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Boundary Element Method for the Mixed BBM-KdV Equation Compared to Non Standard Boundary Conditions

By Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri and Youssef Ouakrim

Submitted: September 24th 2020Reviewed: February 14th 2021Published: June 4th 2021

DOI: 10.5772/intechopen.96617

Downloaded: 52

Abstract

In this chapter, we are interested in the numerical resolution of the mixed BBM-KdV equation defined in unbounded domain. Boundary Element Method (BEM) are introduced to truncate the equation into a considered bounded domain. BEM uses domain decomposition techniques to construct Boundary Condition (BC) as transmission between the bounded domain and its complementary. We then present a suitable approximation of these BC using Discrete Galerkin Method. Numerical simulations are made to show the efficiency of these BC. We also compare with another method that truncates the equation from unbounded to bounded domain, called Non Standard Boundary Conditions (NSBC) which introduces new variables to catch information at the boundary and compose a system to connect all these variables in the bounded domain. Further discussions are made to highlight the advantages of each method as well as the difficulties encountered in the numerical resolution.

Keywords

  • wave equations
  • transparent boundary condition
  • boundary element method
  • non-standard boundary conditions
  • finite difference method

1. Introduction

We consider a combination of two linearized typical dispersive partial differential equations that model solitary waves and all interactions between them, given as follows

tutx+αxxx3utxβtxx3utx+γxutx=0txR+×Ru0x=u0xxRlimx±utx=0tR+E1

such that α, βare dispersion parameters and are positive numbers, while γRis the velocity number. In the case α=0, we obtain the BBM equation [1] and when β=0, we get the KdV equation [2]. Our main purpose is to obtain numerical approximation of Eq. (1) when taken in a bounded domain 0T×abwith suitable boundary conditions with no spurious reflections. For this regard, we use two different techniques that are BEM and NSBC.

The Boundary Element Method (BEM), also known as the Boundary Integral Equation Method (BIEM), is an alternative deterministic method that incorporates a mesh located, only, at domain boundaries and therefore attractive for free surface problems. There are two types of BEM, the direct BEM which requires a closed boundary so that the physical variables (e.g. pressure and normal velocity in acoustics) can only be considered from one side of the surface (interior or exterior), while the indirect (IBEM) can consider both sides of the surface and does not need a closed surface. In the first part of this chapter, we use this technique of BEM to derive the BC to the Eq. (1) in the domain [0, T] × [a, b]. More precisely, we are going to introduce the BEM to establish BC satisfied by the Eq. (1) on two interface points aand bby solving the same equation in the complementary domain R\ab. The BEM has significant advantages over the finite element or difference methods (FEM or FDM), as there is no need for discretizing the domain R\abinto elements. It only uses infinite boundary condition and transmission condition to compute the solution at aand bas integral equations. Consequently, this integral equations will be fixed as the boundary conditions of the problem (1) on the bounded domain [0, T] × ab. Therefore, the boundary condition are approximated as Fredholm Integral Equations of second kind.

Despite the meshing effort is limited and the system matrices are smaller, the BEM also has disadvantages over the Finite Element Method or Difference Finite Method. In fact, the BEM matrices are mostly populated with complex coefficients. Furthermore, singularities may arise in the solution. These deteriorate the efficiency of the solution and must be prevented [2].

The outline of this chapter is organized as follows. In section 2, we describe the BEM for the mixed BBM-KdV equation [3]. Next, we discuss the special case of the BBM equation and give the approximation of the resulting equation Finite Difference Method. Section 3 presents briefly another method to derive boundary conditions for BBM equation called NSBC introduced in [4]. Finally in section 4, comparison of both methods is given with numerical experiments to highlight the transparency of both BC obtained in sections 2 and 3.

