ofHC while keeping H constant, and by doing so, the surface

S_{Ccomes closer to the pointP.It might be thought that the decreased distance fromP to the collocation}

points influences the conclusion “the bigger the collocation cube, the more accurate the equivalent source computation.”

To see whether that is indeed the case, we consider the following experiment: for the valueH = 0.25, we select one of the five parameters for HC from the first example, and we evaluate the error E(P) at the location P = [0, l,0]t

, where l takes one of the nine values 0.6H,0.8H,1.2H,1.4H,1.5H,1.6H,2H, or 3H.Clearly, the first point lies for all five collocation boxes between the surfaces of ci and SC, while

the last point is positioned outside the collocation cube. In Figure 3.3, we plot E(P) as a function of P for the smallest and the largest collocation box, i.e., HC = 2H and HC = 5H, respectively. The quantitative behavior is obvious: starting just outsideci and moving towardS_{C, the accuracy increases}

until the collocation surface is reached. Continuing moving in the same direction, the behavior remains unchanged, i.e., the farther away from the collocation box, the more accurate. Looking back at the first

10−0.8 10−0.6 10−0.4 10−0.2 10−10 10−8 10−6 10−4 10−2 100 l E(P) H_C=2H H_C=5H

Figure 3.3: The errorE(P) as a function ofP = [0, l,0]t

for two different values of HC

experiment, we see that as the collocation cube increases, the more accurate the solution, despite the fact that the point under consideration is closer. This leads to the conclusion that a larger collocation box yields more accurate results.

As we will explain later, ideally, we want to choose HC as small as possible, but on the other hand, the scheme should be as accurate as possible. These two trends conflict each other, and the selection HC = 3H seems to be a reasonable compromise. Thus, HC = 3H is our standard choice if not stated otherwise.

3.1 Parameter value identification 18