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arXiv:qu

ant-ph/0210003v1

1Oct2002

Analytical and numerical solution of coupled KdV-MKdV system

A.A. Halim 1, S.P. Kshevetskii 2, S.B. Leble31,3 Technical University of Gdansk,

ul. G. Narutowicza 11/12, 80-952 Gdansk, Poland.2 Kaliningrad state University, Kaliningrad, Russia

3leble@mif.pg.gda.pl

February 1, 2008

Abstract

The matrix 2x2 spectral differential equation of the second order is considered on x in (, +). Weestablish elementary Darboux transformations covariance of the problem and analyze its combinations.We select a second covariant equation to form Lax pair of a coupled KdV-MKdV system. Thesequence of the elementary Darboux transformations of the zero-potential seed produce two-parametersolution for the coupled KdV-MKdV system with reductions. We show effects of parameters on theresulting solutions (reality, singularity). A numerical method for general coupled KdV-MKdV systemis introduced. The method is based on a difference scheme for Cauchy problems for arbitrary numberof equations with constants coefficients. We analyze stability and prove the convergence of the schemewhich is also tested by numerical simulation of the explicit solutions.

1 Introduction

There are two complementary approaches to integrable systems: analytical and numerical ones to be devel-oped. Even most profound analytical IST method cannot give explicit solution of general Cauchy problemwhile numeric can, but is rather compulsory in use: calculations could need powerful computers. May bemost transparent of analytical methods are based on algebraic structures associated with a problem. Tosuch structures belongs Darboux transformations covariance of Lax representation of nonlinear equationsthat yields a powerful tool for explicit solutions production. We investigate applications of special kind ofsuch discrete symmetry - to be called elementary ones [10]. Its elementarity simply means that a productof such transformations generate the standard one [10, 13]. Here we study combinations of such transformsthat do not coincide with binary ones [12] and hence are not so known.

The main ideas of numerical integration of such integrable systems go up to the famous properties ofthe equations as the Lax pair and infinite series of conservation laws existence [7]. From a point of viewof general theory of such systems some hopes are concerned with a development of the finite-difference orother approximations of the systems. Namely if one could prove a convergence and stability theorems forsuch difference systems (existence of solutions is implied), a way to existence and uniqueness of solutionsis opened [6].

The coupled KdV-MKdV system arises in many problems of mathematical physics. Some integrablesystems are associated with a polynomial spectral problem and have Virasoro symmetry algebras areconsidered [17]. A dispersive system describing a vector multiplet interacting with the KdV field is a

member of a bi-Hamiltonian integrable hierarchy [8]. Recently a multisymplectic numerical twelve pointsscheme was produced. This scheme is equivalent to the multisymplectic Preissmann scheme and is applied

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to solitary waves over long time interval [14]. The coupled KdV-MKdV system is also connected to otherphysical applications [11].

The general system we consider in this work have the following form

nt + m,k

gn,1m,k

mkx + gn,2m,k(

m)2kx + gn,3m,k

mx

kx + g

n,4m,k

mkxx + gn,5m,k

m(k)2x + dnnxxx = 0, (1)

where n, m, k =1,2,...N are the dependent variables numbers. Nonlinear coefficients are gn,lm,k, l =1,2,..5and dn are dispersion coefficients.

In particular, for the system under consideration (N = 3) the variables 1, 2, 3 are denoted by f, u, vto have the integrable system [10]

ft +1

2fxxx +

3

2(uf)x 3

4fxf

2 = 0

ut 14

uxxx 32

uxu + 3vvx +3

4uxf

2 32

(fxv)x = 0

vt + 12

vxxx + 32

vxu 34

(vf2)x + 34

uxxf + 32

uxfx = 0 (2)

The Lax pair is given in [10]. The system exhibits two integrable reductions having explicit solutions,Hirota-Satsuma [4, 1] and a two components KdV-MKdV system [10, 12]. Krishnan [5] showed that ageneralized KdV-MKdV system have solitary wave solutions and investigate the effects of increasing thenonlinearity of one variable on the existence of solitary waves. Some form of KdV-MKdV system haveexplicit solutions in terms of Jacobi elliptic functions [2].

