A self-adaptive moving mesh method for the Camassa-Holm equation

# A self-adaptive moving mesh method for the Camassa-Holm equation

Bao-Feng Feng, Ken-ichi Maruno, and Yasuhiro Ohta  Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539-2999, USA  Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
###### Abstract

A self-adaptive moving mesh method is proposed for the numerical simulations of the Camassa-Holm equation. It is an integrable scheme in the sense that it possesses the exact -soliton solution. It is named a self-adaptive moving mesh method, because the non-uniform mesh is driven and adapted automatically by the solution. Once the non-uniform mesh is evolved, the solution is determined by solving a tridiagonal linear system. Due to these two superior features of the method, several test problems give very satisfactory results even if by using a small number of grid points.

July 12, 2019

###### keywords:
The Camassa-Holm equation, integrable semi-discretization, peakon and cupson solutions, self-adaptive moving mesh method
###### Msc:
65M06, 35Q58, 37K40
journal: Journal of Computational and Applied Mathematics

## 1 Introduction

Since its discovery CH (), the Camassa-Holm (CH) equation

 wT+2κwX−wTXX+3wwX=2wXwXX+wwXXX. (1)

has attracted considerable interest because it describes unidirectional propagation of shallow water waves on a flat bottom. It also appeared in a mathematical search of recursion operators connected with the integrable partial differential equations FF (). By virtue of asymptotic procedures, the CH equation was reconfirmed as a valid approximation to the governing equation for shallow water waves Johnson02 (); Constantin08 (). The CH equation also arises as a model for water waves moving over an underlying shear flow Johnson03 (), in the study of a certain non-Newtonian fluids Busuioc (), and as a model for nonlinear waves in cylindrical hyperelastic rods Dai (). The CH equation is completely integrable (see CH () for the Lax pair formulation and Constantin (); Constantin2 () for the inverse scattering transform), and it has various exact solutions such as solitons, peakons, and cuspons. When , the CH equation admits peakon solutions which are represented by piecewise functions CH (); CHH (); Beals (). When , cusped soliton (cuspon) solutions, as well as smooth soliton solutions, were found by several authors. Schiff (); Kraenkel2 (); Johnson (); Li (); Parker (); Parker2 (); DaiLi (); Matsuno ().

Several numerical schemes have been proposed for the CH equation in the literature. These include a pseudospectral method Kalisch (), finite difference schemes Holden1 (); Coclite1 (), a finite volume method Artebrant (), finite element methods Chiwang (); Matsuo (); Matsuo2 (), multi-symplectic methods Cohen (), and a particle method in terms of characteristics based on the multi-peakon solution Camassa1 (); Lee1 (); Lee2 (); Lee3 (); Holden2 (). We comment that the schemes in Holden1 (); Coclite1 () and in Cohen () can handle peakon-antipeakon interactions. However, it still remains a challenging problem for the numerical integration of the CH equation due to the singularities of cuspon and peakon solutions.

In the present paper, we will study an integrable difference scheme for the CH equation (1) based on an integrable semi-discrete CH equation proposed by the authors Ohta (). The scheme consists of an algebraic equation for the solution and the non-uniform mesh for a fixed time, and a time evolution equation for the mesh. Since the mesh is automatically driven and adapted by the solution, we name it a self-adaptive moving mesh method hereafter.

As a matter of fact, Harten and Hyman has proposed a self-adjusting grid method for one-dimensional hyperbolic problems HH83 (). Since then, there has been significant progress in developing adaptive mesh methods for PDEs Miller (); Dorfi (); Brackbill (); Huang (); Stockie (); TaoTang (). These methods have been successfully applied to a variety of physical and engineering problems with singular or nearly singular solutions developed in fairly localized regions, such as shock waves, boundary layers, detonation waves, etc. Recently, an adaptive unwinding method was proposed for the CH equation Artebrant (). The method is high resolution and stable. However, in order to achieve a good accuracy, a large number of grid points () has to be used. In addition, the designed method is only suitable for the single peakon propagation and peakon-peakon interactions, not for the peakon-antipeakon interaction. As shown subsequently, the self-adaptive moving mesh method gives accurate results by using a small number of grid points () for some challenging test problems.

The remainder of this paper is organized as follows. In Section 2, we present the self-adaptive moving mesh method and show it is consistent with the CH equation as the mesh size approaches to zero. Two time advancing methods in implementing the self-adaptive moving mesh method are presented in Section 3. In Section 4, several numerical experiments, including the propagations of “peakon” and “cuspon” solutions, cuspon-cuspon and soliton-cuspon collisions, are shown. The concluding remarks are addressed in Section 5.