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2. Boundary element method for the mixed BBM-KdV equation

Being in one dimensional space, R, the boundary of any bounded interval reduces to two points. Hence, we use the BEM to find two values that might depend on time. For this regard, we consider a bounded domain ΩT=]0,T[×Ωwhere Ω=]a,b[and a,b,TRsuch thata<b,T>0. Note Σ=aband ΣT=]0,T[×Σ. we take the decomposition R=ΩgΩΩd, such that, Ωg=],a]and Ωd=[b,+[. The corresponding equations to (1) using Dirichlet-to-Neumann domain decomposition write

tutx+αxxx3utxβtxx3utx+γxutx=0inΩTu0x=u0xatΩnu=nwatΣTE2
twtx+αxxx3wtxβtxx3wtx+γxwtx=0inΩgTΩdTw0x=0inΩgΩdw=uinΣgTΣdTlimx+wtx=0at]0,T[E3

The main object of this section is to prove the following result.

Lemma 2.1 The solution of the evolution Eq. (3) satisfies the following integral equations

wtaU2L1λ1s2swxtaU2L1λ1sswxxta=0wtbL11λ1s2wxxtb=0,wxtbL11λ1swxxtb=0E4

where L1fsstands for the inverse Laplace transform of f, denotes the convolution operator and λ1a function of the time co-variable s.

Proof.We apply the Laplace transformation with respect to the time variable tto the exterior problems (3), recall the Laplace transformation

Lwsxw˜sx=0+wtxetsdt,E5

where sstands for the co-variable of time tand verify Rs>0.

We obtain

sw˜sx+αxxxw˜βsxxw˜+γxw˜=0,xb,xa,Rs>0E6

which is a cubic ordinary differential equation whose solutions are of the form are given explicitly by

ŵsx=c1seλ1sx+c2seλ2sx+c3seλ3sx,xR\abE7

where λ1s,λ2s,λ3sdenote the roots of the depressed cubic equation

αλ3βsλ2+γλ+s=0E8

The three solutions are given by

λks=jk1ζsΘ1sjk1ζs+Θ2s,k=1,2,3E9

where the complex jis given by j=exp2/3,

ζs=4αγ3β2s2γ2+18αβs2γ4β3s4+27α2s22332α29αβsγ2β3s3+27α2s54α313,Θ1s=3αγβ2s29α2,Θ2s=βs3α.E10

Assume that Rs>2β3+9γα227α2, then roots of the cubic Eq. (8) possess the following separation property

Rλ1<β3α,Rλ2>β3αandRλ3>β3α.E11

In fact, we consider the change of variable λ=zβ3α. Then the cubic Eq. (8) becomes z3+pz+q=0such that p=3αγβ23α2andq=27α2s+2β39αβγ27α.

Hence under the condition Rq=27α2Rs+2β39αβγ27α, it follows that the roots ri,i=1,2,3of the equation z3+pz+q=0satisfy

Rr1<0,Rr2>0,Rr3>0

Now back to Eq. (7), for xbwe have from the infinite condition that the coefficients c2and c3must vanishe, hence w˜xs=c1ser1sx, deriving over xand using the continuity of win the interface yield

ŵsb1λ12sŵxxsb=0,ŵxsb1λ1sŵxxsb=0E12

Idem for xa, we have c1=0and hence

ŵxxsaλ2s+λ3sŵxsa+λ2sλ3sŵsa=0.E13

As λ1,λ2,and λ3are roots of the cubic Eq. (8) we obtain immediately

λ1sλ2sλ3s=sαandλ2s+λ3s+λ1s=βαE14

Then the Eq. (13) becomes in terms of λ1s

ŵxxsa+λ1s+βsŵxsasαλ1sŵsa=0E15

Now applying the inverse Laplace transform to Eqs. (8) and (10), we infer

wtaαL1λ12ss+λ1sβswxtaαL1λ1sswxxta=0wtbL11λ12swxxtb=0,wxtbL11λ1swxxtb=0E16

Therefore, we get the following result describing the problem in the bounded domain satisfied by the restriction on ΩTof the original problem (1).

