Geometric Numerical Integration for

Peakon b-family Equations

by

Wenjun Cai and Yajuan Sun

Report No. ICMSEC-13-03 December 2013

Research Report

Institute of Computational Mathematics

and Scientific/Engineering Computing

Chinese Academy of Sciences

Geometric numerical integration for peakon b-family

equations

Wenjun Caia, Yajuan Sunb

a Jiangsu Provincial Key Laboratory for NSLSCS,

School of Mathematical Sciences,

Nanjing Normal University, Nanjing 210023, CHINAb LSEC, Academy of Mathematics and Systems Science,

Chinese Academy of Sciences,

P.O. Box 2719, Beijing 100190, CHINA

Abstract

In this paper, we study the family of integrable peakon equations: the Camassa-Holm equation and the Degasperis-Procesi equation which both have very richgeometric properties. Based on these geometric structures, we construct severalnumerical integrators to solve their soliton solutions. Though the Camassa-Holmequation and the Degasperis-Procesi equation have the common properties, theyhave also the significant difference, such as the existence of shock wave solutionfor the Degasperis-Procesi equation. We show that the symplectic pseudospectralFourier spectral integrator can simulate the smooth soliton, peakon soliton verywell. To illustrate the shock wave soliton, we use the splitting technique and com-bine the composition methods. The errors of numerical solution and conservativequantities are shown and also the numerical convergence has been produced.

Keywords: Symplectic integrator, Splitting method, WENO scheme, Multisym-plectic integrator, Peakon, Shockpeakon.

1 Introduction

In the soliton theory, it is important to study the completely integrable nonlinear partialdifferential equations which arise from approximating the shallow water systems. Suchequations usually admit the soliton solutions which have the localized spatial structuresand show the particle-like scattering behavior, and also have the infinite number ofconservation laws. In this paper, our interest is to study a family of third-order dispersivenonlinear equations

ut + c0ux − uxxt + (β + 1)uux − βuxuxx − uuxxx = 0, (1.1)

where β + 1 is the coefficient of advection term uux which often causes a steepening ofwave, β is bifurcation or balance parameter which provides a balance for the nonlinearsolution behavior. The family of PDEs (1.1) includes two important equations: theCamassa-Holm (CH) equation (when β = 2, c0 = 2κ2) [6]; the Degasperis-Procesi (DP)equation (when β = 3, c0 = 3κ3) [13]. Both the CH equation and the DP equation canbe viewed as the model equations of shallow water waves [6, 7, 11, 20]. By presentingthe Lax pair and bi-Hamiltonian structure, the two systems have been proved to becompletely integrable [7, 12]. In the absence of linear dispersion uxxx, with the nonlineardispersion uuxxx one of the novel properties for the DP equation and the CH equation

1

is that they admit the peakon soliton solution [6, 12, 13] which can be derived by usingthe inverse scattering techniques [1, 2, 23, 24].

Although the CH equation and the DP equation have shared some common prop-erties, they are divergent by having the major differences: Lax pair equation and wavebreaking phenomena. As the isospectral problem in Lax pair is the third-order equa-tions for the DP equation when one is second-order equation for the CH equation, theDP equation has more types of solutions than the CH equation. The DP equation hasnot only the peakon solutions, but also has the shockpeakon solutions [10, 22] whichproduce the difficulty for capturing the shock wave numerically. Comparing with theCH equation, there exists only few numerical methods and corresponding numericalanalysis for the DP equation. The available numerical methods include the operatorsplitting schemes [9], the particle method based on the multi-shockpeakon solutions[18], the conservative finite difference schemes [28], Local discontinuous Galerkin (LDG)methods [36], the compact finite difference method [37], etc.

Like other soliton solutions, as a class of conservative PDEs the DP equation and theCH equation have many conservative properties, such as the bi-Hamiltonian structures,infinite number of conservation laws etc. A natural idea of numerical computation isto construct the numerical methods which can carry as much as possible these intrinsicproperties. Geometric numerical integrators are constructed based on this idea [17].Comparing with the non-geometric numerical integrators, geometric numerical integra-tors usually illustrate the remarkable capacity to capture the dynamical behavior of thegiven system over long time [17]. Based on Hamiltonian structure, our purpose of thispaper is to construct the corresponding numerical methods. To simulating the smoothsoliton solutions, we present the pseudo-spectral method in space and symplectic in-tegrator in time. Though the associated matrix is full, it can be solved by means ofFast Fourier Transform (FFT) and reaches the so-called exponential convergence whenthe solution is enough smooth. The resulting numerical method shows good results inpreservation of invariants and long-time simulation. Though the concerned numericalmethods have the nice conservative property, the one fails in simulating the shockpeakonwave solution. By using the splitting technique presented in [9, 14], we split the DPequation as the Benjamin-Bona-Mahony (BBM) equation and the Burgers’ equation.Therefore, we obtain the semidiscretized by composing the multisymplectic method (forBBM equation) and the WENO method of fifth-order (for the Burgers’ equation) inspace. The full discretization is obtain by the splitting method of higher-order.

