## 1. Introduction

In this work, attention is given to a new model system for the study of long waves of small amplitude at the free surface of a perfect fluid. The system can be used in the presence of non-constant bathymetry and lateral boundaries, and the main new feature of the system is that it is straightforward to implement no-slip boundary conditions on a finite domain. The system falls in the general class of Boussinesq systems which have become standard tools in the study of nearshore hydrodynamics.

While the full water-wave problem is given by the Euler equations with free surface boundary conditions [30], it is well known that this problem is difficult to treat both mathematically and numerically. In particular, it is not known whether solutions exist on relevant time scales, and numerical simulations of the full water-wave problem may suffer from serious stability issues. Therefore in practical situations in coastal hydrodynamics, an asymptotic approximation of the Euler equations is often used in order to find simpler systems that still describe the main features of the flow. These simplified systems are usually derived using the long-wave assumption. The simplest in structure of all such long wave systems are the shallow-water wave equations (or Saint Venant equations) which take the form

(1) | ||||

where denotes the free surface elevation and the depth-averaged horizontal velocity of the fluid. Since this system is hyperbolic, there are a number of well developed methods for the approximation of solutions such as TVD methods, Riemann solvers etc, [19, 28], and it is well known that the shallow-water system is able to describe the propagation of tsunamis and flood waves. It is also well known that smooth solutions of (1) preserve the energy functional

which is an approximation of the total energy satisfied by the solutions of the Euler equations. Although the shallow-water system has favourable properties and is widely used, it is restricted to the modeling of very long waves, and is not suitable for the description of coastal phenomena such as solitary waves or periodic wavetrains.

In a seminal contribution, D.H. Peregrine in [23] resolved this issue by deriving a Boussinesq-type system applicable to coastal wave phenomena such as shoaling solitary waves, wave reflection and long-shore currents to name just a few. The Peregrine system is written in dimensional form as

(2) | ||||

and describes the propagation of water waves over a bottom topography () with free surface elevation and a depth-averaged horizontal velocity field .

The first equation in Peregrine’s system (2) is the exact expression of the mass conservation, and is derived from the kinematic free-surface boundary condition. The second equation is derived from the dynamic boundary condition. Although Peregrine’s system looks very convenient due to its simplicity, it appears to have several drawbacks in relation to existence and uniqueness of solutions and numerical discretization. Indeed, it has only recently been proved that the Cauchy problem for the Pergreine system (2) is well-posedness in [14], and it is still unknown whether the system is well-posed in bounded domains . Moreover, as it was shown in [17], numerical discretization of Peregrine’s system in bounded domains can yield suboptimal convergence rates and also low resolution phenomena (i.e. aliasing phenomena) due to its hyperbolic form of the mass conservation.

Several other Boussinesq-type systems with certain favourable properties have been derived as alternatives to Peregrine’s system. One such example, which is of central focus in the present paper, is the class of the BBM-BBM type systems. These systems were first introduced in [7, 5, 4] in one dimension and later in [6] in two dimensions, and they agree asymptotically with the Euler equations in the long-wave small-amplitude regime. In particular in [6] a BBM-BBM system of the form

(3) | ||||

was derived in the case of a flat bottom, and a generalization of this system to the case of general topography was presented in [20].

The main characteristic of these systems is the presence of a dispersive term of mixed type, involving two space derivatives and one time derivative in both equations, as opposed to the Peregrine system which features this term only in one of the two equations. The idea of using mixed-derivative terms goes back to Peregrine [22], and the single KdV-type equation with a mixed-derivative term has become known as the BBM equation [2]. In the context of BBM-type systems, the inclusion of the mixed-derivative term in the first equation has two drawbacks. First, the first equation is no longer an exact mass cosnervation equation, and the mass balance now takes an approximate form [1]. However, mass is still conserved to within the order of approximation, so this is not a serious problem. Secondly, the dispersion relation for the linearized equation is slightly less accurate than the dispersion relation for the Peregrine system [4]. This drawback can be mitigated by inluding higher-order dispersive terms which is the approach followed in the present contribution.

