Numerical Methods for Generalized KdV equations

Mauricio Sepulveda,

Departamento de Ingenierıa Matematica, Universidad de Concepcion. Chile.

E-mail: mauricio@ing-mat.udec.cl.

Octavio Vera Villagran

Departamento de Matematica. Universidad del Bıo-Bıo. Chile.

E-mail: overa@ubiobio.cl.

Resumo: In this paper we study a class of fullydiscrete scheme for the generalized Korteweg-de Vries equation in a bounded domain (0, L).Taking a particular discretization for the non-linear term when p = 4 (that is the criticalcase of the generalized KdV equation) we obtainthank to an estimate in L2(0, T ;H1(0, L)) andthe convergence in L4-strong for the solution ofthe numerical scheme. We provide several nu-merical results to illustrate the performance ofthe method.

1 Introduction

In this work we study a class of fully discretescheme for the critical generalized Korteweg-de Vries equation. Taking a particular discre-tization for the nonlinear term when p = 4(that is the critical case of the generalized KdVequation) we obtain thank to an estimate inL2(0, T ;H1(0, L)) for the solution of the nu-merical scheme, the convergence.

We consider the following initial boundaryvalue problem

ut + uxxx + up ux + ux = 0, (1)

u(0, t) = u(L, t) = 0, (2)

ux(L, t) = 0, (3)

u(x, 0) = u0(x), (4)

with x ∈ (0, L) and t > 0. We are specifi-cally interested in the critical case p = 4. Theequation (1) corresponds to a modified formof the Generalized Korteweg-de Vries equation(GKdV equation henceforth)

ut + uxxx + up ux = 0, (5)

where we added a linear convective term ux

due to the boundary conditions. The genera-lized Korteweg-de Vries equation (5) has been

extensively studied for understanding the in-teraction between the dispersive term and thenonlinearity in the context of the theory of non-linear dispersive evolution equations. Severalauthors proving stability properties of the so-lutions when p < 4 (see Bona et al. [5] and thereference therein). On the other hand, numeri-cal simulation indicate that for p ≥ 4, smoothsolution of the initial-value problem may formsingularities in finite time (see [3, 4]). Whenp = 4, the equation (1) considered in real lineis called critical for various reason. On of thesereason is that the mass remains invariant byscaling in the L2-norm. Local results for datain L2 were established by Kenig, Ponce andVega [12]. On the other hand, in [14] Merleshowed that H1 solutions may blow-up in fi-nite time. Thus the nonlinearity is critical forthe long time behavior of solutions.

The novelty of the present work is to in-troduce a numerical scheme preserving the as-symptotical behaviour property of exponentialdecay with a rate independent of the mesh size,we discretize the nonlinear term of the equationin a particular way, taking into account the in-variance of the mass scaling in the L2-norm.

We etablish a series of estimates for thecontinuous problem (1)-(4) that will be usedfor the numerical scheme in the next section.We start by stating the following existenceresult due to Faminskii [10]

Theorem 1 (See [10], Theorem 1]).Let u0 ∈ L2(0, L) and T > 0 be given. Then,there exists a T ∗ ∈ (0, T ] such that theproblem (1)-(4) admits a unique solution u ∈C([0, T ∗] : L2(0, L)) ∩ L2([0, T ∗) : H1

0 (0, L)).

— 894 —

On the other hand, the following estimateresults is proven in [13]

Proposition 1. Let u be the solution ofproblem (1)-(4) obtained in Theorem 1.

If ||u0||L2(0, L) <√

32 , then

||u||2L2(0, T : H10(0, L)) ≤ c

||u0||2L2(0, L)[1 − 4

9 ||u0||4L2(0, L)

] (6)

where c = c(T, L). Furthermore,

ut ∈ L6/5(0, T : H−2(0, L)). (7)

We start first with a description of thescheme, the wellposedness, some estimate re-sults and the convergence of the scheme.

2 Description of the numerical

scheme

We consider finite differences based on theunconditionally stable schemes described in[9, 18]. Let the discrete space :

XJ = {u = (u0, u1, . . . , uJ ) ∈ IRJ+1 |with u0 = 0 and uJ = uJ−1 = 0}

and

(D+u)j =uj+1 − uj

δx , (D−u)j =uj − uj−1

δx ,

for j = 1, . . . , J − 1, and D = 12(D+ + D−)

the classical difference operators, where δx isthe space-step, and δt is the time-step, for j =0, . . . , J , and n = 0, . . . ,N .