This paper is organized as follows. In section 2, a symplectic pseudo-spectral inte-grator is proposed based on the Hamiltonian structure of DP equation. In section 3, TheDP equation is splitted as a composition of conservative system (BBM equation) andhyperbolic system (Burgers’ equation). The multisymplectic pseudo-spectral method isapplied to the BBM equation while the WENO method of fifth-order is applied to theBurger’s equation. Section 4 is devoted to numerical experiments where we simulate thesoliton solution, the peakon solution, and also the shockpeakon solutions, and providethe numerical errors. We conclude this paper in Section 5. The spectral method hasbeen widely used in numerical computation because of its high accuracy and exponentialconvergency.

2 Symplectic Fourier spectral integrator for the DP equa-

tion

The DP equation is a system of third-order PDEs

ut − uxxt + 4uux − 3uxuxx − uuxxx = 0. (2.1)

2

It was proposed firstly by Degasperis and Procesi [13] as one of a family of equation-s satisfying the asymptotic integrability to third order, then in [12] the completelyintegrability of the system was proved because of the existence of a Lax pair and a bi-Hamiltonian structure. We start this section by reviewing the Hamiltonian formulationof the DP equation.

Denote m = (1− ∂2x)u, then (2.1) can be written as

mt + umx + 3uxm = 0. (2.2)

With Helmholtz operator 1− ∂2x, u(x, t) can be solved by

u(x, t) =1

2

∫ ∞

−∞exp(−|x− y|)m(y, t)dy.

Let B0 and B1 be two operators as

B0 = −∂x(1− ∂2x)(4 − ∂2x), B1 = m2

3 ∂xm1

3 · (∂x − ∂3x)−1m

1

3 ∂xm2

3 ·, (2.3)

where ∂xm · v = (mv)x. The DP equation (2.2) can be rewritten in the infinite Hamil-tonian formulation with bi-Hamiltonian structure [12]

mt = B0δH0

δm= B1

δH1

δm, (2.4)

whereH0 =16

∫∞−∞ u3dx, H1 = −9

2

∫∞−∞mdx, B0 and B1 are two Hamiltonian operators1.

Remark. Note that ∂xm · v = (mv)x, then

B1v =1

3mx(∂x − ∂3x)

−1(2

3mxv +mvx) +m(1− ∂2x)

−1(2

3mxv +mvx).

Denote δHδm be the functional derivative2. By the definition, it is clear that

δH1

δm=

1

2(1− ∂2x)

−1u2,δH2

δm= −

9

2.

The method of lines is a technique which is very often used for solving the PDEsby first discretizing the spatial derivatives and then discretizing the semi-discretizedsystem by the ODEs solver. In this section, we provide the semi-discretized systemby applying the pseudo-spectral Fourier method in space to discretize the DP equationbased on Hamiltonian formulation. The Fourier spectral method (refer to [30] for detail)is to approximate the unknown function by the finite sum of a series of trigonometricfunctions, which are much more suitable for simulating the evolution of solutions withperiodic phenomena. By requiring the approximate solution to satisfy the concernedequations at a set of collocation points, the method often refers to the pseudo-spectralFourier method which, however, has much more wide contents than collocation method.In applying the spectral method to the PDEs, the differential operator is approximatedby the so-called spectral differentiation matrix which has played an important role inthe implementation of spectral method. By using the Fast Fourier Transform (FFT),the Fourier spectral method provides a numerical discretization with the convergency ofso-called infinite order and high efficiency [35].

1A linear operator D which may depend on u and its higher-order derivatives, is called Hamiltonianif it satisfies {F , G} = −{G,F} (skew-adjoint) and {R, {F , G}}+{G, {R,F}}+{F , {G,R}} = 0 (Jacobiindentity) for all functionals F , G,R of u. Here, {·, ·} denotes the Poisson bracket defined by {F , G} =∫∞

−∞

δF

δuD δG

δudx.

2The functional derivative ∂H

∂mis defined by

∫ ∞

−∞

δH

δmδmdx = lim

τ→0(H(m+ τδm)−H(m))/τ .

3

Consider a system of PDEs defined inR×[−L,L] with periodic boundary conditions.Take xj = −L+ hj, 0 ≤ j ≤ N − 1 be the spatial grids with h = 2L

N the space step. Thespectral method is to search an approximate solution in the form of

u(x, t) ≈

N−1∑

j=0

ujϕj(x) := INu(x, t),

where uj = u(xj , t), ϕj(x) = 1/N∑

N

2−1

k=−N

2

exp(−ikπ(xj − x)/L). It is noticed that

ϕj(xj) = 1, ϕj(xl) = 0 for l 6= j, thus

INu(xj , t) = u(xj , t).

