Large time-stepping methods for higher order time-dependent evolution. equations. Xiaoliang Xie. A dissertation submitted to the graduate faculty

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1 Large time-stepping methods for higher order time-dependent evoltion eqations by Xiaoliang Xie A dissertation sbmitted to the gradate faclty in partial flfillment of the reqirements for the degree of DOCTOR OF PHILOSOPHY Maor: Applied mathematics Program of Stdy Committee: Hailiang Li, Maor Professor L. Steven Ho Pal Sacks Jim W. Evans Snder Sethraman Iowa State University Ames, Iowa 8 Copyright c Xiaoliang Xie, 8. All rights reserved.

2 ii DEDICATION To my dear Mother Rilin Zhang and father XiXi Xie

3 iii ACKNOWLEDGEMENTS First and foremost, I wold like to epress my deepest gratitde to Professor Hailiang Li, my academic advisor, for his gidance and spport. Throghot my PhD period, he has encoraged me many times to keep moving forward and to look for creative soltions, and he always provided valable comments whenever needed. His keen scientific insights were indispensable for this dissertation and will always be a sorce of inspiration for my ftre career. Besides, I want to thank Dr.Li for providing great help in the thesis revision, and helping me keep faith in myself even at the most difficlt moments. He cold not even realize how mch I have learned from him, besides math he always teaches me how to become a sccessfl person. I feel really lcky to have had Dr.Hailiang Li as my advisor. Also, I wold like to thank my other committee members, Professors Pal Sacks, Snder Sethraman, L. Steven Ho and Jim Evans as a grop. Besides, cooperation from Dr.Li s other stdents are also important to my work. I appreciate a lot all the helpfl comments from Dr.Zhongming Wang and Dr.Haseena Ahmed. At last, I am profondly gratefl to all those involved in providing me tremendos spport throghot my five years at Iowa State University.

4 CHAPTER. INTRODUCTION Recently there has been significant research interest in the stdy of large time-stepping nmerical methods for different types of Partial Differential Eqations arising in varios applications. Many of the nderlying Partial Differential Eqations are characterized by time dependence, high order derivatives and strong nonlinearity. Their soltions are epected to possess special properties sch as pattern formation, which is not easily observed in a short period of time. Therefore, in solving sch PDEs, long time behavior of nmerial soltions is a critical isse. We consider a class of PDEs of the form t = L() + N(), (.) where L = L(,, D, D,...) is a linear operator which contains high order spatial derivatives and N = N(,, D, D,...) is a nonlinear operator which may also contain high order derivatives. The high order derivatives force small time step in eplicit time-discretization according to the CFL condition. Varios large time-stepping nmerical schemes have been developed for these type of PDEs. The implicit-eplicit scheme is to smartly combine the implicit and eplicit approimation for linear and nonlinear spatial derivative operators. The stability property can be fond in [4]. It has been proposed for reaction-diffsion problems [3], Navier-Stokes eqations [], and the KdV eqation [6, ]. The integrating factor IF method has been developed by Trefethen [33], and Co and Matthews [9]. The idea is to solve high order linear operator eactly by making change of variables. An etension of implicit-eplicit method which is called fast spectral algorithm has been developed by Fornberg and Driscoll [] for prely dispersive eqations and has been generalized by Driscoll [] by sing Rnge- Ktta time-stepping. The main idea is to se different nmerical scheme for the low, medim

5 and high wavenmbers separately. Another is the eponential time differencing (ETD) scheme which has been developed by Co and Matthews for stiff systems [9]. The ETD scheme apply the same integrating factor as in the IF approach, the difference in the integration is over a single time step of length h. Later ETD schemes have been improved to fi the nmerical inaccracy de to cancellation errors, see [].. Goals and Main Reslts Or goal is to design a class of large time-stepping stable methods for different types of higher order PDEs. These will be developed gradally as the compleity of eqations increases. Part I: Linear eqations(n= in (.)) Or key idea for design of large time-stepping nmerical schemes is based on a preconditioning approimation t ɛ = φ, φ = P ( ), (.) where the preconditioning operator (ɛ, D) is selected to make L() φ shares eqal reglarity of. In order for the approimate system to be faithfl we reqire: (i)consistency lim ɛ (ɛ, D) = I; (ii) (ɛ, D) is a linear operator; (iii)ɛ > is a parameter proportional to the time step to be determined. We smmarize the choice of for three different types of linear operator L in Theorem (..). ) (ɛ, ) = I ɛp ( ),ɛ = t ) (ɛ, ) = I ɛp ( ) with t 3) (ɛ, ) = I ɛ(p ( ) c) with t for Re(P (iξ)) =, for all ξ R, ɛ t for Re(P (iξ) <, for all ξ R, ɛ t for Re(P (iξ)) c(c > ), for all ξ R. Where P (iξ) is the Forier transform of P ( ). By choosing in this way, a semi-discrete scheme (.4) is stable in the sense of n+ e c t n. (.3)

