1. Part I: Non-linear dispersive PDE with boundary ?
    • Context & Notations
    • Modelisation of the problem
  2. Part II: Some theory
    • Reformulation of the problem as a boundary problem
    • Well-posedness of the problem
  3. Part III: Numerical point of view
    • Numerical simulations
Floating cylinder
Figure - Freely floating cylinder

Context

  • 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.
Floating cylinder
Figure - Slice of the problem

Notations

  • $\varepsilon = \frac{\text{typ. wave amplit.}}{\text{typ. depth}}$; $\kappa = \frac{\text{typ. depth}}{\text{typ. horiz. scale}}$

  • $\zeta$ - Elevation of the wave

  • $h = 1 + \varepsilon \zeta$ - 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:

$$ \left\{\begin{matrix} \partial_t \zeta + {\rm div}q = 0, \\ (1-\class{fragment highlight-blue}{\kappa^2 \partial_r {\rm div}})\partial_t q + \class{fragment highlight-blue}{\varepsilon {\rm div} \frac{q^2}{h}+ h\partial_r \zeta} = - h \partial_r P, \\ h = 1 + \varepsilon \zeta. \end{matrix}\right. $$

  • 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

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

$$ \Updownarrow $$
PDE system on $(R, +\infty)$ + ODE at the $\class{fragment highlight-blue}{\text{boundary}}$

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

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

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

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

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

    • Commutation formula: $$\mathfrak{R}_0\partial_r = \partial_r \mathfrak{R}_0.$$
    • Bounded operators on $L^\infty$ and $L^2$: if $u \in L^\infty$ (resp $u \in L^2$) then $\mathfrak{R}_{i}u \in L^\infty$ (resp $\mathfrak{R}_{i}u \in L^2$)
  3. Application of $\kappa^2{\rm div}$, use $\partial_t\zeta = - {\rm div}q$ and take the result in $r = R$
$\left\{\begin{matrix} (1-\kappa^2 \partial_r {\rm div})\mathfrak{R}_0 f = f \\ \underline{\mathfrak{R}_0 f} = 0 \end{matrix}\right.$
$$ \left\{\begin{matrix} \class{fragment highlight-blue}{\partial_t \zeta + {\rm div}q = 0}, \\ (1-\kappa^2 \partial_r {\rm div})\partial_t q + \partial_r \zeta = 0, \end{matrix}\right. \quad r \geq R.$$
$$\Downarrow \mathfrak{R}_0 \text{ on the $2^{nd}$ equation}$$
$$\class{fragment highlight-blue}{\partial_t q - \underline{\partial_t q}K(r/\kappa) + \mathfrak{R}_0\partial_r \zeta = 0}, \quad r \geq R \quad \text{where $K$ satisfies: $\left\{\begin{matrix} (1-\kappa^2 \partial_r {\rm div})K = 0 \\ \underline{K} = 1 \end{matrix}\right.$}$$
$$\Longrightarrow \kappa^2 \ddot{\zeta} + \underset{\zeta - \mathfrak{R}_1\zeta}{\underbrace{\kappa^2{\rm div}\partial_r \mathfrak{R}_1 \zeta}} = - \ddot{\delta} G(r/\kappa), \quad r \geq R$$ $$\Longrightarrow \class{fragment highlight-blue}{\kappa^2 \underline{\ddot{\zeta}} + \underline{\zeta} = \underline{\mathfrak{R}_1\zeta} - \ddot{\delta} G(R/\kappa)}$$
Newton's modifed equation:

$$ \class{fragment highlight-blue}{(m + \kappa^2 + \frac{R^2}{8})\ddot{\delta} + \delta = \kappa^2 \underline{\ddot{\zeta}} + \underline{\zeta}.} $$

Definition Well-posedness

A PDE $(E)$ on a space $\Omega \subset \mathbb{R}^n$ is said to be $\textit{well-posed}$ of time order $T^*$ if for any initial condition in $H(\Omega)$, $(E)$ admits a unique solution in $\mathscr{C}^k([0, T), H(\Omega))$ while $T \lesssim T^*$.

Definition The space $H^1_r$

We define the space $H^1_r$ as the functionnal space on $(R, +\infty)$ equipped with the norm $$ || \cdot ||_{H_1^r}^2 = ||\cdot||_{L^2_r}^2 + || {\rm div} \cdot||_{L^2_r}^2 + \kappa^2 ||\partial_r {\rm div} \cdot ||_{L^2_r}^2 $$

Theorem 2 Well-posedness of the Augmented formula

The augmented formula (PDE on $(\zeta, q)$ $(R, +\infty)$ + ODE on $(\delta, \dot{\delta}, \underline{\zeta}, \underline{\dot{\zeta}})$ at $r = R$) is well-posed in $H_r \times H^1_r \times \mathbb{R}^4$ for time order $\class{fragment highlight-blue}{T_{\max} = O(\min(\varepsilon^{-1}\kappa, \varepsilon^{-1/2})}$.