Define FN : INu(x, t) → U(t) = (u0, · · · , uN−1), then FN INu(x, t) = INu(xj , t). Thepseudo-spectral Fourier method is to find the approximate solution INu(x, t) whichsatisfies the given PDE at {xj}

N−1j=0 . It is shown that

∂mINu

∂xm(xj , t) =

N−1∑

k=0

(Dm)j,kuk, m ≥ 1,

where Dm is called the m-differentiation matrices, for m = 1, 2, which are

(D1)j,l =

1

2µ(−1)j+l cot(µ

xj − xl2

), j 6= l,

0, j = l,(2.5)

(D2)j,l =

1

2µ2(−1)j+l+1 1

sin2(µ(xj − xl)/2), j 6= l,

− µ22(N/2)2 + 1

6, j = l.

(2.6)

where µ = πL . Clearly, D1 is skew-symmetric while D2 is symmetric. Applying the

pseudo-spectral Fourier method to u = D(u) δHδu gives the semi-discretized system

d

dtU = D(U)∇H(U),

where D(U)V = FN ◦ IN ◦ D(INu) ◦ F−1N (V ), H(U) =

∫ L−LH(F−1

N (U))dx. It is shownthat the discrete operator D(U) is a N ×N skew-symmetric matrix. In fact, it is easyto check that

< D(U)V,W >N =< FN ◦ IN ◦ D(INu) ◦ F−1N (V ),W >N

=< F−1N (V ),D(INu)INw >

=< INv, IN ◦ D(INu)INw >

= − < V,D(U)W >N ,

where < ·, · >N denotes the inner product in RN , < ·, · > denotes the inner productin the function space. To simulate the solitary solution of the DP equation, we cancompute the equivalent problem on [−L,L] with periodic boundary condition. Basedon the first Hamiltonian formulation (2.4), we can derive the following semi-discretizedsystem:

Ut = ((I −D2)−1D1(4−D2)∇H0(U) (2.7)

4

with ∇H0(U) = −12 [u

20, · · · , u

2N−1]

T . Similarly, based on the second Hamiltonian formu-lation, we derive the following semi-discretized system

(I −D2)Ut = −D1(1−D2)UU − 3(1 −D2)UD1U, (2.8)

with H1(U) = −9/2N−1∑

j=0(1−D2)U .

Applying the symplectic methods to the above semi-discretized system, provides thefull discretization of the DP equation denoted by Scheme I.

3 Operator splitting method for the DP equation

To define the weak solution of the DP equation (2.1), it is usually written in the formof

ut + f(u)x + Px = 0 (3.1)

with (1−∂2x)P = 3κ3u+3f(u) and f(u) = 12u

2. In fact, the DP equation can be splittedas two subsystems [9, 14]:

ut + f(u)x = 0, (3.2)

ut + Px = 0, (3.3)

where (3.3) is called the Benjamin-Bona-Mahhony (BBM) equation , and (3.2) is theBurgers’ equation. There are the classical methods to compute the two systems. To beself-contained, we briefly introduce the methods used in this paper.

Burgers’ equation is a model PDEs in fluid dynamics which usually admit the shockwave solution even the initial value is smooth. The popular shock-capturing schemesusually have the high-resolution, the examples include the high-order total variationdiminishing (TVD) schemes proposed firstly by Harten, the essentially non-oscillatory(ENO) schemes and the weighted essentially non-oscillatory (WENO) scheme etc. (referto [31] and reference therein). Here, in this paper we use the fifth-order WENO schemefor the Burgers’ equation [19].

We consider the following semi-discretized form of (3.2)

duj(t)

dt= −

1

∆x

(

fj+ 1

2

− fi− 1

2

)

, j = 0, · · · , N − 1, (3.4)

where ∆x is the time step, uj ≈ u(j∆x, t), fj− 1

2

and fj+ 1

2

are called the numerical fluxes

which have different choices. To satisfy the monotone condition, we usually split thenonlinear term, e.g., with the Lax-Friedrichs splitting, which gives

f(u) = f+(u) + f−(u),

where f±(u) =1

2(f(u)± αu), α is taken as α = maxu |f

′(u)| over the relevant range of

u. Thus, it is known thatdf+(u)

du≥ 0 and

df−(u)

du≤ 0. As follows, we introduce one

choice of the numerical flux with which (3.4) leads to the fifth-order WENO scheme.Denote vj = f+(uj), Vj = [vj , vj+1, vj+2]

T .Let

v(0)

j+ 1

2

= αVj , v(1)

j+ 1

2

= βVj−1, v(2)

j+ 1

2

= γVj−2

with α = [1/3, 5/6,−1/6], β = [−1/6, 5/6, 1/3], γ = [1/3,−7/6, 11/6].