In this work we present explicit solutions for a system of three equations (2) that have not been specifiedin [10, 12]. We study this two-parameter explicit solutions and show effects of choosing these parameterson the solutions. We demonstrate the use of two arbitrary elementary DTs [ 10] and its special choice that

holds a hereditary of the reduction to built explicit solutions to the KdV-MKdV system (2).Also we modify a numerical method [7, 3] for solution of system (1). It is a difference scheme forCauchy problems for arbitrary number of equations with constants coefficients. The scheme preserves twoconservation laws for the KdV type equations and the order of error of the difference formulas is improved[7, 15]. The convergence is proved and stability is analyzed giving the conditions taken in account inchoosing time and space step sizes [16, 9] .

The present work is organized as follows. Section 2 introduces the matrix spectral equation of thesecond order with 2 x 2 matrix coefficients and two elementary DTs. We select the second equation ofthe Lax pair and derive the compatibility conditions. The product of these two transformations yieldsthe standard DT [10, 13]. Section 3 illustrates how the, first, elementary DT is used to produce solutionto the KdV equation as well as the general evolution equation generated by the compatibility conditions

of the Lax pair. Explicit solutions are introduced for the case of zero initial potentials of the matrixproblem. In section 4 we consider a reduction constraints on the potential of the matrix spectral equation.This reduction gives an automorphism that relates two pairs of solution of the spectral equation for twospectral parameters. We use this results in the compound elementary DTs to produce an explicit solutionto a coupled KdV-MKdV system that results from the compatibility conditions of Lax pair under thisreduction. The effects of these parameters on the solution (reality, singularity) is analyzed. Section 5introduces a numerical method for solving coupled KdV-MKdV system (1). We produce a differencescheme for a Cauchy problem with initial condition rapidly decreasing at both infinities. The main stepsof the scheme convergence and stability analysis is shown while the details are explained in appendix Aand B. The scheme is tested by applying it to integrable coupled KdV-MKdV system and the numerical

results are compared with explicit formulas obtained in section 4.

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2 Lax pair spectral equations and the elementary DTs

Consider a matrix spectral equation of the second order with spectral parameter and 2 2 matrixcoefficients.

xx + Fx + U = 3 (3)

where the vector = (1, 2)Tand the matrix potentials are U = {uij}, F = {fij , fii = 0}, i = 1, 2 while3 = diag(1, 1) is the Pauli matrix.

For equation (3) we perform two elementary Darboux transforms [10]. The first one is

1 = 1x + 111 + 122 , 11 =

1x +1

2f122

/1 , 12 = f12/2,

2 = 2 + 21 1 , 21 = 2/1,

1 = (x + 11) 3 + 12 4 ,

2 = 4 + 21 3. (4)

where (1, 2)

T

and (3, 4)

T

are two solutions of (3) corresponding to different spectral parameters.Substituting the above expressions for

1,

2 into (3) and collecting the coefficients of 1, 2 and theirderivatives we obtain the expressions for new potentials as

f12 = u12 + f12 11,

f21 = 2 21,

u11 = u11 211x

f12 21 f21 12,

u12 = u12x 12xx + 11u12 12

u11 + u22

,

u21 = f21 221x

f2111,

u22 = u22 21u12 u2112

f2112x. (5)

The second elementary DT is performed after the first one and can be obtained by reversing the indices1 2and 2 1 to get, for example, the following potentials

f21 =

u21 +

f21

22,

22 =

2x +1

2

f21

1

/

2,

u22 =

u22 222x

f21

12

f12

21,

12 = 1/

2,

21 =

f21/ 2. (6)

The spectral equation (3) is considered as the first equation of the Lax pair, take the second as

t = xxx + Bx + C, (7)

where B = 32

diagU+ 32

Fx +34

F2

and C = 32Ux 34diagUx 34(f12u21 + f21u12)I + 38(f12,xf21 f12f21,x)3 + 34(u11 u22)3F.

