Equations and modelisation

Equations

The Non-linear Shallow Water equations (NSW) are formulated below with conservative variables:

\[ \left\{ \begin{array}{l} {\displaystyle \frac{\partial h}{\partial t}+\mathrm{div}\left(h\mathbf{u}\right)}=0\\ \\ {\displaystyle \frac{\partial\left(h\mathbf{u}\right)}{\partial t}+\mathrm{div}\left(h\mathbf{u}\otimes\mathbf{u}\right)}=-gh\nabla\left(h+b\right)-fh\mathbf{u}^{\bot}-S_{F}\left(h,\mathbf{u}\right) \end{array}\right. \]

where \(h = h(\mathbf{x}, t)\) stands for the water depth, \(\mathbf{u} = \mathbf{u}(\mathbf{x}, t)\) for the horizontal velocity and \(b\) for the bottom elevation. \(fh\mathbf{u}^{\bot}\) is the Coriolis force and \(S_{F}\left(h,\mathbf{u}\right)\) the friction source term.

For hydraulic simulations, the friction source term is defined by the Manning-Strickler friction source term :

\[ S_{F}\left(h,\mathbf{u}\right)={\displaystyle C_{f}\frac{\left\Vert \mathbf{u}\right\Vert \mathbf{u}}{h^{1/3}}} \]

with \(C_{f}=gn^{2}\) depending on \(n\) the uniform Manning's roughness coefficient.

For oceanic simulations, the friction source term can be written as :

\[ S_{F}\left(h,\mathbf{u}\right)=C_{l}\,\mathbf{u}+C_{b}\left(h\right)\,\left\Vert \mathbf{u}\right\Vert \mathbf{u} \]

Numerical resolution

Hyperbolic part

Friction source term

A time-splliting scheme is used to treat the friction source term, then we focus here on the resolution of

\[ \left\{ \begin{array}{l} {\displaystyle \frac{\partial h}{\partial t}=0}\\ \\ {\displaystyle \frac{\partial\left(h\mathbf{u}\right)}{\partial t}}=-S_{F}\left(h,\mathbf{u}\right) \end{array}\right. \]

Oceanic friction source term

\[ \left\{ \begin{array}{l} {\displaystyle \frac{\partial h}{\partial t}=0}\\ \\ {\displaystyle \frac{\partial\left(h\mathbf{u}\right)}{\partial t}}=-C_{l}\,\mathbf{u}-C_{b}\left(h\right)\,\left\Vert \mathbf{u}\right\Vert \mathbf{u} \end{array}\right. \]

Semi-implicit resolution

The semi-implicit time step resolution writes,

\[ h^{n}\mathbf{u}^{n+1}=h^{n}\bar{\mathbf{u}}^{n+1}-\Delta t\left(C_{l}\,\mathbf{u}^{n+1}+C_{b}\left(h^{n}\right)\,\left\Vert \bar{\mathbf{u}}^{n+1}\right\Vert \mathbf{u}^{n+1}\right) \]

which easely gives the final expression of the velocity,

\[ \mathbf{u}^{n+1}={\displaystyle \frac{h^{n}}{h^{n}+\Delta t\left(C_{l}+C_{b}\left(h^{n}\right)\,\left\Vert \bar{\mathbf{u}}^{n+1}\right\Vert \right)}\,\bar{\mathbf{u}}^{n+1}}=\bar{\mathbf{u}}^{n+1}+\Delta t\,S\left(h^{n},\bar{\mathbf{u}}^{n+1},\Delta t\right) \]

Full implicit resolution

The full implicit time step resolution writes,

\[ h^{n}\mathbf{u}^{n+1}=h^{n}\bar{\mathbf{u}}^{n+1}-\Delta t\left(C_{l}\,\mathbf{u}^{n+1}+C_{b}\left(h^{n}\right)\,\left\Vert \mathbf{u}^{n+1}\right\Vert \mathbf{u}^{n+1}\right) \]

\[ \Delta t\,C_{b}\left(h^{n}\right)\,\left\Vert \mathbf{u}^{n+1}\right\Vert \mathbf{u}^{n+1}+\mathbf{u}^{n+1}\left(h^{n}+\Delta t\,C_{l}\right)=h^{n}\bar{\mathbf{u}}^{n+1} \]