5

Take

v−j+ 1

2

= ω0v(0)

j+ 1

2

+ ω1v(1)

j+ 1

2

+ ω2v(2)

j+ 1

2

,

where ωi are the nonlinear weights given by

ωj =αj

α0 + α1 + α2, αj =

di(ǫ+ βj)2

, j = 0, 1, 2,

with d0 = 3/10, d1 = 3/5, d2 = 1/10, and the smoothness indicators: β0 = 13(a1Vj)2/12+

(b1Vj)2/4, β1 = 13(a1Vj−1)

2/12+(b2Vj−1)2/4, β2 = 13(a1Vj−2)

2/12+(b3Vj−2)2/4. Here,

a1 = [1,−2, 1], b1 = [3,−4, 1], b2 = [1, 0,−1], b3 = [1,−4, 3] are the three-dimensionalvectors, ǫ is a parameter which often is chosen as ǫ = 10−6. Therefore, the numericalfluxes in (3.4) can be taken as

fj+ 1

2

= f+j+ 1

2

+ f−j+ 1

2

, fj− 1

2

= f+j− 1

2

+ f−j− 1

2

, (3.5)

where f+j+ 1

2

= v−j+ 1

2

is the positive flux while applying a mirror image with respect to

j + 12 gives the negative flux f−

j+ 1

2

.

By using the fifth-order WENO scheme, we derived the ODEs (3.4). We apply theTVD Runge-Kutta method to discretize the ODEs. This class of Runge-Kutta methodshas strong stability in any semi-norm (total variation norm, maximum norm, entropycondition, etc.), and is called TVD Runge-Kutta methods in [32] and [33] as they areoriginally usually used in combination with the TVD scheme in space to solve the shockwave equation. They, then, are generalized in [15] and [16]. We describe a third-orderTVD Runge-Kutta method with the following Butcher tableau:

0 0 0 01 1 0 012

14

14 0

1 16

16

23

which applied to ut = L(u) can be rewritten as

u(1) = un +∆tL(un),

u(2) =3

4un +

1

4u(1) +

1

4∆tL(u(1)),

un+1 =1

3un +

2

3u(2) +

2

3∆tL(u(2)).

The subsystem (3.3) is the regularized long-wave (RLW) equation which was firstput forward by Peregrine (1966) to describe the development of long wave behaviour.,then was named after T.B. Benjamin, J.L. Bona and J. Mahoney by P.J. Olver [29]. In[3], the details of existence, uniqueness and stability of the solutions of equation (3.3)had been studied. Different from the KDV equation, the BBM equation is known tohave only three conservation laws, and its solution shows a non-elastic interaction of thesolitary waves. As a conservative system, the BBM equation can be written in infiniteHamiltonian system []

ut = −(1− ∂2x)−1∂x

δH

δu, (3.6)

with the Hamiltonian operator −(1− ∂2x)−1∂x, and here the Hamilton operator is H =

16

∫ L−L u

3dx. It also can be written in the multisymplectic formulation [34]

Mzt +Kzx = ∇zS(z), z ∈ R5 (3.7)

6

with z = (ψ, u, v, w, p)T , S = up−3

2κ3u2 −

1

2u3 +

1

2vw and

M =

0 −12 0 0 0

12 0 −1

2 0 00 1

2 0 0 00 0 0 0 00 0 0 0 0

, K =

0 0 0 0 −10 0 0 −1

2 00 0 0 0 00 1

2 0 0 01 0 0 0 0

.

The BBM equation has the multisymplectic conservation law

∂

∂t(du ∧ dψ + dv ∧ du) +

∂

∂x(2dp ∧ dψ + dw ∧ du) = 0.

Here, we apply the multisymplectic pseudospectral method [5, 8] to the BBM equation(3.7), it reads

Md

dtzj +K

N−1∑

k=0

(D1)j,kzk = ∇zS(zj), j = 0, · · · , N − 1, (3.8)

where zj = (ψj , uj , vj , wj , pj)T . By applying the implicit midpoint rule to the semi-

discretized system (3.8), we derive the full discretization as follows:

Mδ+t znj +K

N−1∑

k=0

(D1)j,kzn+ 1

2

k = ∇zS(zn+ 1

2

j ), (3.9)

where δ+t is the forward difference operator δ+t znj = (zn+1

j − z − jn)/∆t, and zn+ 1

2

j =

(zn+1j + znj )/2. Scheme (3.9) is called multisymplectic as it can preserve the following

discrete multisymplectic conservation law

ωn+1j − ωn

j

∆t+

N−1∑

k=0

(D1)j,kκn+ 1

2

j,k = 0, j = 0, 1, · · · , N − 1, (3.10)

where ωnj = dznj ∧Mdznj and κ

n+ 1

2

j,k = dzn+ 1

2

j ∧Kdzn+ 1

2

k . As D1 is skew-symmetric andκj,k = κk,j, taking the sum of (3.10) from j = 0 to N − 1 gives the global symplecticitywhich is shown as

N−1∑

j=0

ωn+1j − ωn

j

∆t= 0.