Equation (7) is also covariant under transformations (4), (5). The compatibility conditions have thefollowing form

Ft F3x + B2x 3U2x + 2Cx + F Bx 3B3Fx + UB3B3U + F C 3C3F = 0,

Ut

U3x + C2x + UC

3C3U + F Cx

3B3Ux = 0. (8)

and the transformations (4), (5) determine a discrete symmetry of (2)

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3 Solution of two coupled KdV-MKdV equations and KdV

equation via the first elementary DT

For a spectral parameter and a seed potential F, U we obtain the solutions 1, 2 to the pair (3), (7).

Then performing the first elementary DT to obtain the new potentials

F ,

U which are solutions to thesystem (2). For the case of zero seed potential the solutions, 1 and 2 of the system (3), (7) have theform

1 = c1eax+a3t + c2e

(ax+a3t),

2 = d1eiax+(ia)3t + d2e

(iax+(ia)3t). (9)

where c1, c2, d1, d2 are arbitrary constants, a =

and i is the imaginary unit. System (2) reduced (forthe only nonzero elements ) to the following

f21t +1

2

f21xxx +3

4

f21u11x =

3

2

u11u21,

u11t 14

u11xxx 32

u11u11x = 0,

u21t +1

2u21xxx +

3

4u21u11x +

3

2u21xu11 =

3

4u11f21xx +

3

4u211f21. (10)

where f12 = 0, u12 = 0, u22 = 0 and tildes are omitted for simplicity. This system with explicit solutionobtained from (4), (5) as

f21 =

2e(1i)a(a2t+x)

d2e

2ia3t + d1e2iax

/

c2 + c1e2a(a2t+x)

,

u11 = 8a2c1c2e2a(a2t+x) /c2 + c1e2a(a

2t+x)2

,

u21 =2iae(1i)a(a2t+x)

d2e

2ia3t d1e2iax

/

c2 + c1e2a(a2t+x)

. (11)

where c1, c2, d1, d2 are arbitrary constants and a =

.

The second equation in (10) is the KdV equation while the remaining are a two components coupledKdV-MKdV system that was solved by elementary DT.

4 Solution of three coupled KdV-MKdV equations via the com-

pound elementary DTsExistence of different kinds of automorphism causes special constraints [10]. Multiplying (3) by 1 =

0 11 0

to have

1xx + 1Fx + 1U = 13 (12)

but 13 = 31 and consider the conditions 1F = F 1 and 1U = U1 that meansf12 = f21 = f, u11 = u22 = u, u12 = u21 = v. (13)

So (12) becomes

(1)xx + F (1)x + U(1) = 3 (1)

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The above automorphism () 1 () relates two pair of solutions (1, 2) and (3, 4) of (3)corresponding to different values of spectral parameter , as

3()

4()= 1

1()

2() =

2()

1()

Using this result in the elementary DTs (4), (5) and (6) to obtain the expressions for the new potentialsf, u, v. In the case of zero initial potentials these new potentials have the following forms

f = 21(2)x 2(1)x

(1)2 (2)2 ,

u =

(21)x (22)x(1)2 (2)2

x

+ 2

1(2)x 2(1)x

(1)2 (2)22

,

v = 2

1(2)x 2(1)x

(1)2

(2)2

x

+(1(2)x 2(1)x) ((21)x (22)x)

((1)2

(2)2)

2 . (14)

where 1, 2 are as in (9) with c1, c2, d1, d2 are arbitrary constants and a =

.The above expressions are solutions of system (2) that reduced under the reduction conditions (13) tosystem (2).

The choice of the arbitrary constants (c1, c2, d1, d2) affects on the behavior of the solution in formula(4). For example choosing equal constants c1 = c2 = d1 = d2 = 0.5 (we choose the value to be 0.5 tosimplify the resulting formula but the idea valid for any value) the solutions have the form

f = 2a(sin1cosh2 cos1sinh2)/(cosh22 cos21),u = 2a2(sin1cosh2 + cos1sinh2)

2/(cosh22 cos21)2,v = 2a2(cos31cosh2 2sin1sinh2(cos21 + cosh22 + 2) cos1cosh32)

/(cosh22 cos21)2. (15)

where 1 = a3t ax, 2 = a3t + ax,a =

is real.

