1. Part I: Modelisation of the problem
    • Context & Notations
    • Modelisation & Reformulation of the problem
  2. Part II: A first scheme for the numerical simulations
    • Discretization
    • Energy inequalities
  3. Part III: Shallow-water asymptotic & spectral scheme
    • Spectral scheme
    • Comparison with the first method
Floating cylinder
Figure - Freely floating cylinder

Context

  • Cylinder of radius $R > 0$,
  • Axisymetry with respect to the axis of revolution of the cylinder,
  • Conservation of irrotationnality,
  • The fluid fits the bottom of the cylinder under the cylinder,
  • Case of the return to the equilibrium.
Floating cylinder
Figure - Slice of the problem

Notations

  • $\kappa = \frac{\text{typ. depth}}{\text{typ. horiz. scale}}$

  • $\zeta$ - Elevation of the wave

  • $h = h_{\rm eq} + \zeta$ - Height of the water

  • $q = hu$ - Discharge

  • $\delta$ - Position of the solid

  • $P$ - Pressure at the surface

Remarks:

  • In $\mathcal{I}, \dot{\zeta}_i = \dot{\delta}$ and $P_i$ is unknown
  • In $\mathcal{E}, P_e = P_{atm}$ and $\zeta_e$ is unknown

Boussinesq-Abbott equations:

$$ \left\{\begin{matrix} \partial_t \zeta + {\rm d}_r q = 0, \\ (1-\kappa^2 \partial_r {\rm d}_r)\partial_t q + \partial_r \zeta = - \partial_r P, \\ {\rm d}_r = \partial_r + \frac{d-1}{r}. \end{matrix}\right. $$

Newton's equation:

$$m\ddot{\delta} + mg = \int_\mathcal{I} P$$

  • A very (maybe) last notation:
    • $\underline{f_e} = \underset{\rho \to R}{\lim}f(r+\rho)$
  • Transmission conditions:
    • $\underline{q_i} = \underline{q_e}$
    • $\underline{P_i} = \mathcal{L}(\zeta, q, \delta)$
  • Remark:
    • Since $\dot{\zeta}_i = \dot{\delta}$, then $\underline{q_e} = -\frac{R}{2}\dot{\delta}$

Theorem 1 Reformulation

PDE system on $\mathbb{R}^+$ + Transmission conditions

$$ \left\{\begin{matrix} \partial_t \zeta + {\rm d}_r q = 0, \\ (1-\kappa^2 \partial_r {\rm d}_r)\partial_t q + \partial_r \zeta = - \partial_r P, \\ \end{matrix}\right. \quad r \geq 0, \, t > 0.$$

$$ \Updownarrow $$
Augmented formula: PDE system on $(R, +\infty)$ + ODE at the boundary

$$ \left\{\begin{matrix} \partial_t \zeta + \partial_r q = -\frac{q}{r}, \\ \partial_t q + \partial_r \mathcal{R}\zeta = -\ddot{\delta} K(r/\kappa), \\ \underline{q} = -R\dot{\delta}, \end{matrix}\right. \quad r > R, \, t > 0 \quad + \quad \left\{\begin{matrix} \kappa^2\underline{\ddot{\zeta}} + \underline{\zeta} = \underline{\mathcal{R}\zeta} - G(R/\kappa)\ddot{\delta}, \\ \tilde{m} \ddot{\delta} + \delta = \kappa^2\ddot{\underline{\zeta}} + \underline{\zeta}, \end{matrix}\right. \quad t > 0. $$

$$ \left\{\begin{matrix} \partial_t \zeta + {\rm d}_r q = 0, \\ (1-\kappa^2 \partial_r {\rm d}_r)\partial_t q + \partial_r \zeta = 0, \end{matrix}\right. \quad r > R.$$
  1. Reformulation of the Newton's equation

  2. Inversion of the dispersive operator $(1-\kappa^2 \partial_r {\rm d}_r)$:

    • Operators $\mathfrak{R}_0$ & $\mathfrak{R}_1$ defined as:

      $$ \forall f, \mathfrak{R}_0 f = f \Leftrightarrow \left\{\begin{matrix} (1-\kappa^2 \partial_r {\rm d}_r)\mathfrak{R}_0 f = f \\ \underline{\mathfrak{R}_0 f} = 0 \end{matrix}\right., \quad \mathfrak{R}_1 f = f \Leftrightarrow \left\{\begin{matrix} (1-\kappa^2 {\rm d}_r\partial_r)\mathfrak{R}_1 f = f \\ \underline{\partial_r\mathfrak{R}_1 f} = 0 \end{matrix}\right. $$

