Solution on a staggered grid

Solution of the linear gravity wave equation on a staggered grid does not suffer from the spurious mode of the solution on a regular grid. Consider the staggered grid of Fig. 13.

Fig. 13 Staggered grid configuration with variables \(u\) and \(\eta\) displace halp a grid step.

A scheme to solve (66) on the staggered grid, that also uses time staggering, is the following:

(72)\[\begin{split} \frac{\eta_m^{n+1} - \eta_m^{n}}{\Delta t} &= -H \frac{u_{m+\frac{1}{2}}^{n} - u_{m-\frac{1}{2}}^{n}}{\Delta x}\\ \frac{u_{m+\frac{1}{2}}^{n+1} - u_{m+\frac{1}{2}}^{n}}{\Delta t} &= -g \frac{\eta_{m+1}^{n+1} - \eta_{m}^{n+1}}{\Delta x}. \end{split}\]

To eliminate \(u\) from (72), we use the backward formula for the derivatives, thereby obtaining a 2nd order scheme:

(73)\[ \eta_m^{n+1} - 2\eta_{m}^{n} + \eta_m^{n-1} = gH\frac{\Delta t^2}{\Delta x^2} (\eta_{m+1}^{n+1} -2\eta_{m}^{n+1} + \eta_{m-1}^{n+1}). \]

For the staggered grid scheme (73), the spatial difference operator \(D\) is

\[\begin{split} \mathcal{S}(\eta)&=\frac{gH}{\Delta x^2}(e^{ik\Delta x}-2+e^{-ik\Delta x}) \\ &= -4\frac{gH}{\Delta x^2}\sin^2\frac{k\Delta x}{2} = -\omega^2, \end{split}\]

whose null space \(\mathcal{S}(\eta)=0\) is

\[ \frac{k\Delta x}{2}=\pi \implies k=\frac{2\pi}{\Delta x}. \]

This wavenumber is higher than the maximum wave number that a grid with spacing \(\Delta x\) can represent, so the staggered grid does not have a spurious mode.