A Lagrange-Galerkin hp-Finite Element Method for a 3D Nonhydrostatic Ocean Model

Abstract

We introduce in this paper a Lagrange-Galerkin hp-finite element method to calculate the numerical solution of a nonhydrostatic ocean model. The Lagrange-Galerkin method yields a Stokes-like problem the solution of which is computed by a second-order rotational splitting scheme that separates the calculation of the velocity and pressure, the latter is decomposed into hydrostatic and nonhydrostatic components. We have tested the method in flows where the nonhydrostatic effects are important. The results are very encouraging.

Appendices

Appendix 1: Existence and Uniqueness of the Density Problem

This appendix is related to the existence and uniqueness of the variational problem (13) that is used in the numerical scheme to compute the function $\bar{\rho }$ defined in (11). In a general framework, this can be written as follows. Given $f\in L^{2}\left( D\right)$, find $u\in H_{0}( D,\frac{\partial }{\partial z})$ such that

$$\begin{aligned} \int _{D}\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}=\int _{D}f\frac{\partial v}{\partial z}\qquad \text{ for } \text{ all } v\in H_{0}\left( D,\frac{\partial }{\partial z}\right) . \end{aligned}$$

(14)

The space $H\left( D,\frac{\partial }{\partial z}\right)$ defined in (5) is a Hilbert space with norm

$$\begin{aligned} \left\| v\right\| _{H\left( D,\frac{\partial }{\partial z}\right) }=\sqrt{\left\| v\right\| ^{2}+\left\| \frac{\partial v}{\partial z}\right\| ^{2}}. \end{aligned}$$

The subspace $H_{0}( D,\frac{\partial }{\partial z})$ is well defined because following Temam (1977), we can define a trace operator $\gamma :H(D,\frac{\partial }{\partial z}) \rightarrow H^{-\frac{1}{2}}\left( \Gamma _{s}\right)$ the kernel of which is the space $H_{0}( D,\frac{\partial }{\partial z})$. It is easy to see that $\left\| \frac{\partial u}{\partial z}\right\|$ is a norm in $H_{0}( D,\frac{\partial }{\partial z})$. Now applying the Lax-Milgram Lemma, it follows that problem (14) has a unique solution.

Appendix 2: Impact of Extrapolated Coriolis Term on the Overall Stability

Roughly speaking, Eq. (8) is the semi-discrete version of

$$\begin{aligned} \frac{{\rm{D}}\mathbf {u}}{{\rm{D}}t}-\text{ div }\left( A\nabla \mathbf {u}\right) +\mathbf {f}\times \mathbf {u}=\mathbf {r}, \end{aligned}$$

and the finite element formulation of this equation yields

$$\begin{aligned} M\frac{{\rm{D}}\mathbf {u}_h}{{\rm{D}}t}+S\mathbf {u}_h+M\left( \mathbf {f}\times \mathbf {u}_h\right) =M\mathbf {r}_h, \end{aligned}$$

(15)

where M and S are the mass and stiffness matrices, respectively, the coefficients of which are given by

$$\begin{aligned} m_{ij}= & {} \int _{D}\phi _{i}\phi _{j},\\ s_{ij}= & {} \int _{D}A\nabla \phi _{i}\nabla \phi _{j}, \end{aligned}$$

and where $\mathbf {u}_h$ and $\mathbf {r}_h$ are the finite element approximation of $\mathbf {u}$ and $\mathbf {r}$, respectively. Since M is a non-singular matrix, we can write (15) as

$$\begin{aligned} \frac{{\rm{D}}\mathbf {u}_h}{{\rm{D}}t}+M^{-1}S\mathbf {u}_h+\mathbf {f}\times \mathbf {u}_h=\mathbf {r}_h \end{aligned}$$

The stability of the latter equation can be studied as the stability of the simplified ODE

$$\begin{aligned} y'=\lambda y \end{aligned}$$

where $y=u+iv$ and $\lambda =a+bi$ are complex function and number, respectively (note that, for this specific case, a corresponds to the eigenvalues of the matrix $M^{-1}S$ and b to the Coriolis parameter f). This ODE can be discretized using the second-order BDF formula, as in (8):

$$\begin{aligned} \frac{\frac{3}{2}y^{n+1}-2y^{n}+\frac{1}{2}y^{n-1}}{\Delta t}=\lambda \left( 2y^{n}-y^{n-1}\right) . \end{aligned}$$

The stability diagram is shown in Fig. 15a. Since the smallest eigenvalue of $M^{-1}S$ is much smaller than f, $\lambda$ is very close to the imaginary axis, so the extrapolation formula could be unstable.