2.1 The semidiscrete numerical

scheme

We describe first a semi-discrete numericalscheme. We note by uj(t) the approximate va-lue of u(jδx, t), solutions of the nonlinear pro-blem (1)-(4). The approximation of the nonli-near problem (1)-(4) which reads as system ofEDOs :

d

dt[uj ] +

(A(θ)u

)j+ F (u)j = 0, (8)

u0(t) = uJ(t) = uJ−1(t) = 0, (9)

u0 =

∫ xj+ 1

2

xj− 1

2

u0(x)dx, (10)

with j = 1, . . . , J − 1, where xj+ 1

2

= j + 12δx

and xj = jδx. The matrix A(θ) ∈ IR(J−1)×(J−1)

is an approximation of order θ of the dispersiveterm uxxx and the linear convective term ux.For instance, if we want an approximation of1st order in space, we can choose

A(1) = D+D+D− + D. (11)

This forward differences approximation in thedefinition of A(1) is do it in order to have a po-sitive defined matrix I+δtA(1) (see for instance[2, 8, 9, 18]). On the other hand, if we want anapproximation of second order we can take acentral differences approximation as follows

A(2) = D+DD− + D. (12)

We obtain also a positive defined matrix for thematrix I + δtA(2), with A(2) defined in (12).

The nonlinearity u4 ux of the equation (1),is approached by a nonlinear function F (un),where F : IRJ−1 → IRJ−1 is given by the fol-lowing expression

F (u)j = u4j (Du)j − 5

2u3

j

(Du2

)j

+10

3u2

j

(Du3

)j− 5

2uj

(Du4

)j

+(Du5

)j

(13)

for all j = 1, . . . , J − 1.

Remark.

1. Using classical finite difference, F (u)j isan approximation of

u4 ux − 5

2u3

(u2

)x

+10

3u2

(u3

)x

− 5

2u

(u4

)x

+(u5

)x

= u4 ux

which are two ways to write the sameterm. We will see that using the appro-ximation F (u)j given by (13) we obtainH1

0−estimates for the solution of this nu-merical scheme similar to the continuouscase.

2. The approximation (13) use central diffe-rences which it is a second-order appro-ximation of u4 ux. Therefore, it motiva-tes to use a full second-order scheme cho-osing the central differences (12) as ap-proximation of the dispersive and the li-near convective terms. On the other hand,

— 895 —

the second-order scheme has less numeri-cal diffusion that the first-order scheme,with what it is not possible to control somenonlinear terms in the approximation of(13) and get to the limit as we do it forthe first-order scheme. In compensation,we can add a very small diffusive termin order to obtain the convergence of themethod.

2.2 Full discrete implicit numerical

schemes.

In order to discretize in temporal variable thesystem (8)-(10) of ordinary differential equati-ons we choose implicit schemes. We note by un

j

the approximate value of uj(nδt), solutions ofthe nonlinear system (8)-(10), for n = 0, . . . ,N .

2.2.1 Implicit Euler.

The simplest implicit numerical scheme is givenby the implicit Euler of order one, and reads asfollowing

un+1j − un

j

δt

+(A(1)un+1

)j+ F (un+1)j = 0, (14)

un0 = un

J = unJ−1 = 0, (15)

u0 =

∫ xj+1

2

xj− 1

2

u0(x)dx, (16)

with j = 1, . . . , J − 1, where A(1) is defined in(11), and F (·) is defined in (13).

2.2.2 Implicit Runge-Kutta of order 2.

From a practical point of view we need amore performant discretization for our nume-rical examples. Therefore, we consider here atemporal discretization by the 2-stage Gauss-Legendre implicit Runge-Kutta method, whichcorrespond to the table (see [7])

a11 a12 τ1

a21 a22 τ2

b1 b2

||14

14 − 1

2√

312 − 1

2√

314 + 1

2√

314

12 + 1

2√

3

12

12

(17)

The numerical scheme is now specified step bystep for n = 0, . . . ,N . We seek un

j , by way of

the intermediate stages un,ℓj , for ℓ = 1, 2 which

are solution of the 2(J − 1) × 2(J − 1) systemof nonlinear equations

un,ℓj − un

j

δt

+2∑

m=1

aℓ,m

[(A(2)un,m

)j+ F (un,m)j

]= 0,

j = 1, . . . , J − 1,ℓ = 1, 2,

(18)

un,ℓ0 = un,ℓ

J = un,ℓJ−1 = 0, ℓ = 1, 2. (19)

using the formula

un+1j = un

j (20)