We see that the above expression (15) is singular at 2 = 0, 1 = n,n = 0, 1, 2,.... Hence we havesingularity at (x = n

2a, t = n

2a3).

To obtain continuous solutions we can choose c1 = c2, d1 = d2 = r.c1, r is real constant. We againchoose c1 = 0.5 following the previous concept. So (4) have the form

f = 2ar (cosh 2 sin 1 cos 1 sinh 2) /

cosh2 2 r2 cos2 1

,

u = a2

1 r4 r4 cos21 + cosh 22 + r2 sin21 sinh 22

/

cosh2 2 r2 cos2 12

,

v = 2a2r(((7 + 6r2 + 2r2 cos21)cos 1 cosh 2 cos 1 cosh 32)2(1 + r2 + r2 cos21 + cosh 22)sin 1 sinh 2))/(1 + r2 + r2 cos21 cosh 22)2 (16)

where a, 1, 2 as in (15)

Choosing this parameter (r) to be r < 1 gives real nonsingular solutions. The above formula (4) isbuilt from elliptic and periodic functions so it does not preserve its symmetry but its localized as shownin figures (1.a,b) below.

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-2 -1 1 2

-2

-1

1

2

-2 -1 1 2

2

4

6

8

10

12

-2 -1 1 2

-10

-8

-6

-4

-2

Fig.(1.a) Non-singular solutions, f, u and v (r=0.5) , a=2, t=0.

-2 -1 1 2

-0.5

0.5

1

1.5

2

-2 -1 1 2

2

4

6

8

-2 -1 1 2

-4

-2

2

4

Fig.(1.b) Propagation of solutions, f, u and v (r=0.5) , a=2, t=1.Fig.(1) The solutions in (4) does not preserve its symmetry but its localized.

Choosing the parameter r to be r > 1 in formula (4) gives singular solutions as shown figure (2) below.

-2 -1 1 2

-40

-20

20

40 -2 -1 1 2

-400

-300

-200

-100

-2 -1 1 2

100

200

300

400

Fig.(2) Singular solutions, f, u and v (r=2), a=2, t=0.

Moreover the choice of these arbitrary constants (c1, c2, d1, d2 ) as well as the spectral parameter affectsthe reality of the resulting solution. As example for = 2im2, m is real and choosing c1 = c2 = d1 =d2 = 0.5, we get real solution

f = m(cos21sinh2 sinh2 sin1cosh22 + sin1)/(0.25cosh22 0.25)(1 cos21)where 1 = 2mx + 4m3t, 2 = 2mx 4m3t, while choosing c1 = c2 = 1, d1 = d2 = 2 give the followingcomplex solution

f = m (8(5sinh2cos21 + 5sinh2 + 5sin1cosh22 5sin1) 8i(6sinh2+3sin21cosh2 + 3cos1sinh22 + 6sin1))/(17cosh22 + 10 + 36cos1cosh2

8cos21cosh22 + 17cos21)

5 The numerical method

5.1 The difference scheme

For the coupled KdV-MKdV system (1) we introduce a numerical (finite difference) method of solution[7, 15]. This scheme is valid for arbitrary number of equations with constants coefficients and of the form

n,j+1i n,ji

+m,k

(gn,1m,km,ji

k,ji+1 k,ji12h

+ gn,2m,k(m,ji )

2 k,ji+1 k,ji1

2h

+gn,3m,km,ji+1 m,ji1

2h

k,ji+1 k,ji12h

+ gn,4m,km,ji

k,ji+1 2k,ji + k,ji12h

+gn,5m,km,ji

k,ji

k,ji+1 k,ji12h

) + dnn,ji+2 2n,ji+1 + 2n,ji1 n,ji2

2h3= 0 (17)

where i and j are the discrete space and time respectively. The time step is denoted by while h denotesspatial step.

