Conservation of action

In this section, we proof that the discretization Eq. (8.16) is energy conserving. For this, we assume stationarity and neglect the source terms, so $F = 0$. From Eq. (8.16), it follows

  $${\vec{e}}^{(1)} \cdot (\vec{c}_{\vec{x}} N)_1 + {\vec{e}}^{(2)} \... ... \cdot (\vec{c}_{\vec{x}} N)_2 + {\vec{e}}^{(2)} \cdot (\vec{c}_{\vec{x}} N)_3$$ (8.40)

On the other hand, we have

  $$\nabla \cdot (\vec{c}_{\vec{x}} N) = \frac{1}{\Omega} \oint \vec{c}_{\vec{x}} N \cdot \vec{n} d\Gamma$$ (8.41)

The vectors $\vec{c}_{\vec{x}} N$ at edges are taken as averages, and so (see Figure 8.3 for reference)

$$\nabla \cdot (\vec{c}_{\vec{x}} N) \approx \frac{1}{2\Omega} [ (\, (\vec{c}_{\vec{x}} N)_1 + (\vec{c}_{\vec{x}} N)_2 \,) \cdot {\vec{n}}_{12} +$$

$$(\, (\vec{c}_{\vec{x}} N)_2 + (\vec{c}_{\vec{x}} N)_3 \,) \cdot {... ...\, (\vec{c}_{\vec{x}} N)_3 + (\vec{c}_{\vec{x}} N)_1 \,) \cdot {\vec{n}}_{31} ]$$ (8.42)

Using the identity

  $$\vec{n}_{23} = - \vec{n}_{12} - \vec{n}_{31}$$ (8.43)

and

  $$\vec{n}_{12} = -{\vec{e}}^{(2)} \, , \quad \vec{n}_{31} = -{\vec{e}}^{(1)}$$ (8.44)

we have

  $$\nabla \cdot (\vec{c}_{\vec{x}} N) \approx \frac{1}{2\Omega} [ {... ... (\vec{c}_{\vec{x}} N)_2 - {\vec{e}}^{(2)} \cdot (\vec{c}_{\vec{x}} N)_3 ] = 0$$ (8.45)

If the situation is stationary and there are no source terms then the divergence term is zero. Hence, the energy flux vector is divergence free. From this it follows that the closed integral of the flux normal to the faces of the triangle is zero, i.e. source free, if the compact BSBT scheme is applied. This also implies that the wave energy flux is constant along any wave characteristic in between faces $\vec{e}_{(1)}$ and $\vec{e}_{(2)}$ (see Figure 8.2 for reference). The BSBT scheme is thus consistent with local wave characteristics and can be viewed as a semi-Lagrangian scheme. This also holds for non-uniform depth and ambient current.