# Towards a High Order Convergent ALE-SPH Scheme with Efficient WENO Spatial Reconstruction

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Arbitrary Lagrangian–Eulerian Riemann-Based SPH

- A Riemann solver, that can be exact or approximate;
- A reconstruction procedure to infer the values of mass and momentum fluxes at the midpoint of the $ij$ pair, for both the left (L) and the right (R) states of the Riemann problem.

#### 2.2. Rusanov Flux

#### 2.3. High-Order Polynomial Weighted Essentially Non-Oscillatory Reconstruction

#### 2.4. Corrected SPH Estimation of Derivatives

#### Generation of polynomials for the Weighted Essentially Non-Oscillatory Reconstruction

## 3. Results and Discussion

#### 3.1. Moving Least-Squares vs. Corrected SPH Interpolation for the Generation of Reconstruction Polynomials

#### 3.2. Weakly Compressible 2D Vortex

^{2}, ${\rho}_{M}$ = 1.01 kg/m

^{2}, and ${r}_{0}$ = 0.1 m. The size of the circular simulation domain is R = 0.7 m. There are several reasons to select this test case for the preliminary studies of the present WENO-SPH scheme. Firstly, boundary conditions can be easily enforced with the appropriate number of rows of dummy particles around the simulation domain (where the analytical solution is imposed). This allows to avoid the situation where some of the candidate reconstruction stencils lack of sufficient neighbouring particles to build the corresponding polynomials. On the other hand, the test case is challenging as in general vortices are difficult to capture for Lagrangian particle methods.

- RSPH with a piece-wise constant reconstruction. This is the less accurate scheme due to the very large diffusion introduced by the Riemann problem with a trivial reconstruction.
- RSPH with a MUSCL reconstruction. Uses linear reconstructions based on gradients estimated by standard SPH operators, with slope limiters (to prevent oscillatory fields) as first depicted by [30]. The MUSCL reconstruction improves the accuracy of the scheme to the point that anisotropic distributions (like the ones displayed in Figure 4 for the RSPH-WENO scheme) appear, especially for high resolutions. The result is an increase in the discretization error that saturates the decrease in the global error.
- RSPH with WENO reconstruction: The high-order spatial reconstruction further improves the accuracy of the trajectories described by the particles which, in conjunction with the usage of standard divergence SPH operators in Equations (5) and (6), ruins the beneficial effect of the high-order reconstruction.
- RSPH with WENO reconstruction and Eulerian transport. In these simulations, particles remain fixed on the vertices of a Cartesian grid (${\mathit{v}}_{0}=0$). Hence, the gradient of partition of unity is exactly zero in the whole field, unleashing the full potential of the WENO reconstruction to reach a maximum convergence rate of ∼2.7. Considering that the theoretical convergence for the second order reconstructing polynomials used is third order, this result suggests that the overall convergence of the scheme is guided by the order of the WENO polynomials.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SPH | Smoothed Particle Hydrodynamics |

WEC | Wave Energy Converters |

LES | Large Eddy Simulation |

SPHERIC | SPH rEsearch and engineeRing |

MLS | Moving least-squares |

ALE | Arbitrary Lagrangian–Eulerian |

WENO | Weighted Essentially Non-Oscillatory |

ADER | Arbitrary DERivative |

MOOD | Multi-dimensional Optimal Order Detection |

LABFM | Local Anisotropic Basis Function Method |

GPU | Graphics Processing Unit |

CFL | Courant–Friedrichs–Lewy |

RSPH | Riemann-based SPH |

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**Figure 1.**Sketch of the central and the one-sided lateral stencils for a sample randomized 2D particle distribution.

**Figure 2.**Convergence of the L2 norm of the error in the reconstruction of Equation (20) in the squared domain $\left[-1,1\right]\times \left[-1,1\right]$ with side length L = 2 m. Results are displayed for the MLS and for the corrected SPH interpolation methods (with a smoothing length $h=2\Delta x$), as well as for both uniform and randomized particle distributions. Annotations of the order of convergence are included.

**Figure 3.**Pressure distribution of the 2D vortex along the line $\widehat{z}=0$ and at time $\widehat{t}=0.37$, computed with three different schemes: standard SPH with artificial viscosity ($\alpha =0.05$), RSPH with a piece-wise constant reconstruction, and RSPH with a WENO reconstruction. The resolution is $R/\Delta x=39$.

**Figure 4.**Distribution of particles of the 2D vortex simulation (with the RSPH-WENO scheme) at $\widehat{t}=3.7$, with the magnitude of the gradient of partition of unity superimposed. The resolution is $R/\Delta x=92$.

**Figure 5.**Convergence of the L2 norm of the 2D vortex pressure error at $\widehat{t}=0.37$ (the resolution is expressed as the ratio between the domain radius and the particle spacing, Rdp) for five different RSPH schemes.

**Figure 6.**Distribution of particles at $\widehat{t}=3.7$ for the 2D vortex case, with the field of unitary volume variation superimposed, computed by the RSPH-WENO scheme. The resolution is $R/\Delta x=92$.

**Table 1.**Computational time of Riemann-based schemes relative to a standard SPH scheme (for a given resolution).

Riemann-Based Scheme | CPU Time / CPU Time _{Standard SPH} |
---|---|

MUSCL | 2.5 |

WENO | 10.9 |

**Table 2.**Performance comparison of standard SPH and Riemann-based with WENO reconstruction SPH, for the 2D vortex case. Values of the L2 norm of the pressure error at $\widehat{t}=0.37$ are provided.

SPH Scheme | $R/\Delta x$ | $\parallel err\left(\widehat{p}\right){\parallel}_{2}$ | CPU time / CPU Time _{Standard SPH} |
---|---|---|---|

Standard SPH ($\alpha =0.6$) | 160 | 0.020532 | 1 |

WENO-RSPH | 52 | 0.020878 | 0.75 |

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**MDPI and ACS Style**

Antona, R.; Vacondio, R.; Avesani, D.; Righetti, M.; Renzi, M. Towards a High Order Convergent ALE-SPH Scheme with Efficient WENO Spatial Reconstruction. *Water* **2021**, *13*, 2432.
https://doi.org/10.3390/w13172432

**AMA Style**

Antona R, Vacondio R, Avesani D, Righetti M, Renzi M. Towards a High Order Convergent ALE-SPH Scheme with Efficient WENO Spatial Reconstruction. *Water*. 2021; 13(17):2432.
https://doi.org/10.3390/w13172432

**Chicago/Turabian Style**

Antona, Rubén, Renato Vacondio, Diego Avesani, Maurizio Righetti, and Massimiliano Renzi. 2021. "Towards a High Order Convergent ALE-SPH Scheme with Efficient WENO Spatial Reconstruction" *Water* 13, no. 17: 2432.
https://doi.org/10.3390/w13172432