Theorem 1.1 Let α,βbe non negative numbers and γR. The restriction of (1) to Ωis described by the following Initial Boundary Value Problem (IBVP)

tu+αxxxuβtxxu+γxu=0inΩTu0x=u0xatΩnutx=ButxatΣTE17

where Bis derived on ΣTfrom equations

utaαL1λ12ss+βλ1ssuxtaαL1λ1ssuxxta=0
utbL11λ12suxxtb=0,uxtbL11λ1suxxtb=0

We emphasize that those boundary conditions strongly depend on αand βthrough the root λ1s. Some simplifications can be obtained for particular cases allowing direct evaluation of the inverse Laplace transform. Taking for example the BBM equation (for α=0), we can get after applying Laplace transformation to (3),

xw˜sb=γ2βsw˜sbγ2+4βs22βsw˜sb=γ2βw˜sbsγ22β1γ2+4βs2w˜sbs2sγ2+4βs2w˜sb.E18
xw˜sa=γ2βsw˜sa+γ2+4βs22βsw˜sa=γ2βw˜sas+γ22β1γ2+4βs2w˜sas+2sγ2+4βs2w˜sa.E19

In this case, we obtain convolution products with Bessel functions after the Laplace inverse transformation as follows

xwtb=C1ItwtC2J0cItwt+C3J1cwt,
xwta=C1Itwt+C2J0cItwtC3J1cwt.

where we have used the expressions

cc2+s2=LHtJ0ctsLHtJ0cts,andJ0c=J1c,

and the notations c=γ2β,C1=γ2β,C2=γ232β54,C3=2γβ..

Recall that the Bessel functions can be defined by the following integrals

J0ct=1π0πcosctsinτ,J1ct=1π0πcosctsinττ.

From this, we may compute

nwtb=C1Itwt+C2J0cItwtC3J1cwt,
nwta=C1Itwt+C2J0cItwtC3J1cwt.

Thus the boundary operator Bin (2) writes, in the case α=0,

ButxC1Itut+C2J0cItutC3J1cutx=aC1Itut+C2J0cItutC3J1cutx=bE20

Next, we propose an approximation, always for the case α=0, of the BBM equation in ΩTsupplemented with constructed boundary conditions.

2.1 Numerical approximation

This subsection is devoted to the numerical approximation of the obtained IBVP (17) for α=0and Bgiven in (20). Our strategy is to seek numerical simulations that permits to avoid any boundary reflections and in some way renders the fully discrete scheme unconditionally stable.

Let N,Mbe integers, we define time step Δt=TMand spatial step h=baN. The grids tn=nΔt,0nMand xi=a+ih,0iNare used to discretize ΩT. Throughout this paper, we denote uinthe considered approximation of utnxiand the set Nkl=mNkml.

2.1.1 Approximation of the governing equation

We describe a discretization for the BBM equation by the Crank-Nicholson time scheme as follows

utn+1xutnxΔtβxxutn+1xxxutnxΔt+γxutn+1x+xutnx2=0,E21
ut0x=u0x,nN0M1.

For the space finite difference scheme, we use the approximations

xutx12hutx+hu(txh),
xxutx1h2utx+h2u(tx)+u(txh).

The fully discretization then writes,

uin+1uinΔtβui+1n+12uin+1+ui1n+1ui+1n2uin+ui1nh2Δt+γui+1n+1ui1n+1+ui+1nui1n4h=0,inN1N1×N0M1ui0=u0xi,iN0N.E22

