# A Near-Far-Field Model for Bubbles Influenced by External Electrical Fields
## 5 Coupling Near-Field and Far-Field
### Coupled Computation of Near- and Far-Field  
Based on the electric-field, which acts on a smaller time-scale, we have to update the near-field and far-field coupling in a numerical cycle,  
The near-field bubble is computed with the ODE's and with the underlying influence of the E-field. Furthermore, the far-field is computed by the transport equation with level-set methods. We realize the coupling via an interpolation between the mesh-free space of the near-field and the grid-space of the far-field with the given diameters of the bubbles and the trajectories of the transport field  
The numerical cycle is given in Figure 8.  
Figure 7: Final bubble based on the near-field computation and estimation of the bubble parameters (we assume an elliptic curve). In the following, we discuss the detailed coupling of the near-field and far-field, see Figure 9.  
The accuracy of the cycle is given with respect to the step-size of the far-field. Further, we can improve the accuracy of the cycle with higher-order interpolation schemes and higher-order near-field and far-field solvers, see the improvement of the PIC-cycles [15; 25]. Such ideas of different scale-dependent solvers can also be applied to so-called multiscale approaches, see [35]. A stronger coupling of the near- and far-field is given by an idea that is also used in the PIC (particle in cell) algorithms, see [14; 15]. Here, we couple microscopic scales--that is, particle scales with macroscopic scales (i.e., clusters of particles in a macroscopic grid), see [36]. We apply a so called modified PIC algorithm, which we call the BIC (bubble in cell) algorithm. These algorithms use the Young-Laplace equation in the finer scales, while we apply the bubble formation and modification, and the level-set or transport equations in the coarser scales, while we apply the bubble transport, see [37]. The algorithm is given in the following Figure 10.  
We have first order approximation errors for the interpolation between phase-space and grid space, while we apply a linear approximation, see [15]. We could improve the interpolation error with the help of higher order spline-approximations, such as second or third order splines, see [15]. This fine updating of the near- and far-field model allows to obtain more accurate results. Figure 8: Numerical cycle of the near–far-field coupling. Figure 9: Near field and far field coupling in the E-field. We have the conditions:  
\[r =a,\ z=0,\ \theta=\pi/2,\ \text{at}\ s=0, \tag{54}\] \[r =0,\ \frac{dz}{ds}=0,\ \theta=2(\pi/2),\text{at}\ s=L, \tag{55}\]  
where \(a\) is the radius of the bubble, \(L\) is the arc length of the bubble which is a-priori unknown, where we start with \(L=L_{0}=\frac{2\pi a^{2}}{4}\) and we go on with \(L=L_{1}\leq L_{2}\leq\ldots\leq L_{stop}\). We have the following linear equation system:  
\[\left[ \begin{array}{cccc}S_{1}&R_{1}&&&\\ &S_{2}&R_{2}&&\\ &&\ddots&\ddots&\\ &&&S_{N}&R_{N}\\ B_{a}&&&B_{b-1}&B_{b}\end{array} \right]\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{N}\\ y_{N+1}\end{array} \right]=\left[ \begin{array}{c}q_{1}\\ q_{2}\\ \vdots\\ q_{N}\\ d\end{array} \right], \tag{56}\]  
where \(S_{i}=R_{i}=-h_{i}^{-1}I-\frac{1}{2}A(s_{i+1/2})\), \(q_{i}=q(s_{i+1/2}\) with \(h_{i}=s_{i+1}-s_{i}\), \(s_{i+1/2}=s_{i}+\frac{1}{2}h_{i}\). Furthermore, we have matrices and vectors:  
\[B_{a}=\left( \begin{array}{ccc}0&0&0\\ 0&0&0\\ 0&0&0\end{array} \right),B_{b-1}=\left( \begin{array}{ccc}0&0&0\\ 0&1&0\\ 0&0&0\end{array} \right),B_{b}=\left( \begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array} \right),\mathbf{d}=\left( \begin{array}{c}L\\ 0\\ 0\end{array} \right). \tag{57}\]  
* Test example 1: We apply the following test-example with the a quarter of a circle, means \(L=2\pi a^{2}/4\), where \(a=1,2,3,3.5\) and we assume \(N=L/\Delta s\), where we assume \(\Delta s=2\pi\cdot 0.001\) is the arc-length step for the numerical computations. Further, we have \(\frac{\Delta p}{\sigma}=0\).