# Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Model

#### 2.1. Governing Equations

#### 2.2. Numerical Formulation

#### 2.2.1. First Step: Discontinuous Galerkin Solution

#### 2.2.2. New Slope Limiter

^{®}(Sistema Base de Hidrodinámica Ambiental) hydrodynamic model, which is a continuous Galerkin depth-integrated shallow water model. The SisBahia

^{®}interface smooths the nodal oscillations by modifying the computed nodal values of the variables using computed nodal values from adjacent nodes while preserving the amplitude of the phenomenon of interest. Here, the SisBahia

^{®}interface is used to smooth the averaged nodal value in the center of the element and subsequently use this smoothed averaged value to reconstruct the values at the four nodes of the element.

**e**by evaluating Equation (33) at the four nodes. In Equation (33) $\mathit{r}$ is a vector with origin at the arithmetic center of the element extending to any point within the element. The vector $\mathit{r}$ for the node 1 at Figure 2 will be ${x}_{n1}-\overline{x}$, ${y}_{n1}-\overline{y}$, where $\overline{x}and\overline{y}$ are the values at the arithmetic center of the element. This reconstruction implies that the reconstructed variables will have a constant gradient over the element. The limiter is applied after every step of the Runge–Kutta integration in Equation (25).

#### 2.2.3. Second Step: Continuous Galerkin Solution

#### 2.2.4. Third Step

#### 2.2.5. Boundary Conditions

#### 2.2.6. Dry Bed Treatment

#### 2.2.7. Wave Breaking

- The surface variation criterion: a node is flagged if $\frac{\frac{\partial {\xi}_{c}}{\partial t}}{\sqrt{g\left|h\right|}}>\u0263$, with $\u0263$ ∈ [0.3, 0.65] depending on the type of breaker.
- The local slope angle criterion: a node is flagged if $\left|\frac{\partial {\xi}_{c}}{\partial x},\frac{\partial {\xi}_{c}}{\partial y}\right|$ ≥ tan ϕ, with ϕ ∈ [14°, 33°] depending on the flow configuration.

#### 2.2.8. Computational Aspects

## 3. Model Verification and Validation

#### 3.1. Solitary Wave Propagation Along a Constant Depth Channel

#### 3.2. Solitary Wave Runup on a Plane Beach

**A**/

**h**= 0.3 (where A is the solitary wave height and h is the still water depth) ran up a beach with a slope of 1:19.85. In the numerical simulation, square elements with sizes Δx/

**h**= Δy/

**h**= 0.125 are considered, and a Manning coefficient

**n**= 0.01 is used to define the bed surface roughness. A Sommerfeld radiation condition is imposed at the left-side boundary. The solitary wave is initially at half of the wavelength ($L$) from the toe of the beach, with $L$ defined by:

#### 3.3. Solitary Wave Propagation over Fringing Reef

#### 3.4. Series of Regular Waves on a Plane Beach

#### 3.5. Solitary Wave Runup on a Conical Island

#### 3.6. Solitary Wave Transformation, Breaking and Runup over a Three-Dimensional Complex Bathymetry.

^{2}to 0.0025 m

^{2}and the mesh was refined over the 3D reef around x = 16 m to x = 32 m in Figure 14. A Sommerfield radiation boundary condition is applied at the western boundary. The Manning coefficient was set to n = 0.01. The incident solitary wave has a wave height A = 0.39 m, and the water depth in the basin is h = 0.78 m. The dimensionless wave height A/h = 0.39/0.78 = 0.5 indicates a strongly nonlinear wave. The initial setup of the simulation is displayed in Figure 15.

^{®}i7-9750H CPU, at 2.6 GHz and 8 GB of memory. This performance could be improved in future developments if a CUDA

^{®}-GPU version of the model is achieved.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Comparison of numerical results and analytical solutions of free surface elevations at t = 0, 10, 20, 30 and 40 s. Analytical solutions (circles), numerical results (solid lines).

**Figure 4.**${L}_{2}$ error norms for $\xi $ as function of ∆x, ∆y for solitary wave propagation in constant depth.

**Figure 5.**Comparison of simulated and measured surface profiles of a solitary wave runup on a 1:19.85 plane beach. Experimental data (circles), numerical results (solid lines).

**Figure 6.**Comparison of simulated and measured surface profiles of a solitary wave over a fringing reef. Experimental data (circles), numerical results (solid lines).

**Figure 7.**Diagram of the channel with the location of the six investigated sections (From: De Padova et al. [37]).

**Figure 8.**Comparison of simulated and measured free surface elevations for case T1 (spilling wave) (ɣ = 0.2, ϕ = 30°). Experimental data (circles), numerical results (solid lines).

**Figure 9.**Comparison of simulated and measured free surface elevations for case T2 (plunging wave) (ɣ = 0.3, ϕ = 30°). Experimental data (circles), numerical results (solid lines).

**Figure 10.**Schematic sketch of the conical island experiment. (

**a**) Plane view, (

**b**) side view, ο gauge locations (from: Yamazaki et al. [33]).

**Figure 12.**Comparison of simulated and measured free surface elevations for a solitary wave runup on a conical island: A/h = 0.045 (left panel), A/h = 0.096 (middle panel), A/h = 0.181 (right panel). Experimental data (circles), numerical results (solid lines).

**Figure 13.**Comparison of simulated and measured maximum vertical runup heights for a solitary wave runup on a conical island: A/h = 0.045 (

**left panel**), A/h = 0.096 (

**middle panel**), A/h = 0.181 (

**right panel**). Experimental data (circles), numerical results (solid lines).

**Figure 14.**Bathymetry contours and locations of measuring points. Circles (water surface measurement gauges), triangles (acoustic Doppler velocimeters) (from: Wu et al. [8]).

**Figure 15.**Snapshot of the initial setup of the simulation of a solitary wave transformation, breaking and runup over a three-dimensional reef.

**Figure 16.**Comparison of simulated and measured free surface elevations at gauge locations. Measured data (circles), numerical results (solid lines).

**Figure 17.**Comparison of simulated and measured velocities in the x direction at the acoustic Doppler velocimeters locations. Experimental data (circles), numerical results (solid lines).

${\mathit{H}}_{0}\left(\mathbf{cm}\right)$ | T (sec) | ${\mathit{L}}_{0}\left(\mathbf{m}\right)$ | h (m) | ${\mathit{\xi}}_{0}$ | Breaking Type | |
---|---|---|---|---|---|---|

T1 | 11 | 2 | 4.62 | 0.70 | 0.37 | Spilling |

T2 | 6.5 | 4 | 10.12 | 0.70 | 0.74 | Plunging |

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**MDPI and ACS Style**

Calvo, L.; De Padova, D.; Mossa, M.; Rosman, P. Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup. *Computation* **2021**, *9*, 47.
https://doi.org/10.3390/computation9040047

**AMA Style**

Calvo L, De Padova D, Mossa M, Rosman P. Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup. *Computation*. 2021; 9(4):47.
https://doi.org/10.3390/computation9040047

**Chicago/Turabian Style**

Calvo, Lucas, Diana De Padova, Michele Mossa, and Paulo Rosman. 2021. "Non-Hydrostatic Discontinuous/Continuous Galerkin Model for Wave Propagation, Breaking and Runup" *Computation* 9, no. 4: 47.
https://doi.org/10.3390/computation9040047