In practice, we compute the following equivalent scheme whichobtained by eliminatingthe auxiliary variables from (3.9) that

δ+t Un − δ+t D

21U

n +D1

(

3κ3Un+ 1

2 +3

2(Un+ 1

2 )2)

= 0, (3.11)

where Un = (un0 , · · · , unN−1)

T . Denote the numerical solution of Burgers’ equation and

the BBM equation, respectively, by ϕ[1]∆x,∆t, ϕ

[2]∆x,∆t. The resulting numerical discretiza-

tion is derived by combining ϕ[1]∆x,∆t, and ϕ

[2]∆x,∆t by means of the general splitting

technique

Ψ∆x,∆t = ϕ[1]∆x,bm∆t ◦ ϕ

[2]∆x,am∆t ◦ · · · ◦ ϕ

[2]∆x,b1∆t ◦ ϕ

[1]∆x,a1∆t. (3.12)

When taking a1 = a2 = 1/2, b1 = 1, b2 = 0, it is the second-order Strang splitting, whichis called Scheme II in this paper.

7

4 Numerical experiments

In this section, we simulate the peakon and shockpeakon waves of the DP equation byusing two schemes: Scheme I and Scheme II. In our numerical examples, we employthe periodic boundary condition. The computation environment is CPU Core i5 (2.3GHz), 4 GB memory, Mac OS, MATLAB (R2012b).

Denote unj ≈ u(j∆x, n∆t) be the numerical solution. The numerical errors in L1

and L∞ norms are defined by

||en||1 =N−1∑

j=0

|unj − u(j∆x, n∆t)|∆x, ||e||∞ = maxj=0,··· ,N−1

|unj − u(j∆x, n∆t)|.

It is known that the DP equation is completely integrable which shows that it has theinfinite number of conservation laws. Among these conservation laws, the three mostimportant conserved quantities on [−L,L are:

I0 =

∫ L

−Lu3dx, I1 =

∫ L

−L(u− uxx)dx, I2 =

∫ L

−Ludx.

We define the corresponding discrete conserved invariants at time t = n∆t as

In0 =

N−1∑

j=0

(unj )3∆x, In1 =

N−1∑

j=0

(unj −D2unj )∆x, In2 =

N−1∑

j=0

unj∆x,

where D2 is the second-order spectral differential matrix (2.6). The global errors aredefined by

En0 = |In−1 − I0−1|, En

1 = |In0 − I00 |, En2 = |In1 − I01 |.

As mentioned in [22], the conservation laws I−1 and I0 are not conserved in the shock-peakon cases in which waves are discontinuous functions. But in the peakon cases, wehave the following conservation theorem for Scheme I.

Remark. The resulting numerical method preserves the discrete invariants In1 andIn2 ,

Example 4.1. Single peakon solution

When κ = 0, the DP equation has a single peaked traveling wave solution which is

u(x, t) = ce−|x−ct|, (4.1)

where c expresses the wave speed which is chosen as c = 0.25 in our computation. As thesoliton solutions decay exponentially when x approaches to ±∞, the Cauchy problemof the DP equation can be formed into a equivalent model problem in a finite spatialdomain. For the DP equation, the computation domain is taken as [−20, 20].

Fig. 1 shows the numerical solution errors. It is observed that the numerical solutioncomputed by Scheme I has the oscillations near the peakon even we use the fine mesh.This may be explained by the so-called Gibbs phenomenon because the non-smoothnessof the derivative of the solution. However, we can resolve the problem by using Scheme

II.

8

−20 −15 −10 −5 0 5 10 15 200

0.5

1

1.5

2

2.5

3x 10

−3

x

erro

r

−20 −15 −10 −5 0 5 10 15 200

0.5

1

1.5

2

2.5

3x 10

−3

x

erro

r

Fig. 1: Numerical solution errors at t = 1, with ∆x = 40/1024, ∆t = 0.01 (a) byScheme I; (b) by Scheme II.

We illustrate the behavior of the numerical solutions by Scheme I and Scheme II

over [0, 30]. Fig. 2 shows that the wave computed by Scheme I and Scheme II att = 30. Although numerical solution by Scheme I is affected by the Gibbs phenomenon,the magnitude of numerical solution error by Scheme I and Scheme II do not showa significant difference. We also investigate the errors of conserved quantities computedby two schemes.

Fig. 3 shows the errors of three invariants calculated by Scheme I and Scheme II.From Fig. 3, we noticed that the Scheme I can preserve the invariants In1 and In2 whilethe error of I2 is bounded. However, all the three invarints grows linearly for Scheme

II.