## 2 A self-adaptive moving mesh method for the Camassa-Holm equation

It is shown in Ohta () that the CH equation can be derived from the bilinear equations of a deformation of the modified KP hierarchy

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−(12DtDx−1)f⋅f=gh,2cff=(Dx+2c)g⋅h,−2ff=(DtDx+2cDt−2)g⋅h, (2)

through the hodograph transformation

 {X=2cx+loggh,T=t, (3)

and the dependent variable transformation

 w=(loggh)t.

Here , and are Hirota’s D-operator defined as

 Dnxf⋅g=(∂∂x−∂∂y)nf(x)g(y)∣∣∣y=x.

It is proved in Ohta () that the bilinear equations (2) admit a determinant solution , , , where is a Casorati-type determinant of any size. By discretizing the -direction with an uniform mesh size , the following bilinear equations

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−2(1aDt−1)fk+1⋅fk=gk+1hk+gkhk+1,2acfk+1fk=(1+ac)gk+1hk−(1−ac)gkhk+1,−2afk+1fk=((1+ac)Dt−a)gk+1⋅hk−((1−ac)Dt+a)gk⋅hk+1, (4)

admits Casorati-type determinant solution with discrete index which is presented afterwards. Starting from Eq.(4), a semi-discrete CH equation

 (5)

was proposed (see the details in Ohta ()). Here the solution is approximated by at the grid points (). The mesh is a discrete analogue of the hodograph transformation from the -domain with uniform mesh size to -domain. As is seen, it is non-uniform and time-dependent.

The semi-discrete CH equation (5) can be rewritten as

 (6)

by introducing a forward difference operator and an average operator and

 ΔFk=Fk+1−Fkδk,MFk=Fk+Fk+12.

In the present paper, Eq. (5) or Eq. (6) is used as a numerical scheme for the CH equation (1). It is shown to be integrable in Ohta () in the sense that it possesses -soliton solution which, in the continuous limit, approaches -soliton solution of the CH equation. The -soliton solution is of the form

 wk=(loggkhk)t, (7)

with

 fk=τ0(k),gk=τ1(k),hk=τ−1(k),
 τn(k)=∣∣ ∣ ∣ ∣ ∣ ∣∣ψ(n)1ψ(n+1)1⋯ψ(n+N−1)1ψ(n)2ψ(n+1)2⋯ψ(n+N−1)2⋮⋮⋮ψ(n)Nψ(n+1)N⋯ψ(n+N−1)N∣∣ ∣ ∣ ∣ ∣ ∣∣

where

 ψ(n)i=ai,1(pi−c)n(1−api)−keξi+ai,2(−pi−c)n(1+api)−keηi,
 ξi=1pi−ct+ξi0,ηi=−1pi+ct+ηi0.

Next, let us show that in the continuous limit, (), the proposed scheme is consistent with the CH equation. To this end, the equation (6) is rewritten as

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩−2δk+δk−1(Δwk−Δwk−1)+δkMwkδk+δk−1+δk−1Mwk−1δk+δk−1+1c(1−a2c2)=4a2c1−a2c21δkδk−1,∂tδk=(1−δ2k4)(wk+1−wk).

By taking logarithmic derivative of the first equation, we get

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂t{2δk+δk−1(Δwk−Δwk−1)−δkMwkδk+δk−1−δk−1Mwk−1δk+δk−1}2δk+δk−1(Δwk−Δwk−1)−δkMwkδk+δk−1−δk−1Mwk−1δk+δk−1−1c(1−a2c2)=−∂tδkδk−∂tδk−1δk−1,∂tδk=(1−δ2k4)(wk+1−wk).

Thus, we have

The dependent variable is a function of and , and we regard them as a function of and , where is the space coordinate of the -th lattice point and is the time, defined by

 X=X0+k−1∑j=0δj,T=t.

Then in the continuous limit, (), we have

 Δwk→wX,Δwk−1→wX,Mwk→w,Δwk−1→wX,

and

 2δk+δk−1(Δwk−Δwk−1)→wXX.

Further, from

 ∂X∂t=∂X0∂t+k−1∑j=0∂δj∂t=∂X0∂t+k−1∑j=0(1−δ2j4)(wj+1−wj)→w,

we have

 ∂t=∂T+∂X∂t∂X→∂T+w∂X,

where the origin of space coordinate is taken so that cancels . Then the above semi-discrete CH equation converges to the CH equation

 (∂T+w∂X)(wXX−w)wXX−w−1c=−2wX,

i.e.