Fig. 15

Stability region of a explicit second-order BDF method, b predictor–corrector second-order BDF method and c predictor–corrector second-order BDF method with implicit treatment of viscous term (see “Appendix 2”)

To extend the stability region, one can use a prediction-correction treatment of this term, i.e.,

$$\begin{aligned} \frac{\frac{3}{2}\hat{y}^{n+1}-2y^{n}+\frac{1}{2}y^{n-1}}{\Delta t}&= {} \lambda \left( 2y^{n}-y^{n-1}\right) ,\\ \frac{\frac{3}{2}y^{n+1}-2y^{n}+\frac{1}{2}y^{n-1}}{\Delta t}&= {} \lambda \hat{y}^{n+1}. \end{aligned}$$

The stability diagram for the predictor–corrector scheme is shown in Fig. 15b.

A more stable scheme corresponding to the implicit treatment of the viscous term is the following:

$$\begin{aligned} \frac{\frac{3}{2}\hat{y}^{n+1}-2y^{n}+\frac{1}{2}y^{n-1}}{\Delta t}&= {} a\hat{y}^{n+1} + bi\left( 2y^{n}-y^{n-1}\right) ,\\ \frac{\frac{3}{2}y^{n+1}-2y^{n}+\frac{1}{2}y^{n-1}}{\Delta t}&= {} ay^{n+1} + bi\hat{y}^{n+1}. \end{aligned}$$

The stability diagram of which is represented in Fig. 15c.

This predictor–corrector applied to the proposed splitting becomes: for $j=1,2$,

• 1. Set $\mathbf {u}_{e,j}^{n+1}=2\mathbf {u}^{n}-\mathbf {u}^{n-1}$ for $j=1$ or $\mathbf {u}_{e,j}^{n+1}=\mathbf {u}_{e,j-1}^{n+1}$ for $j=2$, and compute $q_{j}^{n+1}\in H^{1}\left( D\right)$ that satisfies:

  $$\begin{aligned} \Delta t\left( \nabla q_{j}^{n+1},\nabla v\right)= & {} \rho _{0}\left( 2\mathbf {u}^{n*}-\frac{1}{2}\mathbf {u}^{\left( n-1\right) **},\nabla v\right) \\&+\rho _{0}\left( -\Delta t\nabla \times \left( A\nabla \right) \times \mathbf {u}_{e,j}^{n+1}-\Delta t\mathbf {f}\times \mathbf {u}_{e,j}^{n+1},\nabla v\right) \\&-\Delta t\left( \nabla r^{n+1},\nabla v\right) +\Delta t\mathbf {g}\left( \rho ^{n+1},\nabla v\right) \end{aligned}$$

  for all $v\in H^{1}\left( D\right)$.

• 2. Compute $\mathbf {u}_{j}^{n+1}\in V$ such that

  $$\begin{aligned} \frac{3}{2}\left( \mathbf {u}_{j}^{n+1},\mathbf {v}\right) +\Delta t\left( A\nabla \mathbf {u}_{j}^{n+1},\nabla \mathbf {v}\right)= & {} \left( 2\mathbf {u}^{n*}-\frac{1}{2}\mathbf {u}^{\left( n-1\right) **}-\Delta t\mathbf {f}\times \mathbf {u}_{e,j}^{n+1},\mathbf {v}\right) \\&-\Delta t\frac{1}{\rho _{0}}\left( \nabla q_{j}^{n+1},\mathbf {v}\right) -\Delta t\frac{1}{\rho _{0}}\left( \nabla r^{n+1},\mathbf {v}\right) \\&+\Delta t\frac{1}{\rho _{0}}\mathbf {g}\left( \rho ^{n+1},\mathbf {v}\right) \end{aligned}$$

  for all $\mathbf {v}\in \mathbf {V}$.

Finally, set $\mathbf {u}^{n+1}=\mathbf {u}_{2}^{n+1}$.