    • Bounded operators on $L^\infty$ and $L^2$
  3. Application of $\kappa^2{\rm d}_r$, use $\partial_t\zeta = - {\rm d_r}q$ and take the result in $r = R$
$$\Rightarrow \left\{\begin{matrix} \partial_t \zeta + {\rm d}_r q = 0, \quad r > R \\ \kappa^2 \ddot{\zeta} + \zeta = \mathfrak{R}_1 \zeta - \frac{R}{2}\ddot{\delta}G(r/\kappa), \quad r \geq R \\ \tilde{m}\ddot{\delta} + \delta = \kappa^2 \underline{\ddot{\zeta}} + \underline{\zeta}, \\ \underline{q} = -\frac{R}{2}\dot{\delta}. \end{matrix}\right. $$
The augmented formula reads: $$ \left\{\begin{matrix} \partial_t \zeta + {\rm d}_r q = 0, \quad r > R \\ \kappa^2 \ddot{\zeta} + \zeta = \mathcal{R} \zeta - R\kappa\ddot{\delta}G(r/\kappa), \quad r \geq R \\ \tilde{m}\ddot{\delta} + \delta = \kappa^2 \underline{\ddot{\zeta}} + \underline{\zeta}, \\ \underline{q} = -R\dot{\delta}, \end{matrix}\right. \quad \text{with} \quad \left\{\begin{matrix} (1 - \kappa^2 {\rm d}_r \partial_r) \mathcal{R}u = u,\\ \underline{\partial_r \mathcal{R}u} = 0. \end{matrix}\right. $$
Therefore the computation of $\zeta$ and $q$ are decoupled:
$$ \left\{\begin{matrix} \kappa^2 \ddot{\zeta} + \zeta = \mathcal{R} \zeta - \kappa R\ddot{\delta}G(r/\kappa), \quad r \geq R \\ M\ddot{\delta} + \delta = \underline{\mathcal{R} \zeta}, \\ \end{matrix}\right. \quad + \quad \left\{\begin{matrix} \partial_t \zeta + {\rm d}_r q = 0, \quad r > R \\ \underline{q} = -\frac{R}{2}\dot{\delta}. \end{matrix}\right. $$
  • Computation of $\mathcal{R} \zeta$ (depending on the dimension)
  • Computation of $\delta$ and $\zeta$
  • Reconstruction of $q$ (Optionnal)
$$ \left\{\begin{matrix} \kappa^2 \ddot{\zeta} + \zeta = \mathcal{R} \zeta - \frac{R}{2}M^{-1} (\underline{\mathcal{R} \zeta} - \delta) G(r/\kappa), \\ M\ddot{\delta} + \delta = \underline{\mathcal{R} \zeta}. \end{matrix}\right. $$
$$ \left\{\begin{matrix} \kappa^2\frac{\zeta^{n+1}_i - 2 \zeta^n_i + \zeta^{n-1}_i}{\Delta t^2} + \left(I - \mathcal{R}\right)\zeta^n_i = M^{-1} ((\mathcal{R}\zeta^n)_0 - \delta^n )G_i, & i \in \{1, ..., N \},\\ \frac{\delta^{n+1} - 2 \delta^n + \delta^{n-1}}{\Delta t^2} = M^{-1} ( (\mathcal{R} \zeta^n)_0 - \delta^n ), \end{matrix}\right. $$

Theorem 3 Energy inequality on the scheme

Let $\kappa > 0$ and $ \Delta t \lesssim \kappa $, $\Delta x \lesssim \kappa$. If $\mathcal{R}$ satisfies certain inequalities then the time scheme satisfies the discrete energy inequality:
\begin{equation} \frac{\mathfrak{E}^{n+1/2} - \mathfrak{E}^{n-1/2}}{\Delta t} \lesssim \frac{1}{\kappa} (\mathfrak{E}^{n+1/2} + \mathfrak{E}^{n-1/2}) \end{equation} with $\mathfrak{E}^{n+1/2} = E_{\rm solid}^{n+1/2} + E_{\rm fluid}^{n+1/2} $
$$ \left\{\begin{matrix} \kappa^2\frac{\zeta^{n+1}_i - 2 \zeta^n_i + \zeta^{n-1}_i}{\Delta t^2} + \left(I - \mathcal{R}\right)\zeta^n_i = M^{-1} ((\mathcal{R}\zeta^n)_0 - \delta^n )G_i, & i \in \{1, ..., N \},\\ \frac{\delta^{n+1} - 2 \delta^n + \delta^{n-1}}{\Delta t^2} = M^{-1} ( (\mathcal{R} \zeta^n)_0 - \delta^n ), \end{matrix}\right. $$
Important points:
  • Easily adaptable with the dimension,
  • Discretization of $\mathcal{R}$ as an eliptic problem,
  • Space's discretization not truely important,
  • PDE but with conditions of the form: $$\Delta t \lesssim \kappa \quad \& \quad \Delta x \lesssim \kappa.$$
Simulations EDO

