A shallow water equation based on displacement and pressure and its numerical solution

Abstract

The primary purpose of this paper is to develop an efficient numerical scheme for solving the shallow water wave problem with a sloping water bottom and wet-dry interface. For this purpose, the Lagrange method and the constrained Hamilton variational principle are used to solve the shallow water wave problem. According to the constrained Hamilton variational principle, a shallow water equation based on the displacement and pressure (SWE-DP) is derived. Based on the discretized constrained Hamilton variational principle, a numerical scheme is developed for solving the SWE-DP. The proposed scheme combines the finite element method for spatial discretization and the simplectic Zu-class method for time integration. The correctness of the SWE-DP and the effectiveness of the proposed scheme are verified by three classical numerical examples. Numerical examples show that the proposed method performs well in the simulation of the shallow water problem with a sloping water bottom and wet-dry interface.

Appendix: Relation with the SWEs in the Euler description

Let the flow velocity in the Euler description be denoted by $V\left( {\xi ,\;t} \right)$. In terms of Eq. (1), we have

$$\frac{\partial }{\partial x} = \left( {1 + u_{x} } \right)\frac{\partial }{\partial \xi }$$

(79)

and

$$\begin{aligned} \bar{d}\left( {\xi ,t} \right) = \bar{d}\left( {x + u\left( {x,t} \right),t} \right) = d\left( {x,t} \right) \hfill \\ \bar{u}\left( {\xi ,t} \right) = \bar{u}\left( {x + u\left( {x,t} \right),t} \right) = u\left( {x,t} \right), \hfill \\ \end{aligned}$$

(80)

where $\bar{\# }\left( \xi \right)$ denotes the variable in the Euler description. The Lagrangian acceleration can be expressed as

$$\ddot{u}\left( x \right) = \frac{{\partial V\left( {\xi ,t} \right)}}{\partial t} + V\frac{{\partial V\left( {\xi ,t} \right)}}{\partial \xi }.$$

(81)

In terms of Eqs. (1) and (16), we have

$$\bar{h}_{x} = \left. {\frac{\partial h\left( a \right)}{\partial a}} \right|_{a = x + u} = \frac{\partial h\left( \xi \right)}{\partial \xi }.$$

(82)

Substituting Eqs. (81) and (82) into the first equation of Eq. (37) and noting Eq. (79), we have

$$\rho d_{0} \left( {\frac{{\partial V\left( {\xi ,t} \right)}}{\partial t} + V\frac{{\partial V\left( {\xi ,t} \right)}}{\partial \xi }} \right) - \rho gd_{0} \frac{\partial h\left( \xi \right)}{\partial \xi } + \frac{1}{2}\frac{{\partial \left( {\beta d} \right)}}{\partial x} = 0.$$

(83)

In terms of the second and third equations of Eq. (37), we can obtain the following equations

$$\frac{\partial \xi }{\partial x} = u_{x} + 1 = \frac{{d_{0} }}{{d\left( {x,t} \right)}} = \frac{{d_{0} }}{{\bar{d}\left( {\xi ,t} \right)}},$$

(84)

and

$$\beta = \rho g\frac{{d_{0} }}{{\left( {1 + u_{x} } \right)}} = \rho g\bar{d}\left( {\xi ,t} \right).$$

(85)

Substituting Eqs. (84) and (85) into Eq. (83) yields

$$\frac{{\partial V\left( {\xi ,t} \right)}}{\partial t} + V\frac{{\partial V\left( {\xi ,t} \right)}}{\partial \xi } + g\frac{{\partial \bar{d}}}{\partial \xi } = g\frac{\partial h\left( \xi \right)}{\partial \xi }.$$

(86)

Meanwhile, the derivative of (84) with respect to time is

$$\dot{u}_{x} d\left( {x,t} \right) + \left( {1 + u_{x} } \right)\dot{d}\left( {x,t} \right) = 0.$$

(87)

Noting that

$$\dot{u}_{x} = \frac{{\partial \dot{u}\left( x \right)}}{\partial x} = \frac{\partial V\left( \xi \right)}{\partial x} = \frac{\partial V}{\partial \xi }\left( {1 + u_{x} } \right),$$

(88)

and

$$\frac{\partial d}{\partial t} = \frac{{\partial \bar{d}}}{\partial t} + \frac{{\partial \bar{d}}}{\partial \xi }V,$$

(89)

Equation (87) can be further rewritten as

$$\frac{{\partial \bar{d}}}{\partial t} + \frac{{\partial \left( {V\bar{d}} \right)}}{\partial \xi } = 0.$$

(90)

Eqs. (86) and (90) constitute the SVE in the Euler description,

$$\left\{ \begin{aligned} \frac{{\partial V\left( {\xi ,t} \right)}}{\partial t} + V\frac{{\partial V\left( {\xi ,t} \right)}}{\partial \xi } + g\frac{{\partial \bar{d}}}{\partial \xi } = g\frac{\partial h\left( \xi \right)}{\partial \xi } \\ \frac{{\partial \bar{d}}}{\partial t} + \frac{{\partial \left( {V\bar{d}} \right)}}{\partial \xi } = 0. \\ \end{aligned} \right.$$

(91)

The above analysis means that the proposed SWE-DP is mathematically equivalent to the SVE when the effect of the vertical velocity is ignored.

Equation (37) can also be expressed in the conservative form. Introduce the two variables:

$$p_{u} = \rho d_{0} \dot{u},\;\;\gamma = {{d_{0} } \mathord{\left/ {\vphantom {{d_{0} } d}} \right. \kern-0pt} d}$$

(92)

where $p_{u}$ is the moment in the x-direction, and $\gamma$ is a dimensionless parameter, that is the ratio of the water depths at times 0 and $t$. Using Eqs. (38) and (92), the first equation in Eq. (37) can be rewritten as

$$\frac{{\partial p_{u} }}{\partial t} + \frac{\partial }{\partial x}\left( {\frac{{\rho gd_{0}^{2} }}{{2\gamma^{2} }}} \right) = \rho gd_{0} \bar{h}_{x} .$$

(93)

In terms of Eq. (92), the incompressible equation, i.e., the third equation in Eq. (37), can be rewritten as

$$\gamma = 1 + u_{x}$$

(94)

Noting from Eq. (92) that $\dot{u} = {{p_{u} } \mathord{\left/ {\vphantom {{p_{u} } {\left( {\rho d_{0} } \right)}}} \right. \kern-0pt} {\left( {\rho d_{0} } \right)}}$, the partial derivative of Eq. (94) with respect to $t$ yields

$$\frac{\partial \gamma }{\partial t} = \frac{{\partial u_{x} }}{\partial t} = \frac{{\partial \left( {u_{t} } \right)}}{\partial x} = \frac{\partial }{\partial x}\left( {\frac{{p_{u} }}{{\rho d_{0} }}} \right)$$

(95)

Hence, Eq. (37) can be expressed as the following conservative form

$$\left\{ \begin{aligned} \frac{{\partial p_{u} }}{\partial t} + \frac{\partial }{\partial x}\left( {\frac{{\rho gd_{0}^{2} }}{{2\gamma^{2} }}} \right) = \rho gd_{0} \bar{h}_{x} \\ \frac{\partial \gamma }{\partial t} + \frac{\partial }{\partial x}\left( {\frac{{ - p_{u} }}{{\rho d_{0} }}} \right) = 0 \\ \end{aligned} \right.$$

(96)

The first equation in Eq. (96) represents the momentum conservation equation, and the second one represents the mass conservation equation. However, it can be observed by comparison with Eq. (91) that there is no so-called convection term in Eq. (96).