While the presence of the Laplacian in the dispersive terms of the mass and momentum equation in these BBM-BBM systems appears to be attractive from the point of view of mathematical analysis and numerical discretization, the initial-boundary value problem for these kind of BBM-BBM-type systems in bounded domains requires zero Dirichlet boundary conditions for the velocity field on the boundary of the domain, [12], in addition to homogenous Neumann boundary conditions for the free surface . These boundary conditions are essentially no-slip wall boundary conditions and are quite restrictive, especially when one considers obstacles or other complicated boundaries of the numerical domain.

In order to address this problem, a new BBM-BBM type system suitable for slip-wall boundary conditions was recently proposed [17]. The system is written in dimensional variables as

(4) | ||||

where

(5) |

for . Here, denotes the horizontal velocity field at height above the bottom, instead of the depth-averaged horizontal velocity used in the Peregrine system (2). For the BBM-BBM-type system of [6] is recovered but with different dispersive term in the second equation. Further simplifications of system (4) can be achieved by considering mild bottom topography.

In the following, we give a detailed explanation why the system (4) is attractive for the study of nearshore surface waves. For the derivation of the system we follow two different approaches: The first one is based on the classical asymptotic method taking as point of departure the full water-wave problem bases on the Euler equations. In the derivation, we pay special attention to incorporate appropriate dispersive terms which yield the correct behaviour in terms of energy conservation. As a consequence, the new system features energy conservation in a similar fashion as the Euler equations. In particular, the solutions of the new regularized system preserve the exact same energy as its non-dispersive counterpart, namely, the shallow-water waves system. Furthermore, we present an alternative derivation based on variational principles. This approach is quite attractive not only for its simplicity in the derivation but also for obtaining physical properties in a straightforward manner. Although the new system is derived with the assumption of the mild bottom topography, it will be shown in Section 5 that it appears to be valid even for more general bottom topographies.

Furthermore, we explore the theoretical background of (4) insofar as it concerns the initial-boundary value problem in a bounded domain with slip-wall boundary conditions. These boundary conditions are necessary to describe water waves propagating in a closed basin and in general when interaction of waves with solid walls. The initial-boundary value problem of the new system with the slip-wall boundary conditions appears to have similar well-posedness properties as other BBM-BBM systems such as the classical BBM-BBM system studied in [12].

We apply the Galerkin finite element method for the spatial discretization of the new system. Due to the difficulty of incorporating the exact boundary conditions into the finite element space, we consider applying the Nitsche method [21]. This method is commonly used in practical problems but very rarely is analyzed. In this paper, we prove that the numerical solution converges, and in some cases with optimal rate of convergence, to the exact solution. We verify these results also numerically.

The paper is organized as follows: First we present the derivation of the system using the two approaches in Section 2. Next, in Section 3, we study the well-posedness of the specific initial-boundary value problem, a necessary ingredient for the justification of a novel model system of equations. The application of the finite element method for the discretization of the new system, its convergence and accuracy are presented in Section 4. Finally in Section 5, we consider several numerical experiments verifying the theoretical findings and demonstrating the applicability of the numerical method.

## 2. Derivation of the new system

In this section we present the derivation of the new system based on the classical asymptotic approach. Furthermore, we present a novel alternative derivation based on variational methods.

### 2.1. Asymptotic reasoning

In what follows we consider characteristic quantities for typical waves in the Boussinesq regime, in particular a typical wave amplitude and length and a typical depth . We will denote the linear wave speed by . We also denote the typical deviation of the bottom topography to be of the form with typical deviation of to be , and the dimensionless variables

(6) |

Then the BBM-BBM system (4) can be written in the nondimensional and scaled form

(7) | ||||

where the parameter , and are all assumed to be small: . Moreover, assuming that , which is equivalent to the small-bottom-variations assumption [9], and by considering the terms of the order , and and higher to be negligible, the BBM-BBM system (7) can be further simplified to