−δt2∑

ℓ=1

bℓ

[(A(2)un,ℓ

)j+ F (un,ℓ)j

],

where A(2) is defined in (11), and F (·)is defi-ned in (13). The application of this full implicitRunge-Kutta Method for the temporal discreti-zation of KdV type equations is not new. It wasapplied in a first time by Bona et al. in [3, 4]using Finite Element Method for the space dis-cretization. In our case, the nonlinear system(18) can be approximately solved using New-ton method or a fixed point method. In thelast section we show some numerical examplesand we describe a performant strategy to solvethis nonlinear system. The choice of full impli-cit schemes (implicit Euler or implicit Runge-Kutta scheme) is important in order to obtainunconditional stability and convergence of themethod. In order to obtain similar estimatesthan [13] for our discrete scheme we will applyrepeatedly the following algebraic identities:

(a2 − b2

)+ (a − b)2 = 2(a − b)a (21)

(a6 − b6

)+ (a − b)6 = 6(a5 − b5)a

− 15(a4 − b4)a2

+ 20(a3 − b3)a3

− 15(a2 − b2)a4

+ 6(a − b)a5 (22)

Notation. We note the following internal pro-duct in XJ

(z , w) =J−1∑

j=1

δx zj wj ,

— 896 —

(z , w)x = (z , xw) =J−1∑

j=1

j δx2 zj wj

for all z,w ∈ IRJ+1, and the norms: |z|2 =√(z , z), |z|x =

√(z , z)x, for all z ∈ IRJ+1.

Additionally, we note the p−norms in XJ defi-ned by:

|z|p .=

J−1∑

j=1

δx |zj |p

1/p

,

|z|∞ .= max

j=1,...,J−1|zj |,

for all z,w ∈ IRJ+1.

First we give some standard identities in-troduced in [9] and used in [2, 8, 18].

Lemma 1. For all z,w ∈ XJ , using theNotation 2.2.2, we have the following identi-ties

(D+z , w) = −(z,D−w) (23)

(D+z , z) =1

2

(−[z1]

2 − δx|D+z|2)

(24)

(D+z , w)x = −(z,D−w)x

+δx(z,D−w) − (z,w), (25)

(D+z , z)x =1

2

(−δx|D+z|2x − |z|2

)(26)

(D+D+D−z , z) =

1

2

[(D−z)1

]2+

δx

2|D+D−z|2, (27)

(D+D+D−z , z)x =

−δx

2

[(D−z)1

]2+

3

2|D−z|2

+δx

2|D+D−z|2x − δx2

2|D+D−z|2 (28)

Proof. Using the identity (10) with a = zj, wj

and b = zj−1, wj−1, multiplying by 1 and jδx,and summing by parts over j = 1, . . . , J − 1,we obtain (23)-(28).

On the other hand, the estimate for thenonlinear terns is given by the choice of (13)and the following lemma,

Lemma 2. For all u ∈ XJ , we have

(u , F (u)) = 0, (29)

(u , F (u))x = −1

6|u|66 (30)

where F (u) is defined by (13).Proof. Using the identity (21) witha = ui, b = ui−1, and and summing overj = 1, . . . , J − 1, we obtain (29) with algebraicarguments. Repeating the same argumentmultiplying by jδx before to summing overj = 1, . . . , J − 1, we deduce (30).

We suppose that we can solve (8)-(10).Then, the estimate results of Lemma 1 andLemma 2 give the following result of stabilityfor the solution of the numerical scheme:

Proposition 2. Let (un)n∈IN a sequencein XJ built by the numerical scheme (8)-(10),with A and F (un) defined by (11)-(13), and

let be T = nδt. If |u0|2 ≤√

32 , then, there exist

a constant C > 0 independent of δx and δt,such that

|un|2 ≤ |u0|2 (31)n∑

k=0

δt∣∣∣D−uk

∣∣∣2

≤ C|u0|22

1 − 49 |u0|42

(32)

|D+D−unδ |2 ≤ δx−1/2

√2

|u0|2 (33)

Proof. First, we multiply the equation (8) byun+1 and we sum over j = 1, . . . , J − 1, using(23), (24), (27) and (29). Thus multiplying byδt and summing over k = 0, . . . , n using theidentity (21) with a = un and b = un+1, weobtain (31) and (33).