−20 −15 −10 −5 0 5 10 15 20−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

x

u(x,

t)

−20 −15 −10 −5 0 5 10 15 20−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

x

u(x,

t)

Fig. 2: Numerical solutions at t = 30, ∆x = 40/1024, ∆t = 0.01 (a) by Scheme I; (b)by Scheme II.

9

0 5 10 15 20 25 3010

−16

10−14

10−12

10−10

10−8

t

error

E−1 E 0 E 1

0 5 10 15 20 25 3010

−16

10−14

10−12

10−10

10−8

10−6

10−4

t

erro

r

E−1 E 0 E 1

Fig. 3: Numerical errors of three invariants, ∆x = 40/1024, ∆t = 0.01 (a) by Scheme

I; (b) by Scheme II.

The computation time for Scheme I and Scheme II is shown in Table. 1. AsScheme I is the pseudo-spectral method, we can use the FFT algorithm which reducesdramatically the computational cost. We also apply the FFT algorithm to solving thesecond subsystem of the DP equation: BBM equation, however the total computation isstill expensive in implementing the Scheme II because of the splitting and compositionprocess.

Table. 1: CPU time for single peakon solution with different grid sizes over [0, 30].

∆x 40/512 40/1024 40/2048

Scheme I 1.2687s 2.4804s 5.4814s

Scheme II 76.5092s 132.4078s 250.5352s

Example 4.2. Two-peakon interaction

We simulate the behavior of two-peakon interaction for the DP equation by takingthe initial condition

u0(x) = m1e−|x−x1| +m2e

−|x−x2|, (4.2)

where m1 = 2,m2 = 1 and x1 = −13.792, x2 = −4. In fact, the analytic expression ofthe general n-peakon solution has been constructed by Lundmark in [22]. When n = 2,the 2-peakon solution is

u(x, t) = m1(t)e−|x−x1(t)| +m2e

−|x−x2(t)|,

where x1(t) = log (λ1−λ2)2b1b2(λ1+λ2)(λ1b1+λ2b2)

, x2(t) = log(b1+b2),m1(t) =(λ1b1+λ2b2)2

λ1λ2(λ1b21+λ2b22+4λ1λ2λ1+λ2

b1b2),

and m2(t) = (b1+b2)2

λ1b21+λ2b22+4λ1λ2λ1+λ2

b1b2with bk(t) = bk(0)e

t/λk satisfying b1 + b2 = ex2 and

b1λ1

+ b2λ2

= m1ex1 + m2e

x2 . Here, λ1 and λ2 are two real nonzero roots of polynomial

invariant 1− (m1 +m2)z +m1m2(1−ex1

ex2)z2, and we assume λ−1

2 < λ−11 .

Here, we only use Scheme I to compute the two-peakon solution defined on [−20, 20]with periodic boundary condition. Fig. 4 illustrates the snapshots of numerical solutionsat t = 0, 4, 8, 12. The interaction of two peakon waves is resolved very well by Scheme

I. Fig. 5 shows the numerical errors of three invariants, which implies that the threeinvariants can be preserved very well over long-time.

10

−20 −15 −10 −5 0 5 10 15 20−0.5

0

0.5

1

1.5

2

x

u(x,

t)

−20 −15 −10 −5 0 5 10 15 20−0.5

0

0.5

1

1.5

2

x

u(x,

t)

Scheme IExact

−20 −15 −10 −5 0 5 10 15 20−0.5

0

0.5

1

1.5

2

x

u(x,

t)

Scheme IExact

−20 −15 −10 −5 0 5 10 15 20−0.5

0

0.5

1

1.5

2

x

u(x,

t)

Scheme IExact

Fig. 4: Snapshots of two-peakon interaction of the DP equation at (a) t = 0; (b) t = 4;(c) t = 8; (d) t = 12.

0 2 4 6 8 10 1210

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

t

erro

r

E−1 E 0 E 1

Fig. 5: Numerical errors of three invariants with initial condition (4.2).

Example 4.3. Soliton solution

The DP equation also admits the multi-soliton solutions which are constructed re-cently in [26, 27] when κ 6= 0. The one soliton solution can be expressed explicitlyby

u(x, t) =8κ3

a1

(a21 − 1)(a21 −14)

cosh ξ1 + 2a1 −1a1

,

11

where ξ1 can be solved from x− c1t−x10 =ξ1κk1

− ln(α1 − 1 + (α1 + 1)eξ1

α1 + 1 + (α1 − 1)eξ1

)

by Newton

iteration, and the constant coefficients are defined by

α1 =

√

(2a1 − 1)(a1 + 1)

(2a1 + 1)(a1 − 1), a1 =

√

1− 14κ

2k211− κ2k21

, c1 =3κ3

1− κ2k21, x10 =

y10κ.