 (∂T+w∂X)(wXX−w)=−2wX(wXX−w−1c). (8)

Setting , we obtain the the CH equation (1).

Note that, in our previous paper Ohta (), we put which gives an alternative form of the CH equation

 wT+2κ2wX−wTXX+3wwX=2wXwXX+wwXXX. (9)

It is shown that they are equivalent under the scaling transformation , . In the present paper, for the convenience in comparing of our results with other papers Johnson (); Li (); Parker (); Parker2 (); DaiLi (); Matsuno (); MatsunoPeakon (), we set .

## 3 Implementation of the self-adaptive moving mesh method

In this Section, we will discuss how to implement the self-adaptive moving mesh method in actual computations. Generally, given an arbitrary initial condition , the initial non-uniform mesh can be obtained by solving the nonlinear algebraic equations by Newton’s iteration method. However, for the propagation or interaction of solitons or cuspons, which are challenging problems numerically, the initial condition can be calculated by (7) from and by putting , which are obtainable from the corresponding determinant solutions. The initial non-uniform mesh can also be calculated by Ohta ()

 δ0k=2(1+ac)gk+1hk−(1−ac)gkhk+1(1+ac)gk+1hk+(1−ac)gkhk+1. (10)

On the other hand, once the non-uniform mesh is known, the solution can be easily obtained by solving a tridiagonal linear system based on the first equation of the scheme.

 alwn+1l−1+blwn+1l+clwn+1l+1=dl, (11)

where

 al=0.5δn+1k−1−2δn+1k−1;  bl=0.5(δn+1k−1+δn+1k)+2δn+1k−1+2δn+1k;  cl=0.5δn+1k−2δn+1k;

and

 dl=4a2c1−a2c2(1δn+1k+1δn+1k−1)−δn+1k−1+δn+1kc(1−a2c2).

In regard to the evolution of , we propose two time advancing methods. The first is the modified forward Euler method, where we assume remains unchanged in one time step. Integrating once, we have

 δn+1k=2cnke(wnk+1−wnk)Δt−1cnke(wnk+1−wnk)Δt+1, (12)

where . The second is the classical 4th-order Runge-Kutta method, where can be viewed as a function of by solving the above tridiagonal linear system. Therefore, in one time step, we have to solve tridiagonal linear system four times.

In summary, the numerical computation in one time-step only involves a ODE solver for non-uniform mesh and a tridiagonal linear system solver. Hence, the computation cost is much less than other existing numerical methods. A Matlab code is made to perform all the computations. Iterative methods, for instance, the bi-conjugate gradient method bicg in Matlab are used to solve the tridiagonal system.

For the sake of numerical experiments in the subsequent section, we list exact one- and two- soliton/cuspon and peakon solutions.

(1). One soliton/cuspon solution: The -functions for the one soliton/cuspon solution are

 g∝1±(c−p1c+p1)eξ1,h∝1±(c+p1c−p1)eξ1, (13)

with , . This leads to a solution

 w(x,t)=2p21cv1(c2+p21)±(c2−p21)coshξ1, (14)
 X=2cx+log(gh),T=t, (15)

where the positive case in Eq.(14) stands for the one smooth soliton solution when , while the negative case in Eq.(14) stands for the one-cuspon solution when . Otherwise, the solution is singular. Thus Eq.(14) for nonsingular cases can be expressed by

 w(x,t)=2p21cv1(c2+p21)+|c2−p21|coshξ1. (16)

Similarly, for the semi-discrete case, we have

 gk∝1+∣∣∣c−p1c+p1∣∣∣(1+ap11−ap1)keξ1,hk∝1+∣∣∣c+p1c−p1∣∣∣(1+ap11−ap1)keξ1, (17)

with , resulting in a solution of the form

 wk(t)=2p21cv1(c2+p21)+|c2−p21|2[(1+ap11−ap1)−ke−ξ1+(1+ap11−ap1)keξ1], (18)

in conjunction with a transform between an uniform mesh and a non-uniform mesh

 δk=2(1+ac)gk+1hk−(1−ac)gkhk+1(1+ac)gk+1hk+(1−ac)gkhk+1.