(8) | ||||

or in dimensional form, after discarding the high-order terms

(9) | ||||

We will refer to this system as simplified BBM-BBM system, which is a generalization of the analogous one-dimensional BBM-BBM system derived in [9]. It is easily seen that the bottom variations practically do not contribute at all in the dispersive terms. As we shall see also later in Section 5, such simplifications diminish the accuracy of the model and make it inappropriate for practical applications such as the shoaling of solitary waves even in the cases where the slope of the seafloor is mild. On the other hand, keeping the high-order terms in the dispersive terms and taking the advantage of the fact that to place the term at a position that will ensure energy conservation, we obtain from (7) the system

(10) | ||||

which, as we shall see later, preserves the same energy functional as the non-dispersive shallow water equations. Numerical experiments have shown that keeping topography variations in the high-order dispersive terms extends the validity of the model in practical problems such as the shoaling of long water waves over general bottoms. Moreover, the model is then more realistic since the actual bottom topography function appears in the equations instead of the typical depth (see e.g. [18]). The asymptotic equivalence of the equations with and enables us to reformulate them appropriately so that the resulting system will be Hamiltonian. For example, after dropping the high-order terms and using dimensional variables the system (10) can be written as

(11) | ||||

Furthermore, assuming moving bottom topographic features described by a bottom function of the form where has a typical magnitude of , the system (11) is written as

(12) | ||||

In this paper we will consider the system (11) in the case where (i.e. ) in a bounded domain with slip wall boundary conditions of the form and on the boundary , where

is the external normal vector to the boundary. We rewrite the BBM-BBM system (

11) in the form of an initial-boundary value problem(13) | |||

(14) |

where the initial state of the problem is specified by the initial conditions

(15) |

and on the boundary we assume physically important slip-wall boundary conditions

(16) |

Equations (13), (14), (15), (16) form an initial-boundary value problem, which we will denote by (IBVP) for the rest of this paper.

###### Remark 2.1.

It is easily seen that . In our case, where , we have that This implies that whenever the bottom is horizontal, the regularization operator coincides with the classical elliptic operator and thus the theory of [12] applies here too. In addition, using the small bottom variations assumption we conclude that this is still valid in the case of a variable bottom. Since the regularization properties of the aforementioned BBM-BBM system are expected to be the same as the original system of [6], we focus our attention to the new one due to its favourable properties when it comes to the application of the slip-wall boundary conditions.

### 2.2. Conservation properties and regularity

Contrary to the classical BBM-BBM (and also Peregrine) type systems in 2D, the aforementioned BBM-BBM system is Hamiltonian. Specifically, any solution of the initial-boundary value problem (13)–(16) conserves the energy

(17) |

in the sense that for all . The energy functional (17) in non-dimensional variables takes the form

(18) |

The conservation of energy gives an upper bound of the -norm of the solution. To show the conservation of energy we write system (13) - (14) in the form

(19) | |||

where and . Then, after integrating by parts and applying the slip-wall boundary conditions at we have

It is noted that the key point for the conservation of energy is the particular choice of the parameter which ensures that .

From (14) we observe that since for any smooth enough function . We conclude that the vorticity of the horizontal velocity is conserved in the sense . Therefore, if the flow, initially, is irrotational, then it remains irrotational with for all .

### 2.3. Variational derivation

The variational derivation of model equations appeared to be attractive not only because of its simplicity but also because of the physical verification of the model and the energy conservation properties that can be obtained in trivial way. Here we follow the methodology introduced in [25, 10]. We first consider the following approximations of the kinetic and potential energies: The shallow-water approximation of the kinetic energy is

and the shallow-water approximation of the potential energy is

where denotes the density of the water. We also consider the non-hydrostatic approximation of the conservation of mass

where denotes the total depth of the water. Then, we define the action integral

where we impose the mass conservation by introducing the Lagrange multiplier , which as we shall see in the sequel coincides with a velocity potential of the horizontal velocity .

The Euler-Lagrange equations for the action integral are then the following

(20) | ||||

(21) | ||||

(22) |

Taking the gradient of all terms in (22) and eliminating using (21) we obtain the approximate momentum conservation equation

(23) |

The new BBM-BBM system is comprised from the approximations of mass conservation (20) and momentum conservation (23) and thus we see immediately that its solutions preserve the approximation of the total energy .