On the other hand, we multiply the equa-tion (8) by j δx un+1 and we sum overj = 1, . . . , J − 1, using (25), (26), (28) and(30). Thus multiplying by δt and summingover k = 0, . . . , n using the identity (10) witha = un and b = un+1, we obtain

n∑

k=0

δt∣∣∣D−uk

∣∣∣2

2≤ (34)

(T + L + δx)

3|u0|2x +

1

9

n∑

k=0

δt |uk|66.

The rest of the proof is a discrete version of theproof of Proposition 1 (see [13]): it is not diffi-cult to prove the following discrete Gagliardo-Nirenberg type inequalities (see [15]):

|u|2∞ ≤ 2 |u|2 |Du|2 (35)

|u|44 ≤ 2 |u|32 |Du|2 (36)

— 897 —

the second term of (34) can be estimated bythe inequalities (35), (36), and (31) as follow

n∑

k=0

δt |uk|66 ≤ 4 |u0|4n∑

k=0

δt∣∣∣Duk

∣∣∣2

2. (37)

Replacing (37) in (34), and using the fact that|Du|22 ≤ |D−u|22 (by triangular inequality), wededuce (32).

Remark. The estimates of Proposition 2are in accord with the stability result ofProposition 1. and they allow to prove aconvergence of the numerical scheme.

3 Numerical Examples

3.1 Computing strategy

The operator A defined in (11) is well definedas a linear application XN → IRN+1, in thesense that we do not need additional point onthe outside of [0, L] to compute Au. Thus Ais represented by a penta-diagonal matrix of(N + 1) × (N + 1) :

Au =

γ1 ε1 ζ1

β2 γ2 ε2 ζ2 0α3 β3 γ3 ε3

. . .. . .

. . .. . .

. . .. . .

. . .. . .

. . . ζn−2

0 . . . βn−1 γn−1 εn−1

αn βn γn

where αi, βi, γi, εi and ζi, for i = 1, . . . , n arecoefficients easy to compute.In the case of the implicit Euler approximation,the nonlinear system (14) can be write as

(I + A)un+1 = un − δt F (un+1) (38)

where A = δt (diag(aδ) + A). The nonlinearsystem (38) can be approximately solved usingNewton method or a fixed point method.In both cases, we have in each iteration tosolve a linear system with a define positivepenta-diagonal matrix. Taking into accountthe structure of the matrix (I + A) it is easy toapply a LU decomposition based on a simplemodification of the Thomas algoritm for apenta-diagonal matrix.

In the case of the 2-stage Gauss-Legendreimplicit Runge-Kutta method, the nonlinearsystem (18) can be write as

(I + a11 A)un,1 + a12 Aun,2 =

un − δt a11 F (un,1) − δt a12 F (un,2)

a21 Aun,1 + (I + a22 A)un,2 =

un − δt a21 F (un,1) − δt a22 F (un,2)

where aij are defined in the Butcher table (17),for i, j = 1, 2. We can be rewrite this nonlinearsystem as an uncoupled system in its linear partas(12 I + 6A + A2

)un,1 =

(12 I + 2

√3A

)un − δt (3 I + A)F (un,1)

−δt (3 − 2√

3)F (un,2) (39)(12 I + 6A + A2

)un,2 =

(12 I − 2

√3A

)un − δt (3 + 2

√3)F (un,1)

−δt (3 I + A)F (un,2) (40)

Using now Newton method or a fixed pointmethod, we have in each iteration to solvea linear system with an 9-diagonal matrix(12 I + 6A + A2

). Taking into account the

structure of the 9-diagonal matrix it is easy toapply again, a simple modification of the Tho-mas algoritm.

3.2 Interaction between two solitons

An exact solution for the generalized KdVequation

ut + uxxx + up ux + ux = 0, (0, L) × (0, +∞),

with x ∈ IR can be write as a traveling-wavesolution (soliton) of the form

u(x, t) =α

cosh2/p [β p (x − (4β2 + 1) t − x0)]

where α and x0 are arbitrary constants and β =[αp

2 (p+1) (p+2)

]1/2(see for instance, Ablowitz and

Segur [1]).Obviously in a bounded domain there is not

soliton, but the solution can be approximate byone if it is not close to a boundary. We chooseL = 30) and x0 = 50 during T = 100 [sec]to avoid any numerical reflection due to theboundaries.