(2). Two soliton/cuspon solutions: The -functions for the two soliton/cuspon solution are

with , , , . The parametric solution can be calculated through

 w(x,t)=(loggh)t,X=2cx+log(gh),T=t, (19)

whose form is complicated and is omitted here. Note that the above expression includes the two-soliton solution (, ), the two-cuspon solution (, ), or the soliton-cuspon solution (, ).

Similarly, for the semi-discrete case, we have

 gk∝1+∣∣∣c−p1c+p1∣∣∣(1+ap11−ap1)keξ1+∣∣∣c−p2c+p2∣∣∣(1+ap21−ap2)keξ2 +∣∣∣(c−p1)(c−p2)(c+p1)(c+p2)∣∣∣(p1−p2p1+p2)2(1+ap11−ap1)k(1+ap21−ap2)keξ1+ξ2,
 hk∝1+∣∣∣c+p1c−p1∣∣∣(1+ap11−ap1)keξ1+∣∣∣c+p2c−p2∣∣∣(1+ap21−ap2)keξ2 +∣∣∣(c+p1)(c+p2)(c−p1)(c−p2)∣∣∣(p1−p2p1+p2)2(1+ap11−ap1)k(1+ap21−ap2)keξ1+ξ2,

with , . The solution can be calculated through

 w(x,t)=(loggkhk)t, (20)

with a transform

 δk=2(1+ac)gk+1hk−(1−ac)gkhk+1(1+ac)gk+1hk+(1−ac)gkhk+1. (21)

Again, the explicit form of the solution is complicated and is omitted here.

(3). Peakon solutions: In the continuous CH equation, it is possible to construct peakon solutions from soliton solutions by taking the peakon limit CHH (); LiOlver (); Schiff (); Johnson (); Parker (); PKMatsuno (); MatsunoPeakon ().

For the continuous case, we can express the 1-soliton solution as

 w=2p21κv11+p21κ2+(1−p21κ2)coshξ1,

where , , . Taking the peakon limit , , , the solution , where , gives the 1-peakon solution MatsunoPeakon (). In Fig.1, one can see that the 1-soliton solution approaches to the 1-peakon solution as approaches to 0. Figure 1: 1-soliton solution for the CH equation: the left: p1=0.5,c=1; the right (close to the peakon limit): p1=99,c=100.

We can also consider the peakon limit for the semi-discrete CH equation. For the semi-discrete case, we can express the 1-soliton solution as

 wk=2p21κv11+p21κ2+1−p21κ22[(1+ap11−ap1)−ke−ξ1+(1+ap11−ap1)keξ1],

where , , . The peakon limit for the semi-discrete CH equation is again , , Taking the peakon limit, the solution , where , approaches to a solution which approaches to the peakon solution of the CH equation as taking the continuous limit. In Fig.2, one can see that the 1-soliton solution approaches to the 1-peakon like solution as approaches to 0. Taking the continuous limit, this solution approaches to the 1-peakon solution of the CH equation. Figure 2: 1-soliton solution for the semi-discrete CH equation: the left: p1=0.5,c=1,a=0.1; the right (close to the peakon limit): p1=99,c=100,a=0.005.

## 4 Numerical experiments

In this section, we apply our scheme to several test problems. They include: 1) propagation and interaction of nearly-peakon solutions; 2) propagation and interaction of cuspon solutions; 3) interactions of soliton-cuspon solutions; 4) non-exact initial value problems.

### 4.1 Propagation and interaction of nearly-peakon solutions

Example 1: One peakon propagation. It has been shown in PKMatsuno (); MatsunoPeakon () that the analytic -soliton solution of the CH equation converges to the nonanalytic -peakon solution when (). To show this, we choose one soliton solution with parameters , . Thus the speed of the soliton () is . Its profile is plotted and is compared with one peakon solution in Fig. 3. These two solutions are indistinguishable from the graph. The error in , where , is calculated to be , and the discrepancy for the first conserved quantity is less than . Therefore, this soliton solution can be viewed as an approximate peakon solution with amplitude .

The propagation of the above designed approximate peakon solution is solved by the self-adaptive mesh scheme with two different time advancing methods: the modified forward Euler method (MFE) and the classical Runge-Kutta method (RK4). The length of the interval in the -domain is chosen to be and the number of grid is . For the above parameters of one-peakon solution, the length of the computation domain turns out to be about . Figures 4 (a)-(d) display the numerical solutions at , together with the self-adjusted mesh. It can be seen that the non-uniform mesh is dense around the crest. The most dense part of the non-uniform mesh moves along with the peakon point with the same speed. With the same grid points , the relative errors in -norm and the first conserved quantity are computed and compared in Table LABEL:t:peakon. Here, , where and represent the numerical and analytical solutions at the grid points , respectively. indicates the relative error in , where stands for the counterpart of by the numerical solution. Trapezoidal rule on the non-uniform mesh is employed for the evaluation of the integrals.