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5.2 Stability analysis of the scheme

We prove stability with respect to small perturbations of initial conditions [16, 9]. It is the boundness ofthe discrete solution with respect to small perturbation of the initial data. We give here the main stepswhile the details are presented in Appendix A. We can write

dn,j+1i = Tn,j+1i,r d

n,jr = T

n,j+1i,r T

n,ji,r d

n,j1r =

r

Tni,r

rdn,or

where dn,j+1i is perturbations of the discrete solution, dn,or small perturbation of the initial data and

Tn,j+1i,r is a differentiable operator. Stability required the boundedness of r

Tni,r

ri.e Tr is bounded.

We found that Tj+12 ea(,h),a(, h) = 2 max

l,n,m,k

gn,lm,k

maxi,m,k

m,jx,i

k,jx,i

+ [ max

l,n,m,k

gn,lm,k

maxi,m,k

m,jx,i

k,jx,i

+1

h maxl,n,m,kgn,lm,k maxi,m,k m,ji k,ji +

3

h3maxn |dn|]2 (18)The scheme is stable if a(, h) const. We have here a conditional stability. That is we require that

0 more faster than h 0 . Namely we need

(constant) . h6

5.3 Convergence proof for the scheme

We prove that the solution of (17) converges to the solution of (1) if the exact solution is continuously

differentiable one [16, 9]. We introduce here the main points for the scheme convergence and give thedetails in Appendix B.

ji is the difference solution of (17), uji is the exact solution. Hence the error v

ji is given by v

ji =

ji uji .

Introducing L2 norm defined by Vj =

i

n

vn,ji

2h

1/2

The scheme converges when the norm of that error Vj 0 as (, h 0)We found that Vj+1 P(M) O ( + h2), where P(M) is a polynomial in the bounded constantM = e

aj1

ea1 and a as in (18). Hence the convergence proved.

5.4 Numerical calculations and test

The coupled KdV-MKdV system (2) is solved numerically using scheme (17) with initial condition from(4) at t = 0 and the results are compared with the explicit formulas (4). The percentage errors are shownin the following plots.

First mode (f), % Error, t=.1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

-4 -2 0 2 4

x axis

%Error

Second mode (u), % Error, t=.1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

-4 -2 0 2 4x axis

%E

rror

Third mode (v), % Error, t=.1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

-4 -2 0 2 4

x axis

%Error

Fig.3 percentage errors of the numerical solutions relative to the explicit solutions.

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The results of the test confirms the validity of the numerical scheme we propose. It also illustrates theerrors of evaluation that could be estimated by the resulting inequalities of the scheme convergence proof.

6 Conclusion

Darboux transformations covariance of Lax representation of nonlinear equations is a powerful tool forexplicit solutions production. Here we investigate applications of special kind of such discrete symmetry -to be called elementary ones. We use these elementary DT to produce explicit solutions for coupled KdV-MKdV system. The iteration of DT can be formulated in form of determinants representations [10, 12].A numerical method for general coupled KdV-MKdV system is introduced. It is a difference scheme forCauchy problems for arbitrary number of equations with constants coefficients. We analyze stability andprove the convergence of the scheme. The scheme keeps two conservation laws chosen in analogy with KdVtype equations. Analyzing stability and proving the convergence beside comparing the numerical resultswith explicit formulas allow us to use the numerical scheme to systems with arbitrary coefficients that ispresumably non-integrable. Obviously the coupled KdV systems are successfully treated by our scheme [ 3].

Acknowledgment We thank S. B. Kshevetskii for useful discussions about numerical scheme forthe problem under consideration.

References

[1] Dodd R and Fordy A, On the integrability of a system of coupled KdV equations, Phys. Lett. A89 (1982),168170.

[2] Guha-Roy C, On explicit solutions of a coupled KdV-MKdV equation, International Journal of Modernphysics B3(6) (1989), 871875.

[3] Halim A A, Kshevetskii S P and Leble S B, On numerical integration of coupled Korteweg-de Vries System,Applied Mathematics Letters (to be published).

[4] Hirota R and Satsuma J, Soliton solution of coupled KdV equations, Phys. Lett. A85 (1981), 407409.

[5] Krishnan EV, Remarkes on a system of coupled nonlinear wave equations, Journal of Mathematical physics31(5) (1990), 11551156.

