Extended water wave systems of Boussinesq equations on a finite interval Theory and numerical analysis_2

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Extended water wave systems of Boussinesq equations on a ﬁnite interval:

Theory and numerical analysis

Dionyssios Mantzavinos aand Dimitrios Mitsotakis b,∗

aDepartment of Mathematics, University of Kansas, U.S.A.

bDepartment of Mathematics and Statistics, Victoria University of Wellington, New Zealand

Dedicated to the memory of Vassilios A. Dougalis

Abstract

Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of

nonlinear and dispersive water waves of signiﬁcant interest such as solitary and tsunami waves. The

initial-boundary value problem on a ﬁnite interval for this family of systems is studied both theoretically

and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via

the uniﬁed transform of Fokas. The corresponding analysis reveals two types of admissible boundary

conditions, thereby suggesting appropriate boundary conditions for the nonlinear Nwogu system on

a ﬁnite interval. Then, well-posedness is established, both in the weak and in the classical sense,

for a regularized Nwogu system in the context of an initial-boundary value problem that describes

the dynamics of water waves in a basin with wall-boundary conditions. In addition, a new modiﬁed

Galerkin method is suggested for the numerical discretization of this regularized system in time, and its

convergence is proved along with optimal error estimates. Finally, numerical experiments illustrating

the eﬀect of the boundary conditions on the reﬂection of solitary waves by a vertical wall are also

provided.

Keywords: Boussinesq systems, initial-boundary value problem, uniﬁed transform of Fokas,

well-posedness, Galerkin/ﬁnite element method

2020 MSC: 35G46, 35G61, 65M60

1. Introduction

Nonlinear and dispersive water waves are described by the Euler system of equations [44]. As

Euler’s equations still remain one of the hardest problems to solve (even numerically), various simpliﬁed

systems of partial diﬀerential equations have been suggested as alternative approximations, especially

for long waves of small amplitude because of their applications. Waves of this type are also called

weakly nonlinear and weakly dispersive waves. Tsunamis, solitary waves, internal waves and even

atmospheric waves fall into the regime of weakly nonlinear and weakly dispersive waves [44]. The

aforementioned simpliﬁed systems should obey the laws of physics and mathematics; importantly, they

should be well-posed when supplemented with physically sound boundary conditions (Newton’s principle

of determinacy), admit classical solitary wave solutions, preserve symmetries and reasonable forms of

invariants such as energy, and agree with laboratory experiments. Compliance with such fundamental

laws is along the lines of scientiﬁc rigor and justiﬁes the use of such systems in practical applications.

After the pioneering work of Boussinesq [8, 9], several Boussinesq systems have been introduced

to describe the propagation of weakly nonlinear and weakly dispersive waves, including Peregrine’s

∗dimitrios.mitsotakis@vuw.ac.nz

Preprint submitted to Journal de Math´ematiques Pures et Appliqu´ees October 10, 2022

arXiv:2210.03279v1 [math.AP] 7 Oct 2022

system [34]. Peregrine’s system, also known as classical Boussinesq system in the case of ﬂat bottom

topography, was rederived in one dimension along with a whole class of asymptotically equivalent

systems known as the abcd-systems [6]. The Cauchy problem on the real line for this class of Boussinesq

systems was studied in [36, 7, 35]. In scaled and non-dimensional variables, the abcd family of systems

takes the form ηt+ux+ε(ηu)x+σ2(auxxx −bηxxt) = 0 ,

ut+ηx+εuux+σ2(cηxxx −duxxt) = 0 ,(1.1)

where xdenotes the horizontal spatial independent variable, tis the time, η=η(x, t) is the free surface

elevation above a ﬂat bottom located at depth D0=−1, u=u(x, t) is the horizontal velocity of the

ﬂuid measured at depth θD0with θ∈[0,1], ε, σ are small parameters characterizing the nonlinearity

and the dispersion of the waves, and a, b, c, d are parameters given by

a=1

2θ2−1

3ν, b =1

2θ2−1

3(1 −ν),

c=1

21−θ2µ, d =1

21−θ2(1 −µ),µ, ν ∈R,

so that a+b+c+d= 1/3. The parameters εand σ2in the Boussinesq regime are of the same order,

and the Stokes number is S=ε/σ2=O(1). Hence, as usual, throughout this work we take ε=σ2for

simplicity and without loss of generality.