Remarking that \(\mathbf{u}^{n+1}=\alpha\bar{\mathbf{u}}^{n+1}\), the solution is given by a quadratic equation,

\[ \Delta t\,C_{b}\left(h^{n}\right)\,\left\Vert \mathbf{u}^{n}\right\Vert \,\alpha^{2}+\left(h^{n}+\Delta t\,C_{l}\right)\,\alpha-h^{n}=0 \]

with the final solution,

\[ \mathbf{u}^{n+1}={\displaystyle \frac{2\,h^{n}}{h^{n}+\Delta t\,C_{l}+\sqrt{\left(h^{n}+\Delta t\,C_{l}\right)^{2}+4\,\Delta t\,h^{n}\,C_{b}\left(h^{n}\right)\,\left\Vert \bar{\mathbf{u}}^{n+1}\right\Vert }}\,\bar{\mathbf{u}}^{n+1}=\bar{\mathbf{u}}^{n+1}+\Delta t\,S\left(h^{n},\bar{\mathbf{u}}^{n+1},\Delta t\right)} \]

Manning-Strickler source term

\[ \left\{ \begin{array}{l} {\displaystyle \frac{\partial h}{\partial t}=0}\\ \\ {\displaystyle \frac{\partial\left(h\mathbf{u}\right)}{\partial t}}={\displaystyle C_{f}\,\frac{\left\Vert \mathbf{u}\right\Vert \mathbf{u}}{h^{1/3}}} \end{array}\right. \]

Semi-implicit resolution

The semi-implicit time step resolution writes,

\[ h^{n}\mathbf{u}^{n+1}=h^{n}\mathbf{u}^{n}-\Delta t\,{\displaystyle C_{f}\,\frac{\left\Vert \mathbf{u}^{n+1}\right\Vert \mathbf{u}^{n+1}}{\left(h^{n}\right)^{1/3}}} \]

which easely gives the final expression of the velocity,

\[ \mathbf{u}^{n+1}={\displaystyle \frac{\left(h^{n}\right)^{4/3}}{\left(h^{n}\right)^{4/3}+C_{f}\,\Delta t\,\left\Vert \mathbf{u}^{n}\right\Vert }\,\mathbf{u}^{n}} \]

Full implicit resolution

The full implicit time step resolution writes,

\[ h^{n}\mathbf{u}^{n+1}=h^{n}\mathbf{u}^{n}-\Delta t\,{\displaystyle C_{f}\,\frac{\left\Vert \mathbf{u}^{n+1}\right\Vert \mathbf{u}^{n+1}}{\left(h^{n}\right)^{1/3}}} \]

\[ \Delta t\,C_{f}\,\left\Vert \mathbf{u}^{n+1}\right\Vert \mathbf{u}^{n+1}+\left(h^{n}\right)^{4/3}\left(\mathbf{u}^{n+1}-\mathbf{u}^{n}\right)=0 \]

Again, remarking that \(\mathbf{u}^{n+1}=\alpha\mathbf{u}^{n}\), the solution is given by a quadratic equation,

\[ \Delta t\,C_{f}\,\left\Vert \mathbf{u}^{n}\right\Vert \alpha^{2}+\left(h^{n}\right)^{4/3}\alpha-\left(h^{n}\right)^{4/3}=0 \]

with the final solution,

\[ \mathbf{u}^{n+1}={\displaystyle \frac{2\,\left(h^{n}\right)^{2/3}}{\left(h^{n}\right)^{2/3}+\sqrt{\left(h^{n}\right)^{4/3}+4\,\Delta t\,C_{f}\,\left\Vert \mathbf{u}^{n}\right\Vert }}\,\mathbf{u}^{n}} \]

Coriolis force