Fully discrete stability and dispersion analysis for a linear dispersive internal wave model

Abstract

In the context of modeling internal water waves, some strongly nonlinear reduced models exhibit a nonlocal term involving a Hilbert-type transform. These models accurately account for the physical dispersion regarding the Euler equations. To perform a careful stability analysis and detect whether numerical solutions are reliable or spurious, it is necessary to adapt the classical von Neumann analysis to account for nonlocal dispersive terms. We address this issue by a fully discrete analysis of the one-dimensional linear model, in the flat bottom case. We find a formula for the amplification factor that provides estimates concerned with numerical stability and dispersion (namely, phase errors). Subsequently, we contrast the numerical properties of the original dispersive problem with that of the underlying non-dispersive case, namely a linear hyperbolic system. The stability estimates corroborate the fact that physical dispersion provided by a nonlocal (singular integral) term allows for less restrictive stability conditions.

Proof of Theorem 2

Our proof of Theorem 2 is based on the Fourier Series framework provided in Iório and Iório (2001). Similar arguments have been previously employed in Bona et al. (2002) for a class of Boussinesq systems.

Proof

Proceeding as in Lemma 2, we find

$$\begin{aligned} \Vert [\eta (t,\cdot ), u(t,\cdot )]^{T} \Vert ^2_{s,s+1/2} = \frac{1}{2l}\sum _{k=-\infty }^\infty \left[ 1 + \kappa ^2\right] ^{s}\left\| A(\kappa ) \left[ \hat{\eta }(k),\hat{u}(k)\right] ^T\right\| ^2_2, \end{aligned}$$

where $\kappa =k\pi /l$ and the matrix function $A(\kappa )$ is defined in Eq. (38). Moreover, it follows from Eq. (8) that

$$\begin{aligned} \left\| A(\kappa )\left[ \hat{\eta }(k),\hat{u}(k)\right] ^T\right\| _2 \le \left\| A(\kappa )G\left( t,\kappa \right) A^{-1}(\kappa )\right\| _2 \left\| A(\kappa )\left[ \hat{\eta }^0(k),\hat{u}^0(k)\right] ^T\right\| _2, \end{aligned}$$

whereas $G\left( t,\kappa \right)$ can be written as $G\left( t,\kappa \right) =V(\kappa ) \,\tilde{G}(t,\kappa ) \,V^{-1}(\kappa )$, where

$$\begin{aligned} V(\kappa ) = \begin{bmatrix} 1&\quad 0\\ 0&\quad v(\kappa ) \end{bmatrix}, \quad \tilde{G}(t,\kappa ) = \begin{bmatrix} \cos (\omega (\kappa ) t)&\quad i\sin (\omega (\kappa ) t)\\ i\sin (\omega (\kappa ) t)&\quad \cos (\omega (\kappa ) t) \end{bmatrix}. \end{aligned}$$

(42)

We have $\Vert A(\kappa )G\left( t,\kappa \right) A^{-1}(\kappa )\Vert _2 \le \Vert A(\kappa )V(\kappa )\Vert _2\Vert (A(\kappa )V(\kappa ))^{-1}\Vert _2$, i.e.,

$$\begin{aligned} \Vert A(\kappa )G\left( t,\kappa \right) A^{-1}(\kappa )\Vert _2 \le \max \left\{ v(\kappa )\root 4 \of {1+\kappa ^2},\frac{1}{v(\kappa )\root 4 \of {1+\kappa ^2}}\right\} . \end{aligned}$$

(43)

Taking into account that the function $\phi$ defined in Eq. (5) satisfies $|\kappa |\le \phi (\kappa )\le 1+|\kappa |$, we can find the following upper bound for the inequality (43):

$$\begin{aligned} \Vert A(\kappa )G\left( t,\kappa \right) A^{-1}(\kappa )\Vert _2 \le \max \left\{ \sqrt{\frac{\rho _1}{\rho _2}\frac{1}{\sqrt{\beta }}},\root 4 \of {2}\sqrt{1+\frac{\rho _2}{\rho _1}\frac{h_1}{h_2}\max \{1,\delta \}}\right\} . \end{aligned}$$

The inequality (13) follows from choosing $C$ as the above upper bound. Analogously, for the persistence property (14) we can prove that

$$\begin{aligned} \left\| A(\kappa )\partial _t^r\left[ \hat{\eta }(k),\hat{u}(k)\right] ^T\right\| ^2_2&\le \left\| A(\kappa )\partial _t^rG\left( t,\kappa \right) A^{-1}(\kappa )\right\| ^2_2\left\| A(\kappa )[\hat{\eta }^0(k),\hat{u}^0(k)]^T\right\| ^2_2\\&\le \tilde{C}\left[ 1+\kappa ^2\right] ^{r/2} \left\| A(\kappa )\left[ \hat{\eta }^0(k),\hat{u}^0(k)\right] ^T\right\| ^2_2 \end{aligned}$$

where the constant $\tilde{C}>0$ does not depend on $t$, thus using the Weierstrass M-test we conclude that $[\partial _t^r \eta (t,x), \partial _t^r u(t,x)]^{T} \in C([0,\infty ); H^{(s-r/2,s+1/2-r/2)})$. $\square$