Some important representatives of the abcd family of systems (1.1) include systems that are well-

posed and also admit classical solitary waves as special solutions. Examples of such systems are the

Bona-Smith system (a= 0, b > 0, c≤0, d > 0), which includes the coupled Benjamin-Bona-Mahony

BBM-BBM system as a special case (c= 0), the classical Boussinesq system (a=b=c= 0, d > 0), the

Nwogu system (a < 0, b=c= 0, d > 0), and the regularized Nwogu system or “reverse” Bona-Smith

system (a < 0, b > 0, c= 0, d > 0). It should be noted that the Nwogu system was derived in [33]

as an alternative to the Peregrine system with the ability to choose the coeﬃcients a, b, c, d so that the

linear dispersion relation becomes optimal compared to the corresponding linear dispersion relation of

the Euler equations.

In practical applications and numerical simulations, systems like those mentioned above are posed

in bounded domains. This fact highlights the need for appropriately formulated initial-boundary value

problems. An important issue in this direction is the choice of appropriate boundary conditions. In

particular, in the case of both the regularized and the non-regularized Nwogu systems, it is not a priori

clear how many boundary values — and of which type — must be speciﬁed as data for a well-posed

problem on the ﬁnite interval. One of the main results of this work is the identiﬁcation of appropriate

boundary conditions on the ﬁnite interval for the following generalized Nwogu system:

ηt+ux+ε(ηu)x+ε(auxxx −bηxxt)=0,

ut+ηx+εuux−εduxxt = 0 ,(x, t)∈(−L, L)×(0, T ),(1.2)

with a < 0, b≥0, c= 0, d > 0, which contains both the regularized and the original Nwogu system,

for b > 0 and b= 0 respectively.

Appropriate boundary conditions for the generalized system (1.2) are identiﬁed through solving

the linear counterpart of that system via the uniﬁed transform. This method was ﬁrst introduced by

Fokas in [18] (see also the monograph [19] and the review article [14]) and has since been employed

for the analysis of linear as well as initial-boundary value problems in various settings — see, for

example, [22, 24, 17, 20, 23, 25, 26, 27, 21, 28]. Recent developments have led to the advancement of the

uniﬁed transform to systems of PDEs, in particular via the work [13]. Exploiting the framework laid out

in [13] along with recent progress noted in [29] on the linearization of the classical Boussinesq equation

on the half-line, here we employ the uniﬁed transform to derive a novel, explicit solution formula for

the linearization of the generalized Nwogu system (1.2) on a ﬁnite interval:

ηt+ux+ε(auxxx −bηxxt) = 0 ,

ut+ηx−εduxxt = 0 ,(x, t)∈(−L, L)×(0, T ).(1.3)

While the initial conditions accompanying this system are the usual ones, namely η(x, 0) and u(x, 0)

given, we perform our analysis without initially specifying any boundary conditions. Instead, we discover

the conditions that lead to an explicit solution formula (and hence to a well-formulated problem) through

the application of the uniﬁed transform. In particular, our analysis indicates that one of the following two

pairs of boundary values must supplement system (1.3) as boundary conditions (see also Remark 2.1):

{u(±L, t), uxx(±L, t)}or {η(±L, t), ux(±L, t)}.(1.4)

This ﬁnding provides strong theoretical evidence that the boundary conditions (1.4) should lead to a

well-posed problem for the nonlinear generalized Nwogu system (1.2) and, in particular, for the original

Nwogu system on a ﬁnite interval.

We note that our analysis, via the uniﬁed transform, of the linear system (1.3) is carried out for

nonzero boundary conditions of the form (1.4). Nevertheless, one of the most important initial-boundary

value problems for Nwogu-type systems is the one with wall-boundary conditions on the boundaries of

a basin. The well-posedness of this initial-boundary value problem with reﬂection boundary conditions

for the Peregrine system and its linearization was studied in [24, 1, 29]. Speciﬁcally, it was proven that

for Peregrine’s system only the classical homogeneous Dirichlet wall-boundary conditions

u(−L, t) = u(L, t)=0,(1.5)

are required for well-posedness, ensuring that there is no ﬂow through the boundaries. Analogous

initial-boundary value problems have been studied in detail for Bona-Smith systems in [3], while their

special case of BBM-BBM systems was studied in [5]. There, it was shown that in addition to the

wall-boundary condition (1.5) homogeneous Neumann boundary conditions for ηare also required:

ηx(−L, t) = ηx(L, t)=0.(1.6)

Although these conditions are not satisﬁed by Peregrine’s system, they are satisﬁed by the solutions of

the Euler equations when wall-boundary conditions are imposed [30] (see also Appendix Appendix A).

Thus, satisfying both boundary conditions (1.5) and (1.6) reﬂects a more accurate description of water

waves in a basin.