2.1.2 Approximation of the boundary condition

The constructed boundary conditions (BC) contains time convolutions that are non-local and introduces many difficulties, for example, using a direct implementation leads to long and low accuracy. Several techniques have been used to overcome these problems by trying to localize the BC, see [5, 6, 7, 8] for more details. The resulting localized BC are easy to implement and more efficient but tends to depend sensitively on the initial data. In our case, we utilize the Discrete Galerkin Method. The BC are formulated as Fredholm integral equations of second kind. The basic idea is to write the boundary condition on (20) in the form

nuta0tK¯1tsusads=0,E23
nutb0tK¯2tsusbds=0.E24

where, the introduced Kernels K¯1,K¯2represent a linear combination of the two Bessel functions of order 0 and 1 at the time ts. After a space discretization we obtain

utx00tK1tsusx0ds=utx1,E25
utxN0tK2tsusxNds=utxN1,E26

where K1=hK¯1and K2=hK¯2. Both resulting Eqs. (25) and (26) can be identified to the linear integral equation

ytDKtsyss=zt,tD.E27

The Eq. (27) is a Fredholm integral equation of second kind, where Dis a closed bounded set in Rm, with m1. The approximation of such integral equation could be made by a discrete Galerkin method using the quadrature rule of Gauss-Legendre as presented in [9]. Based on this, the BC can be similarly discretized while considering the domain Das the time interval 0t. Precisely, we use the Gauss Legendre Quadrature of order q, labeled GLQqwith zeros ξjand weights wjbeing in the interval 11for jN0q. Let iN0N1, we introduce the following transformation

Fi:11titi+1ξti1ξ2+ti+11+ξ2E28

The approximation of the BC is now given by

utn+1x00tn+1K1tn+1susx0ds=utn+1x1E29
utn+1xN0tn+1K2tn+1susxNds=utn+1xN1E30

that is

u0n+10tn+1K1tn+1susx0ds=u1n+1E31
uNn+10tn+1K2tn+1susxNds=uN1n+1E32

For the seek of simplicity, we rewrite the integral terms of (31) and (32) in the form

0tn+1Kitn+1susxkds=Ai0tn+1usxkds+Bi0tn+1J0ctn+1s0surxkdrds+Di0tn+1J0ctn+1susxkdsAiI1tn+1xk+BiI2tn+1xk+DiI3tn+1xk,

such that ik102N, A1=A2=hC1,B2=B1=hC2and D1=D2=hC3, all constants Ciare defined in (??). Thus, basing on the approach presented in [9], the GLQqapplied to the integrals previously defined is described by the following, for nN0M1and k0N,

I1tn+1xk=Δt2i=0nj=0qwjuFiξjxk.E33

The second integral is more complicated since it involves two composing integrals, using Gauss-Legendre quadrature twice yields

I2tn+1xk=Δt24i=0nj=0qwjJ0tn+1Fiξjl=0jm=0qwmuFlξmxkE34

and the remained integral is approximated by

I3tn+1xk=Δt2i=0nj=0qwjJ1tn+1FiξjuFiξjxk.E35

From approximations (33)(35), the numerical solution on the interface of (17) can be given by

u0n+1=u1n+1+f1n+1,E36
uNn+1=uN1n+1+f2n+1.E37

We accomplish this by simply adding (36) and (37) to the discretization of the interior governing Eq. (22). We obtain an implicit scheme that we illustrate by the following system in matrix form

I+A˜+B˜Un+1=I+ABUn+Fn,nN0M1,u0n+1=u1n+1+f1n+1,uNn+1=uN1n+1+f2n+1.E38

with

Un=u1nuN1nI=1001,Fn=Cd1+Cd2f1nu1n00Cd1Cd2f2nuN1n,
A=Cd121011012,B=Cd201011010

and

A˜=Cd12Cd11011012Cd1,B˜=Cd2Cd2101101Cd2.

where the discretization constants are Cd1=βh2,Cd2=γΔt4h.

