## 3 Methodology Description
### Numerical Approach
#### 3.2.1 Additional Model Corrections: Particle Shifting and Free-Surface Tracking  
The problems covered in this work involve large pressure gradients that can lead to large areas with negative gauge pressures. Under these circumstances, it is possible that numerical cavitation will appear. This is a phenomenon known in SPH as tensile instability, and it is still a great challenge for the method. There are several sources in the literature that deal explicitly with tensile instability (see e.g., [37; 38]). Normally, the solution consists of adding a correction term to the momentum equation. Recently, several works have followed a different approach based on the application of the particle shifting technique (PST) introduced by [39]. Then, in this work a shifting algorithm with a tensile instability correction term will be used, in the same fashion as was proposed by [40]. As will be shown, this will be enough to avoid numerical cavitation in the simulations. A similar approach to the problem has been followed by [26] with satisfactory results. The PST will also improve the accuracy of the scheme, as it will keep the particle distribution regular. However, the system in (7) needs to be rewritten in an Arbitrary Lagrangian Eulerian (ALE) framework. This implies new terms linked to the particle shifting velocity to be included in the governing equations, as discussed by [40; 41]. Hence, the particle shifting velocity is defined according to [40] as:  
\[\delta\bar{u}_{i}=-U_{ref}\,2\,h\sum_{j}\left[1+0.2\left(\frac{W_{ij}}{W(dr)} \right)^{4}\right]\
abla_{i}W_{ij}V_{j}\,. \tag{11}\]  
The shifting velocity points toward the areas where particle distribution is irregular, and its intensity is modulated by the constant \(U_{ref}\left(2\,h\right)\), which is problem dependent. In this work, \(U_{ref}=Ma\cdot c_{s}\). Normally, shifting velocity remains low enough, but an upper limit is established in the present work, being \(0.25\,U_{ref}\,\delta\bar{u}_{i}\,\parallel\delta\bar{u}_{i}\parallel\). Finally, shifting velocity is added to the particle velocity and integrated in time to obtain the corrected position of the particles, such that:  
\[\frac{D\bar{\mathbf{r}}_{i}}{Dt}=\bar{\mathbf{u}}_{i}+\delta\bar{\mathbf{u}}_{i}, \tag{12}\]  
In flows involving a free surface, an additional control is needed for the PST. In this case, a similar approach to the one carried out by [38] is established:  
\[\delta\mathbf{u}_{i}=\left\{ \begin{array}{rl}0&\text{if }\lambda_{i}<0.55\text{ and }i \in\mathcal{F}\\ \kappa_{i}(\mathbf{I}-\mathbf{n}_{i}\otimes\mathbf{n}_{i})\delta\mathbf{u}_{i}^{*}&\text{if } \lambda_{i}\geq 0.55,\mathbf{n}_{i}\cdot\delta\mathbf{u}_{i}^{*}\geq 0\text{ and }i\in \mathcal{F}\\ \delta\mathbf{u}_{i}^{*}&\text{if }\lambda_{i}\geq 0.55,\mathbf{n}_{i}\cdot\delta\mathbf{u}_{i}^ {*}<0\text{ and }i\in\mathcal{F}\\ \delta\mathbf{u}_{i}^{*}&\text{if }i\
otin\mathcal{F}\end{array} \right. \tag{13}\]  
where \(\mathcal{F}\) is the region close to the free surface (defined at a distance \(h\)), and a scalar coefficient \(\kappa_{i}\) is defined such that:  
\[\kappa_{i}=\left\{ \begin{array}{rl}0&\text{if }\underset{j}{\max}\left[ \arccos\left(\mathbf{n}_{i}\cdot\mathbf{n}_{j}\right)\right]\geq 15^{\circ}\\ 1&\text{otherwise}\end{array} \right. \tag{14}\]  
A free-surface tracking algorithm to detect free-surface particles is implemented in accordance to the work developed by [40; 42].