[6] Krzhivitski A and Ladyzhenskaya O A, A grid method for the Navier-Stokes equations, Dokl. Akad. NaukSSSR 167 309-311 (Russian); translated as Soviet Physics Dokl. 11 (1966), 212213.

[7] Kshevetskii S P, Analytical and numerical investigation of nonlinear internal gravity waves, Nonlinear Pro-cesses in Geophysics 8 (2001), 3753.

[8] Kupershmidt B A, A coupled Korteweg-de Vries equation with dispersion, J. Phys. A: Math. Gen. 18 (1985),15711573.

[9] Lankaster P, Theory of matrices, Academic Press, New York, 1969.

[10] Leble S B and Ustinov N V, Darboux transforms, deep reduction and solitons, J. Phys. A: Math. Gen. (1993),50075016.

[11] Leble S B, Nonlinear waves in waveguides, Springer-Verlag Berlin, Germany (1991).

[12] Leble S B and Ustinov N V, Korteweg-de Vries-modified Korteweg-de Vries systems and Darboux transformsin 1+1 and 2+1 dimensions, J. Math. Phys. 34(4) (1992), 14211428.

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[13] Matveev V B and Salle M A, Darboux transforms and solitons, Springer-Verlag, Berlin (1991).

[14] Ping Fu Zhao and Meng Zhao Qin, Multisymplectic geometry and Multisymplectic Preissmann scheme forthe KdV equation, J. Phys. A: Math. Gen., 33(18) (2000), 36133626.

[15] Shaohong Zhu, A difference scheme for the coupled KdV equation, Communication in Nonlinear Science &

Numerical Simulation, 4(1) (1999), 6063.

[16] Tannehill J C, Anderson D A and Pletcher R H, Computational fluid mechanics and heat transfer, Taylor &Francis, Washington, (1997).

[17] Wen-Xiu Ma and Zi-Xiang Zhou, Coupled integrable systems associated with a polynomial spectral problemand their Virasoro symmetry algebras, Progress of Theoretical Physics 96(2) (1996), 449457.

7 Appendix A

Stability analysis of the scheme

We prove stability with respect to small perturbations (because we consider nonlinear equations) of initialconditions. Strictly speaking it is the boundness of the discrete solution in terms of small perturbation ofthe initial data. Consider the differential

Tn,j+1i,r =

n,j+1i /n,jr

, dn,jr =

n,ji2

n,ji1

n,ji

n,ji+1

n,ji+2

tand define the norm dj =

r

n

(dn,jr

We can write dn,j+1i = Tn,j+1i,r d

n,jr = T

n,j+1i,r T

n,ji,r d

n,j1r =

r

Tni,r

rdn,or

where dn,j+1i is perturbations of the discrete solution, dn,or small perturbation of the initial data.

Stability required the boundedness of r

Tni,r

ri.e Tr is bounded. We calculate T from (17) as follow

Tn,j+1i,r = i,r m,k

( gn,1m,k

2h

m,ji (i+1,r i1,r) + i,r

k,ji+1 k,ji1

+gn,2m,k2h

m,ji

2(i+1,r i1,r) + 2m,ji i,r

k,ji+1 k,ji1

+gn,3m,k2h

m,ji+1 m,ji1

(i+1,r i1,r) + (i+1,r i1,r)

k,ji+1 k,ji1

+gn,4m,k2h

m,ji (i+1,r 2i,r + i1,r) + i,r

k,ji+1 2k,ji + k,ji1

+

gn,5m,k

2h

m,j

i

k,j

i (i+1,r i1,r) + m,j

i

k,j

i+1 k,j

i1

i,r +

k,j

i

k,j

i+1 k,j

i1

i,r

)

dn2h3

[i+2,r 2i+1,r + 2i1,r 2i2,r] (19)

Rewriting (19) in terms of identity (E), symmetric (S) and anti-symmetric (A) matrices

Sj+1 maxl,n,m,k

gn,lm,k max

i,m,k

m,jx,i k,jx,i

,Aj+1 h maxl,n,m,k

gn,lm,k max

i,m,k

m,ji k,ji

+ 3h3maxn |dn| ,where jx,i =

ji+1ji1

2h , n , m , k = 1, 2, ..N, l = 1, 2,...5.

