# Coupling phase-resolving coastal wave models
## The linear-hybrid case

<figure><img src="image_1.png" /><figcaption>PRESENTER:
**José** Galaz</figcaption></figure>

**BACKGROUND:** Instead of a 3d model, the hybrid model represents wave shoaling, breaking and runup by **coupling** Boussinesq-type and Saint-Venant models. It has gained wide adoption: It simply **switches-off/on** dispersive terms but it has three problems:

- Oscillations

- Mesh convergence (lack of)

- Instabilities

**IS THE HYBRID TRULY RELIABLE?**

**IS THE HYBRID'S PDE WORTH SOLVING?**

## METHODS

1. Through Laplace transform, the transmission conditions implicit in the hybrid model can be identified: Dirichlet + Neumann.

2. An iterative method (Schwarz Waveform Relaxation, SWRM) is used to prove the well-posedness by decomposing the domain into two coupled models with transmission conditions and **iterating on the traces of transmission conditions**.

3. If the initial condition is supported in one side of the interface, then the **coupling error** can be described as the difference between the hybrid and the **one-way coupling**, controlled by a “reflection coefficient”

## RESULTS

- The SWRM converges in 3 iterations, so the problem is well-posed: **YES**

- The coupling error solves a Cauchy problem with a filtered-reflected initial condition

- The SV to B case is always worse due to the larger reflection coefficient and the greater phase-speed

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

Take a picture to get my contact info

<figure><img src="image_4.png" /></figure>

# The linear hybrid Boussinesq-

## SaintVenant model is well-posed

## and produces $O(h^2)$ reflections

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

Use the **one-way coupling** as reference

<figure><img src="image_7.png" /></figure>

$$u - u^{3D} = (u - u^*) + (u^* - u^{3D})$$

Total error

Coupling error
= reflections

The linear hybrid

$$ \begin{cases} \partial_t \eta + h_0 \partial_x u = 0 & \text{in } \mathbb{R} \times ]0, T[ \\ (1 - x \frac{h_0^2}{3} \partial_x^2) \partial_t u + g \partial_x \eta = 0 & \text{in } \mathbb{R} \times ]0, T[ \\ \eta(\cdot, 0) = \eta_0, \quad u(\cdot, 0) = u_0 & \text{in } \mathbb{R} \end{cases} $$

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Deep water and SVB is worse</figcaption></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

José Galaz, Maria Kazolea, Antoine Rousseau
Inria team LEMON
Inria team CARDAMOM

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>
# Coupling phase-resolving coastal wave models
## The linear-hybrid case

<figure><img src="image_1.png" /><figcaption>PRESENTER:
**José** Galaz</figcaption></figure>

**BACKGROUND:** Instead of a 3d model, the hybrid model represents wave shoaling, breaking and runup by **coupling** Boussinesq-type and Saint-Venant models. It has gained wide adoption: It simply **switches-off/on** dispersive terms but it has three problems:

- Oscillations

- Mesh convergence (lack of)

- Instabilities

**IS THE HYBRID TRULY RELIABLE?**

**IS THE HYBRID'S PDE WORTH SOLVING?**

## METHODS

1. Through Laplace transform, the transmission conditions implicit in the hybrid model can be identified: Dirichlet + Neumann.

2. An iterative method (Schwarz Waveform Relaxation, SWRM) is used to prove the well-posedness by decomposing the domain into two coupled models with transmission conditions and **iterating on the traces of transmission conditions**.

3. If the initial condition is supported in one side of the interface, then the **coupling error** can be described as the difference between the hybrid and the **one-way coupling**, controlled by a “reflection coefficient”

## RESULTS

- The SWRM converges in 3 iterations, so the problem is well-posed: **YES**

- The coupling error solves a Cauchy problem with a filtered-reflected initial condition

- The SV to B case is always worse due to the larger reflection coefficient and the greater phase-speed

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

Take a picture to get my contact info

<figure><img src="image_4.png" /></figure>

# The linear hybrid Boussinesq-

## SaintVenant model is well-posed

## and produces $O(h^2)$ reflections

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

Use the **one-way coupling** as reference

1. Solve the first model everywhere
$$ \mathcal{L}_{-} \bar{u}_{-} = 0, x \in \mathbb{R} $$

<figure><img src="image_7.png" /></figure>

$$ u - u^{3D} = (u - u^*) + (u^* - u^{3D}) $$

Total error
Coupling error
= reflections

Model error

## The linear hybrid

$$ \begin{cases} \partial_t \eta + h_0 \partial_x u = 0 & \text{in } \mathbb{R} \times ]0, T[ \\ (1 - x \frac{h_0^2}{3} \partial_x^2) \partial_t u + g \partial_x \eta = 0 & \text{in } \mathbb{R} \times ]0, T[ \\ \eta(\cdot, 0) = \eta_0, \quad u(\cdot, 0) = u_0 & \text{in } \mathbb{R} \end{cases} $$

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Deep water and SVB is worse</figcaption></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

José Galaz, Maria Kazolea, Antoine Rousseau
Inria team LEMON
Inria team CARDAMOM

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>