## 3 Results
### Test Problems  
We tested our algorithm on a set of simple, but representative examples that were well suited to demonstrate our method, and not far from real world situations or arguably even more difficult. The instances lived in a square \([0,1]^{2}\) or \([0,1]\times[-1,1]\), and the origin and the destination were \(x_{O}=[0,0]^{T}\) and \(x_{D}=[1,0]^{T}\), respectively. All values were chosen dimensionless. The wind fields were constructed in a such a way that the straight line from the origin to the destination was particularly unfavorable. The wind speed was limited to \(\bar{w}=\frac{1}{2}\bar{o}\), which was rather strong, but not unrealistic. Even though not formally required, the graph nodes were positioned on a uniform Cartesian grid, such that the diagonal distance of two adjacent nodes was \(2h\). For the sake of simplicity the graph structure will be described based on the node spacing in x-direction \(h_{x}=\sqrt{2}h\). This directly defined \(l=\sqrt{h}\) by (14). In the first test problem (a), the wind field was a laminar flow of opposing parallel currents, namely,  
\[w(x)= \begin{bmatrix}\bar{w}\min(\max(2\frac{x_{D}^{2}}{H}-1,-1),1)\\ 0\end{bmatrix} ,\]  
with \(H=0.5\), see Figure 2; a similar problem is discussed in [45]. Due to its simplicity, this problem had only one distinct minimum, a property that we make use of in the next section. Figure 2: Test problem (a) with \(H=0.5\), \(x_{O}=(0,0)\), \(x_{D}=(1,0)\), \(\bar{w}=\frac{1}{2}\bar{o}\). Blue dots: network-graph with \(h=l^{2}\) and \(1/h_{x}=1\) and \(6\), respectively, red: discrete optimal trajectory, green: continuous optimal trajectory. In problems (b)-(d), the wind \(w\) was the sum of an increasing number of vortices \(w_{i}\), each of which was described by  
\[w_{i}(x)=\left[ \begin{array}{c}-s\widehat{w}(r)\sin(\alpha)\\ s\widehat{w}(r)\cos(\alpha)\end{array} \right],\]  
where \(s\) is the spin of the vortex (\(s{=}{+}1\): counter-clockwise, \(s{=}{-}1\): clockwise), \(r=\|x-z\|_{2}\) is the distance from the vortex center \(z_{i}\), \(\alpha\) is the angle with respect to the center and the x-axis with \(\tan(\alpha)=\frac{(x-z_{i})_{2}}{(x-z_{i})_{1}}\) and the absolute vortex wind speed \(\widehat{w}\) is a function of \(r\) and the vortex radius \(R\):  
\[\widehat{w}(r)=\left[ \begin{array}{cc}\widehat{w}\exp\left(\frac{r^{2}}{r^{ 2}-R^{2}}\right)&\text{if }r\leq R\\ 0&\text{otherwise}\end{array} \right]. \tag{18}\]  
Problem (b) involved one large vortex with \(R=1/2\) at \(z=[0.5,-0.1]^{T}\), see Figure 3. This caused [PERSON]'s method to converge to a suboptimal trajectory (above the vortex) if initialized with the straight trajectory. That was the case if the DisCOptER algorithm was used with a minimally sparse graph with \(h_{x}=1\) (see Figure 3, middle). We observed that the discrete shortest path passed below the vortex for any \(h_{x}>1\). From there the globally optimal solution shown in the Figure 3 (left 3) was found for \(h_{x}=1/6\). Problem (c) involved 15 vortices with \(R=1/8\). One was centered at \(z=[0.5,-R/2]^{T}\), the others were regularly aligned as seen in Figure 4. Vortices with positive spin (clockwise) were colored green, vortices with negative spin (anti-clockwise) were colored red. Due to the turbulence of this wind field, a plain application of [PERSON]'s method was not guaranteed to converge. As an example, (Figure 4, left 2) shows again the result with the trivial initialization. Further there are several local minima, Figure 4 (left 3 and 4) show two of them. A relatively high graph density (\(h_{x}\leq 1/17\)) was required to find the globally optimal trajectory (Figure 4, left 5). Problem (d) involved 50 regularly aligned vortices of radius \(R=1/16\) (Figure 5). This was clearly an exaggeration, and no commercial plane would ever try to traverse a wind field like this. We used the instance to show that the proposed algorithm outperformed existing methods even under the most adverse conditions.