## 3. Well-posedness

In this section we study briefly the well-posedness of the BBM-BBM system (13)–(14). For simplicity we assume flat bottom topography and with the same initial and boundary conditions as before. Moreover, for theoretical purposes we assume that the domain is smooth (at least piecewise smooth with no reentrant corners). The equations are simplified and can be written as

(24) | ||||

We will seek weak solutions of the initial-boundary value problem (IBVP). For this reason we will use the usual Sobolev space consisting of weakly differentiable functions, and the space . We equip the space with the usual -norm defined for all to be and the space with the norm for all . We will also denote the usual inner product of by , and we will use the space and for any .

###### Remark 3.1.

Denoting , we define the spaces

and

It is known that for a domain with appropriately smooth boundary, we have

For details on the properties of these particular spaces we refer to [15].

###### Remark 3.2.

We will also consider the spaces

equipped with the norm

(25) |

These spaces are practically the departure spaces of the operator . We reserve the notation to denote the classical Sobolev space . Furthermore, we define the negative norms

while denotes the standard dual norm in the Sobolev space .

We define the bilinear forms and as

(26) | |||

(27) |

Then the weak formulation of the problem (IBVP) is defined as follows: Seek such that

(28) | ||||

Before stating the main result of this paragraph, we define the mappings and as follows

(29) |

and

(30) |

The mappings and are well defined. Indeed, it’s not hard to see that they are continuous in and , respectively, in the sense that and , where denotes the inequality for an unspecified positive constant , independent of . Specifically, we have the following lemma:

###### Lemma 3.1.

###### Proof.

The continuity of can be proven easily using the Cauchy-Schwarz inequality

and thus . Similarly one can prove the inequality as well. In addition, since (30) holds for all , by choosing , (where is the space of infinitely differentiable functions, with compact support on ), yields that and

hence in , (see also [15], Thm. 2.9). Therefore, , and due to Remark 3.1 we conclude . ∎

###### Remark 3.3.

Alternatively, we can reach the same conclusion by observing that is the solution so that we have .

Now we are ready to prove the main result of this section.

###### Theorem 3.1.

###### Proof.

With the help of the mappings and we write (26) and (27

) as a system of ordinary differential equations in the distributional sense

(33) | |||

(34) |

or in the more compact form

(35) |

where and

(36) |

If and then and due to Grisvard’s theorem [16] and thus the function is well-defined. Moreover, since maps its argument into and into we deduce that is on , with derivative given by

(37) |

The continuity of follows from the continuity of and : Let , then using Lemma 3.1 we have,

where we have used the following Gagliardo-Nirenberg inequality valid in two space dimensions,

Taking we deduce that is continuous. Thus, from the theory of ordinary differential equations in Banach spaces (cf. e.g. [3, 8]), we have that for any initial conditions , there exists a maximal time and a unique solution of the initial-boundary value problem (IBVP).

To prove that the maximal time is independent of , first we observe that the solution of (IBVP) satisfies the following energy conservation:

(38) |

Defining

we rewrite (38) in the form

Using Hölder’s inequality we have

(39) |

From (39) and using the Gagliardo-Niremberg inequality, it follows

which implies

The last inequality gives the a priori bound

(40) |

Since

we have that for and thus

on a time interval where independent of . Therefore, the maximal time of existence of the solution can be extended up to . Hence, we conclude that for , the maximal time is independent of . ∎

###### Remark 3.4.

Due to the regularity properties of the operator , we conclude that if the initial conditions then there exists a maximal time and a unique solution of (IBVP) for .

###### Remark 3.5.

The previous analysis carries over to the case of a sufficiently smooth bottom .
In particular, assuming small bottom variations the details of the analysis can be
carried through as in the case of flat bottom, as long as we multiply the momentum equation with .
The weak formulation of the initial-boundary value problem (13)–16)
will be the following:

Seek such that

(41) | ||||

where

(42) | |||

(43) |

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