— 898 —

We consider an interaction of two solitonswith p = 4. For that, we take the initial condi-tion:

u(x, 0) =α1

cosh2/p [β1 p (x − (4β2 + 1) t − x1)]

+α2

cosh2/p [β2 p (x − (4β2 + 1) t − x2)]

with α1 = 1.0, α2 = 0.5, betai =[αp

i

2 (p+1) (p+2)

]1/2, i = 1, 2, and x1 = 50.0 and

x2 = 80. We make a simulation for L = 300.0;T = 100; J = 10000; n = 100000; δt = T/n;δx = L/J (see Figure ??).

We remark that the norm L2 of the initialcondition of a soliton is given by

‖u0‖L2(0,L) ≈ 151/4

√π

2≈ 2.4665092 >

√3/2

0 50 100 150 200 250 300

0

20

40

60

80

100

120

−1

0

1

2

Figura 1: Interaction between two solitons.

Acknowledgment

MS has been supported by Fondecyt project# 1070694 and Fondap in Applied Mathe-matics (Project # 15000001). OVV ackno-wledge support by Direccion de Investigacionde la Universidad del Bıo-Bıo. DIPRODE pro-jects, # 061008 1/R. Finally, OV and MS ack-nowledge support by CNPq CONICYT Pro-ject, # 490987/2005-2 (Brazil) and # 2005-075(Chile).

Referencias

[1] M. J. Ablowitz and H. Segur Solitons andthe Inverse Scattering Transform. SIAMStudies in Applied Mathematics 4. Phila-delphia, 1981.

[2] E. Bisognin, V. Bisognin, M. Sepulveda,O Vera, Coupled system of Korteweg deVries equations type in domains with mo-ving boundaries. Journal of Computatio-nal and Applied Mathematics. In Press.DOI: 10.1007/s10665-007-9162-6

[3] J.L. Bona, V.A. Dougalis, O.A. Kara-kashian, W.R. McKinney, Conservative,high-order numerical schemes for the ge-neralized Korteweg-de Vries equation. Phi-los. Trans. Roy. Soc. London Ser. A 351(1995), no. 1695, 107-164.

[4] J.L. Bona, V.A. Dougalis, D.E. Mitsota-kis, Numerical solution of KdV-KdV sys-tems of Boussinesq equations. I. The nu-merical scheme and generalized solitarywaves. Math. Comput. Simulation 74(2007), no. 2-3, 214-228.

[5] J.L. Bona, P.E. Souganidis, W. Strauss,Stability and instability of solitary wavesof Korteweg-de Vries type. Proc. Roy. Soc.London Ser. A 411 (1987), no. 1841, 395–412.

[6] J.C. Butcher, Implicit Runge-Kutta pro-cesses. Mathematics of Computation, 18(1964) 50-64.

[7] J.C. Ceballos, M. Sepulveda, O. Vera, TheKorteweg-de Vries-Kawahara equation ina bounded domain and some numerical re-sults. Applied Mathematics and Compu-tation. Vol 190, (2007) 912-936.

[8] T. Colin and M. Gisclon. An initial-boundary-value problem that approxi-mate the quarter-plane problem for theKorteweg-de Vries equation, NonlinearAnalysis, 46(2001)869-892.

[9] A. V. Faminskii, On an initial boundaryvalue problem in a bounded domain forthe generalized Korteweg-de Vries equa-tion, Funct. Diff. Eq., 8(2001)1-2, 183-194.

— 899 —

[10] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the ge-neralized Korteweg-de Vries equation viathe contraction principle, Comm. PureAppl. Math., 46 (1993), 527–620.

[11] F. Linares and A. F. Pazoto, On the ex-ponential decay of the critical generalizedKorteweg-de Vries equation with localizeddamping, Proc. of the Amer. Math. Soc.Vol. 135, 5(2007) 1515-1522

[12] F. Merle, Existence of blow-up solutions inthe energy space for the critical generalizedKdV equation, J. Amer. Math. Soc. 14(2001), 555–578.

[13] L. Nirenberg, On elliptic partial differen-tial equations, Ann. Scuola Norm. Sup.Pisa Sci. Fis. Mat. 13 (1959), 116–162.

[14] M. Sepulveda and O. Vera. Numericalmethods for a transport equation perturbedby dispersive terms of 3rd and 5th order.Scientia Series A: Mathematics (New Se-ries) Vol 13 (2006), 13-21.

— 900 —