For a general PDE of unknown $u$ with initial condition $u_0$ under the form $$ \partial_t u + Au = \mathcal{F}, $$

the formal solution is given by the Duhamel's formula: $$ \forall t > 0, t < T^*, \, u(t) = e^{tA}u_0 + \underset{\text{Contractant ?}}{\underbrace{\int_0^t e^{(t-s)A}\mathcal{F}(s)ds}}. $$

Goal: for a nonlinear PDE, obtaining a contractant estimate for $\mathcal{F} \class{fragment}{= \mathcal{F}[u]}$

  • $$ \left\{\begin{matrix} \partial_t u + \partial_r \mathfrak{F}_\kappa [u, Z] = \varepsilon \mathcal{F}[u, Z], \\ \partial_t Z + \mathfrak{G}(u, Z) = \varepsilon \mathcal{G}(u, Z). \end{matrix}\right. $$
    $$ \left\{\begin{matrix} \partial_t u^{k+1} + \partial_r \mathfrak{F}_\kappa [u^{k+1}] = \varepsilon \mathcal{F}[u^k, Z^k],\\ \partial_t Z^{k+1} + \mathfrak{G}(u^{k+1}, Z^{k+1}) = \varepsilon \mathcal{G}(u^{k}, Z^{k}). \end{matrix}\right. $$
  • Goal: for $u_1, u_2$, $Z_1, Z_2$, control of type $$ ||\mathcal{F}[u_1, Z_1](t) - \mathcal{F}[u_2, Z_2](t)||_{Y} \leq C t||(u_1, Z_1) - (u_2, Z_2)||_{X} $$ and $$ |\mathcal{G}[u_1, Z_1](t) - \mathcal{G}[u_2, Z_2](t)| \leq \tilde{C} t||(u_1, Z_1) - (u_2, Z_2)||_{X} $$ for $C, \tilde{C} > 0$ independant of $t$.
  • In this case the nonlinearities are contractant while $t < \max(C^{-1}, \tilde{C}^{-1})$.
$\longrightarrow$ With our $\mathcal{F}$ and $\mathcal{G}$, it leads to the well-posedness of time order $$T = O(\min(\varepsilon^{-1}\class{fragment highlight-red}{\kappa}, \varepsilon^{-1/2}))$$

Theorem 3

In the linear regime, $\delta \in H^2(\mathbb{R}^+)$ but: $$ \forall \beta \in (0, 2), \, \delta, \dot{\delta}, \ddot{\delta} \notin L^2(t^{\beta}dt, \mathbb{R}^+). $$ Also, for large time scales: $$ \forall k \in \{0, 1, 2\}, \, \forall 0 < \rho \ll 1, t^{-1/2-\rho} \lesssim |\delta^{(k)}(t)| $$
In the linear regime, the augmented formula reads: $$ \left\{\begin{matrix} \partial_t \zeta + {\rm div}q = 0, \quad r > R \\ \kappa^2 \ddot{\zeta} + \zeta = \mathfrak{R}_1 \zeta - \frac{R}{2}\ddot{\delta}G(r/\kappa), \quad r \geq R \\ \tilde{m}\ddot{\delta} + \delta = \kappa^2 \underline{\ddot{\zeta}} + \underline{\zeta}, \\ \underline{q} = -\frac{R}{2}\dot{\delta}. \end{matrix}\right. $$
Therefore the computation of $\zeta$ and $q$ are decoupled:
  • Computation of $\mathfrak{R}_1 \zeta$
  • Computation of $\delta$ and $\zeta$
  • Reconstruction of $q$ (Optionnal)
$$ \left\{\begin{matrix} \kappa^2 \ddot{\zeta} + \zeta = \mathfrak{R}_1 \zeta - \frac{R}{2}M^{-1} (\underline{\mathfrak{R}_1 \zeta} - \delta) G(r/\kappa), \\ M\ddot{\delta} + \delta = \underline{\mathfrak{R}_1 \zeta}. \end{matrix}\right. $$
$$ \left\{\begin{matrix} \kappa^2\frac{\zeta^{n+1}_i - 2 \zeta^n_i + \zeta^{n-1}_i}{\Delta t^2} + \left(I - \mathfrak{R}_1\right)\zeta^n_i = M^{-1} ((\mathfrak{R}_1\zeta^n)_0 - \delta^n )G_i, & i \in \{1, ..., N \},\\ \frac{\delta^{n+1} - 2 \delta^n + \delta^{n-1}}{\Delta t^2} = M^{-1} ( (\mathfrak{R}_1 \zeta^n)_0 - \delta^n ), \end{matrix}\right. $$
$$ \left\{\begin{matrix} \kappa^2\frac{\zeta^{n+1}_i - 2 \zeta^n_i + \zeta^{n-1}_i}{\Delta t^2} + \left(I - \mathfrak{R}_1\right)\zeta^n_i = M^{-1} ((\mathfrak{R}_1\zeta^n)_0 - \delta^n )G_i, & i \in \{1, ..., N \},\\ \frac{\delta^{n+1} - 2 \delta^n + \delta^{n-1}}{\Delta t^2} = M^{-1} ( (\mathfrak{R}_1 \zeta^n)_0 - \delta^n ), \end{matrix}\right. $$
Important points:
  • Discretization of $\mathfrak{R}_1$ as an elliptic problem,
  • Space's discretization not truely important,
  • CFL of the form: $$\Delta t \lesssim \kappa \quad \& \quad \Delta x \lesssim \kappa.$$

Figure - Simulation

  • Well-posedness with time existence uniform in $\kappa$ in order to obtain the shallow-water asymptotic,
  • Well-posedness with a larger time existence in the linear case,
  • A spectral scheme for the small values of $\kappa$,
  • Non-linear case for the numerical analysis,
  • Flux scheme for the numerical simulations,
  • Model without the symetries.
  • Later again: Ice sheet model to understand the ice sheet fracturation.

Article: Freely floating cylinder on a 3D fluid governed by the Boussinesq equations in the axisymmetric without swirl case, G.Beck, E.Contentin, L.Martaud (2025)

Proceeding: The return to the equilibrium of a floating cylinder in the Boussinesq regime, G.Beck, E.Contentin, L.Martaud (2026)

Thank you!

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Alan Riquier, postdoctoral at Fields Institute.