We choose κ = 0.511, κk1 = 0.8, y10 = 0. Fig. 6 illustrates the exact solution andthe numerical solution at t = 10 computed by Scheme II with ∆t = ∆x = 40/256. Wecan see clearly that although we take a coarse mesh, the performance of the numericalsolution is also very good which is due to the high accuracy of Scheme II. We list theconvergence order of Scheme II in Table. 2 by using the smooth soliton solution anduniform meshes with N cells at time t = 1.

−20 −15 −10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

u(x,t

)

Scheme IIExact

Fig. 6: Comparison of the numerical solution and exact solution at t = 10.

Table. 2: Convergence order of Scheme II in space.

N L1 Order L∞ Order

64 5.1775e-02 1.4110e-02128 8.3838e-03 2.6266 2.1144e-03 2.7384256 3.5374e-04 4.5669 1.0478e-04 4.3348512 2.8145e-06 6.9737 2.6112e-06 5.3265

Example 4.4. Wave breaking

For the DP equation, there exist the blow-up phenomena with certain initial profileswhich theoretical results have been established in [21]. To investigate the wave breakingphenomena, we choose the following initial condition

u0(x) = sech2(d(x− x0)), (4.3)

where d is a parameter which expresses the width of the initial profile. With this initialcondition, the momentum density can be calculated as

m0(x) = u0(x)− u0,xx(x) = sech2(d(x− x0))(

1− 4d2 +6d2sech2(d(x− x0)))

. (4.4)

According to the sign of m0(x), we can determine when the wave breaking may happen.From (4.4), it is noticed that m0(x) > 0 if d < 1

2 , the sign of m0(x) will change when

12

d > 12 . To confirm further, we take two initial values with d = 0.3 and d = 2, respectively.

As the shock waves usually appear after wave breaking, we use Scheme II to simulatethe case of d = 2 and Scheme I to simulate another case. Fig. 7 and Fig. 8 show thenumerical solutions and errors of three invariants when d is taken as d = 0.3 and d = 2,respectively. We take the space step ∆x = 40/1024 and the time step ∆t = 0.01. It isshown that in Fig. 8 the magnitude of E1 is still very small, but the invariant I−1 is nolonger conserved. This is consistent with the theoretical result in [22].

−40−20

020

40

0

10

20

30−0.5

0

0.5

1

1.5

2

x

t

u(x,

t)

0 5 10 15 20 25 3010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

t

erro

r

E−1 E 0 E 1

Fig. 7: Left: 3D plot of numerical solution when d = 0.3; Right: Errors of threeinvariants.

−40 −30 −20 −10 0 10 20 30 40−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u(x,t)

t=0t=2t=20

0 5 10 15 2010

−16

10−15

10−14

10−13

10−12

t

erro

r

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t

erro

r

Fig. 8: Top: Numerical solutions at different times when d = 2; Left bottom: Error ofinvariant I1; Right bottom: Error of invariant I−1.

13

Another example for the wave breaking phenomenon is provided in [9, 14] with theinitial condition

u0(x) = e0.5x2

sin(πx). (4.5)

The computation domain is taken as [−2, 2]. We use N = 512 be the number of gridpoints and ∆t = 0.001. In Fig. 9, the shock peakon profiles at t = 0, 0.12, 0.18, 0.3, 0.5and 1.1 are shown. In Fig. 10, it is shown that the invariant I1 is preserved up to round-off error.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

4

x

u(x,

t)

(a) t=0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

4

x

u(x,

t)

(b) t=0.12

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

4

x

u(x,

t)

(c) t=0.18

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

3

4

x

u(x,

t)

(d) t=0.3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

x

u(x,

t)

(e) t=0.5

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

u(x,

t)

x

(f) t=1.1

Fig. 9: Shock formation of the DP equation with initial condition (4.5).

14

0 0.2 0.4 0.6 0.8 110

−18

10−17

10−16

10−15

10−14

t

erro

r

Fig. 10: Error En1 with the initial condition (4.5).

Example 4.5. Peakon-antipeakon interaction

Lundmark [22] has studied in detail the solutions involving peakons and antipeakons,and showed when a peakon collides with an antipeakon the jump discontinuities whichis often called “shockpeakon”, will appear. As follows, we will confirm numericallythis theoretical results for both the symmetric (m1 +m2) = 0 case and nonsymmetric(m1 +m2) 6= 0 cases with m1,m2 the coefficients in (4.2). As the solution is not verysmooth, we simulate the wave by using Scheme II.

For the symmetric case, we choose the initial condition (4.2) with m1 = 1,m2 =−1, x1 = −5, x2 = 5. The solution is a two-peakon wave before collision. At tc ≈ 5, thecollision happens which produces a stationary shock decaying shockpeakon wave.