Example 2: Two peakon interaction. For , we initially choose two approximate peakon solutions moving with velocity , and , respectively. Their interaction is numerically solved by MFE and RK4, respectively, with a fixed grid number of . Figure 5 displays the process of collision at different times. Table LABEL:t:peakon2 presents the errors in -norm and . It could be seen that, in spite of a small number of grid points and a large time step, RK4 simulates the collision of two approximate peakons with good accuracy. Figure 5: Numerical solution for collision of two nearly-peakon with p1=198.9975, p2=199.4995 and c=200.0: (a) t=0.0; (b) t=10.0; (c) t=15.0; (d) t=20.0; (e) t=30.0.

In regard to the propagation and interaction of approximate peakon solutions, we summarize as follows:

1. Due to the integrability of the scheme and the self-adaptive feature of the non-uniform mesh, the -norm is small and the first conserved quantity is preserved extremely well even for a small number of grid points.

2. The errors is mainly due to the time advancing methods. MFE is first order in time, so it produces relatively large and , roughly changing in proportional with time. RK4 is fourth-order in time, so up to , and are of the orders and for a grid number of and a time step .

### 4.2 Propagation and interaction of cuspon solutions

The classical 4th-order Runge-Kutta method fails whenever the cuspon solution is involved. It seems that a kind of instability occurs in this case, whose theoretical reason is still unclear. Therefore, only MFE is employed to conduct the numerical experiments hereafter.

Example 3: One-cuspon propagation. The parameters taken for the one-cuspon solution are , . The number of grid is taken as in an interval of width of in the -domain. Through the hodograph transformation, this corresponds to an interval of width in the -domain. Figure 6(a) shows the initial profile and the initial mesh. Figures 6(b)-(d) display the numerical solutions (solid line) and exact solutions (dotted line) at , together with the self-adjusted mesh. It can be seen that the non-uniform mesh is dense around the cuspon point, and moves to the left in accordance with the movement of the cuspon point. Table 3 exhibits the results of relative errors in -norm and . Figure 6: Numerical solution of one single cuspon solution: (a) t=0.0; (b) t=2.0; (c) t=3.0; (c) t=4.0.

Example 4: Two-cuspon interaction. The parameters taken for the two-cuspon solutions are , , . Figures 7(a)-(d) display the process of collision at several different times, along with the exact solution. Meanwhile, the self-adaptive mesh is also shown in the graph. It can be seen that two cuspon solutions undertake elastic collision, regaining their shapes after the collision is complete. As mentioned in DaiLi (), the two cuspon points are always present during the collision. The grid points are automatically adapted with the movement of the cuspons, and are always concentrated at the cuspon points. In compared with the exact solutions, we can comment that the numerical solutions are in a good agreement with exact solutions. As far as we know, what is shown here is the first numerical demonstration for the cuspon-cuspon interaction. Figure 7: Numerical solution for the collision of two-cuspon solution with p1=11.0, p2=10.5, c=10.0: (a) t=13.0; (b) t=14.8; (c) t=16.6; (d) t=25.0.

### 4.3 Soliton-cuspon interactions

Here we show two examples for the soliton-cuspon interaction with . In Fig.8, we plot the interaction process between a soliton of and a cuspon of at several different times where the soliton and the cuspon have almost the same amplitude. It can be seen that when the collision starts (), another singularity point with infinite derivative () occurs. As collision goes on (), the soliton seems ’eats up’ the cuspon, and the profile looks like a complete elevation. However, the cuspon point exists at all times, especially, at , the profile becomes one symmetrical hump with a cuspon point in the middle of the hump. Figure 8: Numerical solution for cuspon-soliton collision with p1=9.12, p2=10.98 and c=10.0: (a) t=0.0; (b) t=12.0; (c) t=14.4; (d) t=14.6; (e) t=14.8; (f) t=17.0; (g) t=25.0.