Figure - Simulations with different values of $\kappa$

Energie EDO kappa petit

Figure - Simulations with $\kappa = 0.001$

$$ \left\{\begin{matrix} \kappa^2 \ddot{\zeta} + \zeta = \mathcal{R} \zeta - \frac{R}{2}M^{-1} (\underline{\mathcal{R} \zeta} - \delta) G(r/\kappa), \\ M\ddot{\delta} + \delta = \underline{\mathcal{R}\zeta}. \end{matrix}\right. $$
The Fourier cos transform applied to the equation of $\zeta$: $$ \forall k \in \mathbb{N}, \, \kappa^2 \ddot{\widehat{\zeta}_k} + \kappa^2 \omega_k^2 \widehat{\zeta}_k = c\kappa^2(\underline{\mathcal{R} \zeta} - \delta) \frac{1}{1+\kappa^2 k^2}, \quad \omega_k^2 = \frac{k^2}{1+\kappa^2 k^2} $$
$\Rightarrow$ expression of $\widehat{\zeta}_k$ for all $k$,
Also $\forall u, \underline{\mathcal{R}u} = \int_0^{+\infty} \frac{\widehat{u}(k)}{1+\kappa^2k^2}dk$,
$\Rightarrow$ expression of $\underline{\mathcal{R}\zeta}$ for all $t$ as $$ \forall t > 0, \, \underline{\mathcal{R}\zeta} = \underline{\mathcal{R}\zeta}^{\rm homog} + \int_0^t K(s-t)(\underline{\mathcal{R} \zeta}(s) - \delta(s))ds $$
Discretization: $$ \forall k \in \mathbb{N}, \, \left\{\begin{matrix} \widehat{\zeta}^{n+1}_k = \cos(\omega_k\Delta t) \widehat{\zeta}_k^{n} + \frac{\sin(\omega_k\Delta t)}{\omega_k}\widehat{\dot{\zeta}}_k^{n} + \frac{c}{1+\kappa^2 k^2} (\underline{\mathcal{R} \zeta}^n - \delta^n) \frac{1 - \cos(\omega_k\Delta t)}{\omega_k^2}, \\ \widehat{\dot{\zeta}}_k^{n+1} = -\omega_k\sin(\omega_k\Delta t) \widehat{\zeta}_k^{n} + \cos(\omega_k\Delta t)\widehat{\dot{\zeta}}_k^{n} - \frac{c}{1+\kappa^2 k^2} (\underline{\mathcal{R} \zeta}^n - \delta^n)\frac{\sin(\omega_k\Delta t)}{\omega_k}. \end{matrix}\right. $$
\begin{equation} \left\{\begin{matrix} \delta^{n+1} = \cos(\Delta t \sqrt{M})\delta^{n} + \frac{\sin(\sqrt{M}\Delta t)}{\sqrt{M}}\dot{\delta}^{n} + (1 - \cos(\sqrt{M}\Delta t))\underline{\mathcal{R} \zeta}^n,\\ \dot{\delta}^{n+1} = -\sqrt{M}\sin(\sqrt{M}\Delta t) \delta^{n} + \cos(\sqrt{M}\Delta t)\dot{\delta}^{n} + \sin(\sqrt{M}\Delta t)\underline{\mathcal{R} \zeta}^n \end{matrix}\right. \end{equation}
Comparaison

Figure - Comparaison of the methods with $\kappa = 0.3$

Comparaison

Figure - Spectral method with $\kappa = 10^{-5}$

Convergence curve of the ODE computation to the raffined Spectral scheme, on $||\delta_{\rm EDO} - \delta_{\rm spec}||_{l^1_{\Delta t}}, ||\delta_{\rm EDO} - \delta_{\rm spec}||_{l^2_{\Delta t}}, ||\zeta_{\rm EDO} - \zeta_{\rm spec}||_{l^2_{\Delta x}}$.
Comparaison
Convergence curve of $\zeta$ in the case where $\delta$ is given by $t\mapsto e^{-t/\tau}$, $\tau > 0$ a given parameter, and $\zeta$ has an exact solution.
Comparaison
  • Non-linear regime,
  • Flux scheme,
  • 2D spectral method.

Thank you!

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Alan Riquier, postdoctoral at Fields Institute.