For the nonsymmetric case, we choose the initial values with m1 = 2,m2 = −1, x1 =−5, x2 = 5. The collision occurs at tc ≈ 3.3628. After collision, there appears a shock-peakon wave which moves to the right as the peakon wave is stronger than the antipeakonwave.

Fig. 11 and Fig. 12 show the numerical solutions and the exact solution for thesymmetric and nonsymmetric cases. The numerical results show that the numericalsolution and the exact solution are excellently matched, and the numerical solutioncan also simulate the collision very well. We also illustrate the numerical errors of theinvariant I1 for both cases in Fig. 13. The numerical result states that the I1 can bepreserved by Schemes II) for the symmetric solution while for the nonsymmetric case,the numerical error I1 grows linearly with very small range.

−15 −10 −5 0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u(x,

t)

(a)

−15 −10 −5 0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u(x,

t)

Scheme IIExact

(b)

15

−15 −10 −5 0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u(x,

t)

Scheme IIExact

(c)

−15 −10 −5 0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u(x,

t)

Scheme IIExact

(d)

Fig. 11: Peakon-antipeakon interaction in the symmetric case with ∆x = 30/1024,∆t =0.01: (a) t = 0; (b) t = 3; (c) t = 6; (d) t = 10.

−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1

1.5

2

2.5

x

u(x,

t)

(a)

−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1

1.5

2

2.5

x

u(x,

t)

Scheme IIExact

(b)

−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1

1.5

2

2.5

x

u(x,

t)

Scheme IIExact

(c)

−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1

1.5

2

2.5

x

u(x,

t)

Scheme IIExact

(d)

Fig. 12: Peakon-antipeakon interaction in the nonsymmetric case with ∆x = 30/1024,∆t = 0.01: (a) t = 0; (b) t = 2; (c) t = 4; (d) t = 6.

16

0 2 4 6 8 1010

−18

10−17

10−16

10−15

10−14

t

erro

r

(a)

0 1 2 3 4 5 610

−16

10−15

10−14

10−13

t

erro

r

(b)

Fig. 13: Error En1 of peakon-antipeakon interaction: (a) Symmetric case; (b) Nonsym-

metric case.

Example 4.6. Shockpeakon-shockpeakon interaction

As the DP equation can admit a shockpeakon wave, we can also investigate theinteraction of two shock peakon waves. We adopt the following initial condition whichis introduced in [14]

u0(x) = 2e−|x+5| − sgn(x+ 5)e−|x+5| − e−|x−5| − 0.5sgn(x− 5)e−|x−5|. (4.6)

The evolutionary behavior of two shock peakon waves is plotted in Fig. 14. The initialwave is a two shockpeakon wave, which will merge into a single shockpeakon wave attc ≈ 3.5. The global error of conserved quantity I1 is plotted in Fig. 15 which shows alinear growth even though the error is not very large.

−20 −15 −10 −5 0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

x

u(x,

t)

t=0t=2t=3t=tc

(a)

−20 −15 −10 −5 0 5 10 15 20−0.5

0

0.5

1

1.5

2

2.5

x

u(x,

t)

t=tct=6t=8t=10

(b)

Fig. 14: Shockpeakon-shockpeakon interaction: (a) before collision; (b) after collision.

17

0 2 4 6 8 1010

−16

10−15

10−14

10−13

10−12

t

erro

r

Fig. 15: Error En1 with the initial condition (4.6).

5 Concluding remarks

To solve the problem of DP equation with different initial values, we have proposed twoclass of integrators: Scheme I and Scheme II. It is noticed that Scheme I is thesymplectic pseudo-spectral method constructed based on the Hamiltonian formulationof the DP equation. Hence, by Fast Fourier Transformation, Scheme I can give avery accurate result with cheap cost in simulating the behavior of periodic and smoothsolitary waves. As the DP equation can be split into the Burgers’ equation and BBMequation, we can use the splitting technique to derive Scheme II which is obtainedby combining the WENO scheme and multisymplectic pseudospectral method. Despitethe computation cost, Scheme II can be implemented in simulating both the smoothsolitary wave and the shock wave solution. Scheme I is outstanding for solving thesmooth soliton solution and even peakon solution because of its less computation costand superior conservative properties, it fails for solving the shock wave solution. Scheme

II is obtained based on the decomposition of the DP equation: the Burgers’ equationand the BBM equation. By applying the fifth-order WENO scheme to the Burgers’equation, and a new multisymplectic integrator for the BBM equation, using the Strangsplitting technique gives Scheme II which has the better ability to simulate the shockwave solution.

Acknowledgement.

This research was supported by the National Natural Science Foundation of China(60931002, 11271357), the Foundation for Innovative Research Groups of the NNS-FC 11021101, the National Basic Research Program of China 2010CB832702 and theITER-China Program 2014GB124005.

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20