Tj+12 =(Tj+1) Tj+1 = (E Aj+1 + Sj+1) (E+ Aj+1 + Sj+1)

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1 + 2 Sj+1 + (Aj+1 + Sj+1)2 ea(,h),

a(, h) = 2 maxl,n,m,k g

n,lm,k maxi,m,k

m,jx,i

k,jx,i + [ maxl,n,m,k g

n,lm,k maxi,m,k

m,jx,i

k,jx,i

+1

hmaxl,n,m,k

gn,lm,k max

i,m,k

m,ji k,ji

+ 3h3

maxn

|dn|]2

We have here a conditional stability. That is we require that 0 more faster than h 0. Namely weneed

(cons tan t) . h6

8 Appendix B

The scheme convergenceWe prove the convergence by proving that the norm of the error (between the difference solution and theexact solution) vanishes as the mesh is refined. Let ji the difference solution of (17), u

ji the exact solution.

The error vji is given by vji =

ji uji .

The scheme converges when the norm Vj 0 as (, h 0) where the norm is defined as Vj =i

n

vn,ji

2h

1/2

substitute in (17) by ji = vji + u

ji keeping in mind that for

ji equation (17) is O ( + h

2) and usingthe operator T defined by

vn,ji [m,k

(gn,1m,k(um,ji

vk,ji+1 vk,ji12h

+ vm,jiuk,ji+1 uk,ji1

2h) + gn,2m,k((u

m,ji )

2 vk,ji+1 vk,ji1

2h

+(vm,ji )2u

k,ji+1 uk,ji1

2h) + gn,3m,k(

um,ji+1 um,ji12h

vk,ji+1 vk,ji12h

+vm,ji+1 vm,ji1

2h

uk,ji+1 uk,ji12h

)

+gn,4m,k(um,ji

vk,ji+1 2vk,ji + vk,ji12h

+ vm,jiuk,ji+1 2uk,ji + uk,ji1

2h) + 2gn,5m,k(u

m,ji v

k,ji

vk,ji+1 vk,ji12h

+ vm,ji uk,ji

uk,ji+1 uk,ji12h

)) + dnvn,ji+2 2vn,ji+1 + 2vn,ji1 vn,ji2

2h3]

=r

Tj+1ir v

n,jr

So we obtainvn,j+1i =

r

Tj+1ir vn,jr + f

n,jm,k,i (20)

where fn,jm,k,i =m,k

gn,1m,kvm,ji

vk,ji+1vk,ji1

2h + gn,2m,k(v

m,ji )

2 vk,ji+1v

k,ji1

2h + gn,3m,k

vm,ji+1vm,ji1

2h

vk,ji+1vk,ji1

2h

+ gn,4m,kvm,ji

vk,ji+12vk,ji +v

k,ji1

2h+ 2gn,5m,kv

m,ji v

k,ji

vk,ji+1vk,ji1

2h+ O ( + h2)

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fj =

i

fn,jm,k,i

2h

1/2

|gn,1m,k|maxh3/2

Vj2 + |gn,2m,k|maxh2

Vj3 + |gn,3m,k|maxh5/2

Vj2 + |gn,4m,k|maxh5/2

Vj2

+|gn,5m,k|max

h2

Vj

3+ O ( + h2)

|gn,lm,k|maxh2 Vj3 + O ( + h2) ,gn,lm,k

max= max

n,m,k,1gn,lm,k

Using Schwartz inequality so (20) becomesV j+1 Tj+1 Vj + fj

Tj+1 Tj Vj1 + (Tj+1 fj1 + fj) eaj Vo + ea(j1) fo + ea(j2) f1 + ... + fj eaj Vo + M

gn,lm,kmax

Vj+13 + M O ( + h2) , M = eaj1ea1Using Vo = 0, the above inequality has the solution

Vj+1

P(M) O ( + h2) , P(M) is a polynomial in M.

Since M is bounded then Vj+1 0 as , h 0 and the convergence proved.