The physical relevance of homogeneous (zero) boundary conditions as illustrated above motivates

the study of well-posedness for the nonlinear regularized Nwogu system, namely system (1.2) with b > 0,

supplemented with such conditions on a ﬁnite interval. In particular, the second main result of this

work establishes the well-posedness of that system in the case of reﬂective boundary conditions (the

Dirichlet problem was analyzed in [3]). We note that, although in this work we restrict ourselves to the

case of a ﬂat bottom, it is anticipated that the variable bottom case can be handled by using similar

concepts. The particular initial-boundary value problem with reﬂective boundary conditions that we

consider for the regularized Nwogu system can be written in dimensionless and scaled variables as

ηt+ux+ε(ηu)x+ε(auxxx −bηxxt)=0,

ut+ηx+εuux−εduxxt = 0 ,(x, t)∈(−L, L)×(0, T ),

η(x, 0) = η0(x), u(x, 0) = u0(x),

u(−L, t) = u(L, t) = 0 , uxx(−L, t) = uxx(L, t)=0,

(1.7)

where a < 0 and b, d > 0 such that a+b+d= 1/3. As noted earlier, in the case where a= 0

and b, d > 0, the system reduces to the BBM-BBM system which was analyzed extensively in [5],

while for a < 0, b= 0 and d > 0 the system becomes the well-known Nwogu system [33]. Observe that,

because of the boundary conditions on u, the second (momentum) equation in (1.7) yields the additional

conditions (1.6), which are satisﬁed implicitly, and thus there is no need for them to be explicitly stated.

Furthermore, as shown in Appendix Appendix A, the second set of boundary conditions uxx(±L, t)=0

in (1.7) are also satisﬁed by the solutions of Euler’s equations.

After proving that the nonlinear system (1.7) is well-posed in the Hadamard sense locally in time, we

study its numerical discretization with Galerkin/ﬁnite element method. Wall-boundary conditions for

the numerical solution of the Nwogu system were ﬁrst suggested in [41, 42, 43] but without theoretical

justiﬁcation. In the special case of the linearized Nwogu system, one can establish well-posedness with

the particular wall-boundary conditions using Galerkin approximations [10]. The presence of the third-

order spatial derivative uxxx in Nwogu-type systems, like the term ηxxx in the case of the Bona-Smith

system, makes their numerical discretization with Galerkin methods challenging. This diﬃculty, for

example, can be related to the requirement of well-deﬁned second derivative of the velocity component

uof the numerical solution. While Lagrange elements guarantee only local smoothness, smooth cubic

splines (at least) are left to be used for the standard Galerkin method accompanied with suboptimal

convergence results [4, 15]. As a remedy to this problem, a modiﬁcation of the standard Galerkin

method for the Nwogu system was suggested in [42], allowing the use of Lagrange elements. While

the convergence of that particular method is still unclear, a similar modiﬁed Galerkin method was

studied and proven to be convergent in the case of the Bona-Smith system [16]. Here, we develop the

analogous modiﬁed Galerkin/ﬁnite element method for the regularized Nwogu system (1.7), and we

show that its semidiscrete Galerkin approximations converge to the analytical solutions of (1.7) with

optimal convergence rate.

Structure. This work is organized as follows. In Section 2, using the uniﬁed transform of Fokas we

obtain an explicit solution formula for the linearization (1.3) of the generalized Nwogu system (1.2) on

the ﬁnite interval (−L, L). Our analysis shows that this problem (and hence its nonlinear counterpart)

is well-formulated if either {u(±L, t), uxx(±L, t)}or {η(±L, t), ux(±L, t)}are prescribed as boundary

conditions. In Section 3, we establish local Hadamard well-posedness for the regularized Nwogu

system (1.7) in the case of wall-boundary conditions u(±L, t) = uxx(±L, t) = 0 (the well-posedness

of (1.7) with η(±L, t) = ux(±L, t) = 0 was proved in [3]). In Section 4, we provide the derivation and

analysis of a modiﬁed Galerkin method for the numerical solution of the regularized system (1.7). The

convergence of the method is also veriﬁed experimentally, while a demonstration of the reﬂection of

solitary waves illustrates the practical use of the initial-boundary value problem with wall-boundary

conditions. Finally, some brief concluding remarks are given in Section 5, while the use of the particular

set of wall-boundary conditions is justiﬁed in Appendix Appendix A by showing that solutions to

the Euler equations satisfy the same wall-boundary conditions even in the case of variable bottom

topography.