3. Non standard boundary conditions for the BBM equation

In [4], we have presented a new method to derive transparent boundary conditions for the BBM equation. These boundary conditions have the advantage of being local in time but needs an additional function construct the BC which means bigger system to be solved. We recall that the problem designed to be the restriction in ΩTof the BBM initial Eq. (1) with α=0is given by

tuβtxxu+γxu=fin]0,T]×]a,b[tvβtxxv+γxv=gin]0,T]×]a,b[xu=von]0,T]×abβtxvγv=tufon]0,T]×abu0x=u0xonabv0x=v0xonabE39

4. Numerical examples

We take an initial condition as solitary wave like function locally supported in Ω. The evolution of the solutions are plotted in different time steps before, under and after traveling the right boundary of the considered bounded domain. We save a reference solution that is numerically calculated in a broaden domain of Ωwith Dirichlet boundary condition. We compute infinite error between numerical solutions using both formulations presented in this paper and the reference solution. We denote GLQifor approached solution with BEM and Gauss Legendre Quadrature in (2) for i0,1,2, while NSBC refers to numerical solution with non standard boundary conditions given in (3). We define the following errors

uurefta=supt0Tutaurefta,E40
uureftb=supt0Tutbureftb,E41
uureftx=suptx0T×]a,b[utxureftx.E42

Let β=γ=1, T=10and u0x=sech2x. The considered initial data is locally supported in the interval Ω=]10,10[, since u010=u0108,25.109. We fix h=102and we vary the time step Δt. For a better comparison of these methods, we compute CPU time, in seconds, needed for each one to obtain numerical solution.

Table 1 shows that both methods give a good approximation of the restriction to ΩTof the reference solution. NSBCgive better approximation than GLQ. We can remark a slow convergence of GLQiwith respect to iand also time step. However, NSBC gives a good approximation in as much as Δtgoes to zero. Furthermore, GLQiis more expensive in CPU time when iincreases than NSBC due to the presence of non local convolutions in time in the boundary condition.

BCdtuuref(ta)uuref(tb)uuref(tx)CPU time(s)
GLQ01021.06×1046.33×1026.03×1024
GLQ11.06×1046.3×1025.9304×1029
GLQ21.06×1045.895×1025.9187×10215
NSBC7.39×1055.8×1036.8×1035
GLQ01031.06×1045.92×1025.86×10239
GLQ11.06×1045.86×1025.857×10262
GLQ21.06×1045.852×1025.85×10280
NSBC6.7×1051.41×1031.4×10350
GLQ01041.06×1045.855×1025.858×102436
GLQ11.06×1045.853×1025.852×1021040
GLQ21.06×1045.85×1025.85×1023205
NSBC5.9×1067.35×1047.3×1041015

Table 1.

Infinite errors using different boundary conditions.

We also plot in Figure 1, captions at different times of either reference solution and approximated solutions using NSBC and GLQifor i=2. We can see that NSBC follows the refrence solution better than GLQ especially at last times in the right figure. One remarks that no reflections turn back to the bounded domain when the wave is going out from the right boundary using both methods.

Figure 1.

Reference solution and approximated solutions NSBC andGLQifori=2at different times forΔt=103.

5. Conclusion

We have compared two methods of deriving and approaching boundary conditions for the BBM equation. We presented the BEM for a general equation that is the mixed BBM-KdV equation and that shows the hardness to put easy implemented BC. Furthermore, being non local in time, BC seems to be low accurate and slowly convergent as presented in numerical example. However, this point opens many possibilities trying to improve the accuracy of such BC whether by improving the approximation of convolution product, that comes from Inverse Laplace transformation, via quadrature or exploring a numerical equivalent to such operation such as Ztransformation. We have proposed an other manner to derive local BC that gives better approximation than non local BC. All these conclusions have been made in one space dimension but nothing can be said about the comparison in higher dimension to decide which method is more adapted, this matter will be our interest in future works.

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Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri and Youssef Ouakrim (June 4th 2021). Boundary Element Method for the Mixed BBM-KdV Equation Compared to Non Standard Boundary Conditions [Online First], IntechOpen, DOI: 10.5772/intechopen.96617. Available from:

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