In Fig.9, we present another example of a collision between a soliton () and a cuspon () where the cuspon has a larger amplitude () than the soliton (). Again, when the collision starts, another singularity point appears. As collision goes on, the soliton is gradually absorbed by the cuspon. At , the whole profile looks like a single cuspon when the soliton is completely absorbed. Later on, the soliton reappears from the right until , the soliton and cuspon recover their original shapes except for a phase shift when the collision is complete. Figure 9: Numerical solution for cuspon-soliton collision with p1=9.12, p2=10.5 and c=10.0: (a) t=0.0; (b) t=9.0; (c) t=10.0; (d) t=10.3; (e) t=10.6; (f) t=11.5; (g) t=16.0. Figure 10: Numerical solution starting from an initial condition (19) with c=10: (a) t=0.0; (b) t=10.0; (c) t=20.0; (d) t=30.0.

### 4.4 Non-exact initial value problems

Here, we show that the integrable scheme can also be applied for the initial value problem starting with non exact solutions. To the end, we choose an initial condition whose mesh size is determined by

 δk=2ch(1−0.8sech(2kh−Wx/2)), (22)

then, the initial profile can be calculated through the second equation of the semi-discretization, which is plotted in Fig.10 (a). Figures 10 (b), (c) and (d) show the evolutions at , respectively. Note that in this computation. It can be seen that a soliton with large amplitude is firstly developed, and moving fast to the right. By , a second soliton with small amplitude is to be developed.

Next, we increase the value of to 90, which implies a very small dispersion term, corresponding to the dispersionless CH equation. The initial profile and the evolutions at are shown in Fig.11. It is seen that four nearly-peakons are developed from the initial profile at . Later on, an array of nearly-peakons of seven and eight are developed at , respectively. This result is similar to the result for the KdV type equations with a small dispersion, i.e. the peakon trains are generated. (For the KdV type equations, soliton trains are generated. For example, see Kamchatnov (); El () for numerical simulations and Karpman () for a theoretical analysis for the KdV equation.) A theoretical analysis for the dispersionless CH equation to explain the above intriguing numerical result is called for. Figure 11: Numerical solution starting from an initial condition (19) with c=90: (a) t=0.0; (b) t=50.0; (c) t=150.0; (d) t=200.0.

## 5 Concluding Remarks

In the present paper, we have proposed a self-adaptive moving mesh method for the CH equation, which based on an integrable semi-discretization of the CH equation. It has the properties: (1) it is integrable in the sense that the scheme itself admits the -soliton solution approaching to the -soliton solution of the CH equation in the limit of mesh size going to zero; (2) the mesh is non-uniform and is automatically adjusted so that it is concentrated in the region where the solution changed sharply, for example, the cuspon point; (3) once the non-uniform mesh is evolved, the solution is determined from the evolved mesh by solving a tridiagonal linear system. Therefore, either from the accuracy or from the computation cost, the proposed method is expected to be superior than other existing numerical methods of the CH equation. This is indeed true. The numerical results in this paper indicate that a very good accuracy is obtained.

Two time advancing methods, the modified forward Euler method and the classical 4th-order Runge-Kutta method, are used to solve the evolution of non-uniform mesh. The Runge-Kutta method gains much better accuracy than the modified forward Euler method. However, it fails for the computations of cuspons. Using the self-adaptive moving mesh method for the CH equation, we have obtained interesting numerical computation results starting with non-exact solutions. When is very small, the peakon train is generated from the non-exact initial condition.

As further topics, it is interesting to construct integrable discretizations, or, the self-adaptive moving mesh methods for a class of integrable nonlinear wave equations possessing soliton solutions with singularities such as peakon, cuspon or loop solutions. For example, such equations include the short pulse equation which was derived as a model for the propagation of ultra-short optical pulses in nonlinear media SP (),

 uXT=u+16(u3)XX, (23)

and the Degasperis-Procesi (DP) equation DP ()

 uT+3κ3uX−uTXX+4uuX=3uXuXX+uuXXX. (24)

It is worth pointing out that the authors have constructed semi- and full-discretization for the short pulse equation, which is another example of the self-adaptive moving mesh method FMO09 (), in which we have succeeded in computing the one- and two-loop soliton propagations and interactions.

## Acknowledgments

The work of B.F. was partially supported by the U.S. Army Research Office under Contract No. W911NF-05-1-0029. The work of Y.O. was partly supported by JSPS Grant-in-Aid for Scientific Research (B-19340031, S-19104002). Y.O. and K.M. are grateful for the hospitality of the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge where this article was completed during the programme Discrete Integrable Systems (DIS). The authors are grateful to the anonymous referee for valuable comments.

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