Notation. Throughout this work, we denote by L2:= L2(−L, L) the Hilbert space of measurable,

square-integrable real-valued functions on (−L, L). For any integer s≥0, we denote by Hs:=

Hs(−L, L) the classical Sobolev space of s-times weakly diﬀerentiable functions on (−L, L),

Hs=v∈L2:∂j

xv∈L2for all j= 0,1, . . . , s,

accompanied with the usual norm kvks:= Ps

j=0 RL

−L∂j

xv(x)2dx1/2

, where ∂j

xdenotes the j-th

partial derivative with respect to x. Note that H0=L2, while the norm of L2will be denoted by k·k.

For m≥0, we will also consider the Banach space Cs:= Cs(−L, L) of real-valued s-times continuously

diﬀerentiable functions deﬁned on [−L, L], equipped with the norm

kvkCs:= sup

0≤j≤s

sup

x∈[−L,L]∂j

xv(x).

We write A.Bif A≤CB with C > 0 a constant independent of discretization parameters such as ∆x

or other crucial parameters.

2. Explicit solution of the linear problem

In this section, we employ the uniﬁed transform of Fokas in order to solve the linear counterpart (1.3)

of the generalized Nwogu system (1.2) on the ﬁnite interval (−L, L). Our analysis reveals that the

combinations (1.4) are two possible choices of admissible boundary conditions for this linear problem. As

such, these two combinations should also lead to a well-posed problem at the nonlinear level, supporting

the theoretical and numerical ﬁndings of Sections 3 and 4. The ﬁrst set of data in (1.4) describes wall-

boundary conditions and so it can be used for studying the reﬂection of water waves on a vertical wall,

while the second set of data corresponds to the wave maker problem.

Importantly, we note that: (i) The calculations of this section remain valid for b= 0, which is

the value corresponding to the original Nwogu system (1.7). In particular, setting b= 0 (equivalently,

β= 0) in the solution formulas (2.14) yields the corresponding solutions to the linearization of the

Nwogu system (see problem (2.15) in Remark 2.2). (ii) Furthermore, our analysis and the resulting

solution formulas hold for general nonzero boundary conditions. (iii) In addition, since swapping ηwith

utransforms the generalized Nwogu system into the Bona-Smith system, the formulas (2.14) derived

here provide the solution also for the linearized Bona-Smith system formulated with nonzero boundary

conditions on a ﬁnite interval.

2.1. Derivation of the global relation

We begin by noting that throughout this section we assume suﬃcient smoothness and decay as

needed for our computations to hold. Setting α=−aε > 0, β=bε >0, δ=dε > 0 allows us to write

the linear generalized Nwogu system (1.3) in the form

ηt+ux−αuxxx −βηxxt = 0 ,

ut+ηx−δuxxt = 0 ,(x, t)∈(−L, L)×(0, T ).(2.1)

While we supplement system (2.1) with the usual initial conditions

η(x, 0) = η0(x), u(x, 0) = u0(x),(2.2)

we do not yet specify any boundary conditions as it is not a priori clear what choices of boundary data

lead to a well-formulated problem. Instead, we introduce the notation

gj(t) := ∂j

xu(−L, t), hj(t) := ∂j

xu(L, t), j = 0,1,2,

g3(t) := η(−L, t), h3(t) := η(L, t),(2.3)

for the various boundary values that arise in our analysis and defer the prescription of some of these as

boundary conditions to a later point.

Let the ﬁnite-interval Fourier transform pair of a function f∈L2(−L, L) be deﬁned by

f(k) = ZL

x=−L

e−ikxf(x)dx, k ∈C, f(x) = 1

2πZk∈R

eikx b

f(k)dk, x ∈(−L, L),(2.4)

and note that, since xis bounded, b

f(k) is an entire function of kvia a Paley-Wiener-type theorem (e.g.

see Theorem 7.2.3 in [39]). Taking the Fourier transform (2.4) of system (2.1) while noting that the

second component of (2.1) yields ηx(−L, t) = δg0

2(t)−g0

0(t) and ηx(L, t) = δh0

2(t)−h0

0(t), we obtain the

vector ODE

vt(k, t) + M(k)b

v(k, t) = A(k, t), k ∈C\n±i

√δ,±i

√βo,(2.5)

where v= (η, u)T,Mis a 2 ×2 matrix given by

M(k) = ik 01 + αk21 + βk2−1

1 + δk2−10!,

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ExtendedwaterwavesystemsofBoussinesqequationsonaniteinterval:TheoryandnumericalanalysisDionyssiosMantzavinosaandDimitriosMitsotakisb;aDepartmentofMathematics,UniversityofKansas,U.S.A.bDepartmentofMathematicsandStatistics,VictoriaUniversityofWellington,NewZealandDedicatedtothememoryofVassiliosA.Dou...

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Extended water wave systems of Boussinesq equations on a finite interval Theory and numerical analysis_2

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