6 3 We also discssed the effect of spatial discretization on the stability of the flly discrete scheme, we define A to be the matri derived from the discretization of P ( ) to have the following reslts: ) When P (iξ) for ξ R and t ɛ t, a symmetric negative definite matri A will make flly discretized scheme nconditionally stable in the sense of n+ n. ) When P (iξ) is pre imaginary and ɛ = t, a skew-hermitian matri A will make flly discretized scheme nconditionally stable in the sense of n+ = n. Where A is the spatial discretization matri corresponding to linear operator P ( ). Or stdy is mainly focsed on stability isses, and we know for linear eqations, as long as we have a stable scheme it is also a convergent scheme de to the La Eqivalence Theorem. By the stdy of the linear eqations we bild a solid fondation for nonlinear eqations, we always apply the reslt from this part for the nonlinear eqations. Part II: Semi-linear eqations. When both N and L eist and the high order derivatives appear on the linear operator and the nonlinear operator is of low order, we regrop the operators and then split the linear and nonlinear part, applying the reslt for linear operator L according to the discssion for linear eqations, and deal with the nonlinear term wisely to get large time-stepping nmerical schemes. Different nonlinear operator N shold be treated differently to get a stable nmerical scheme. We give three eamples of nonlinear eqations of this type, the Korteweg-de Vries, the Swift-Hohenberg eqation and the generalized Kramoto- Sivashinsky eqation. The KdV eqation For the KdV eqation t + + =, first we apply the operator discssed in part I, then we se the Strang splitting techniqe, and apply /3 dispersive scheme on to

7 4 obtain: () () + t () n t/ () + D D + D + n =, 6 ( )( + ) =, = () n+ () t/ n+ + () + D D + D + (), =, =,,..., N, (.4) where N is the grid point vale, () and () are N dimensional vectors, t is the time step, is the length of the partition, and D D + D n = n + n + +n n h 3. The nice featre of this scheme is to preserve both mass and energy at discrete level, i.e. (n+ ) = (n ), and (n+ ) = (n ). Meanwhile it is second order in both time and space. In a recent work [], a second order finite volme method was introdced in order to preserve both mass and energy evolving two eqations of both and. The scheme (.4) shares the same properties as in [], and mch easier to implement. Several nmerical tests are performed by scheme (.4)to show the accracy, stability and robstness of long-time simlation. The Swift-Hohenberg type eqations We consider the Swift-Hohenberg type eqation t = L + df d, (.5) where L = is linear operator and F is a nonlinear fnction of. The eqation (.5) admits a nonincreasing energy fnctional Ψ = sbect to the Dirichlet bondary condition Ω { ( ) F ()}ddy, (.6) = =, (, y) Ω. (.7)

8 5 We apply the reslt in part I on linear operator L and se divided difference techniqe on F to obtain n+ n t ( = L θ n+ + ( θ) n) + F (n+ ) F ( n ) n+ n. (.8) We smmarize the reslt of how to choose θ in semi-discrete scheme (.8) to make it stable in the following: ).If θ =, the scheme (5.6) is stable in the sense of Ψ[ n+ ] = Ψ[ n ( n+ n ) ] ddy Ψ[ n ]. (.9) t for arbitrary t. ). If θ >, then the scheme (5.6) is stable in the sense of for all t θ. Ω Ψ[ n+ ] Ψ[ n ], The nmerical reslt is consisted with the theoretical reslt (5..), and in or nmerical tests we find the Dirichlet bondary condition = = is essential to captre the pattern formation for -D Swift-Hohenberg eqation. The Kramoto-Sivashinsky eqations For the generalized Kramoto-Sivashinsky eqation t + + (a() ) + (b ()(b() ) ) + (c() ) =, R, (.) where a,b,c are continos fnctions of, we establish the L stability of the finite difference scheme n+ n t + 6h (v + + v + v )(v + v ) + D + (a(v )D v ) +D (b (v )D + D b(v )) + D + D (c(v )D + D v ) =, (.) where v = n+ + n. Nmerical tests for three different types of Kramoto-Sivashinsky eqations are condcted to show the accracy and long time stability of this scheme.

9 6 Part III: Fll nonlinear eqations. Net we consider the case both N and L in (.) eist and the high order derivative appears on the nonlinear operator N. There has been significant research interest in these kinds of eqations. In [34], efficient nmerical schemes with particlar emphasis on the se of large time steps have been developed for epitaial growth models of thin films. In [4], a large time-stepping method for the Cahn-Hilliard eqation has been developed and analyzed. We consider the model eqation which is the gradient flow of the fnctional t = ( ), Ω R, (.) I[] = Ω ( ) d. (.3) We develope two schemes which are stable for large time steps in the sense of keeping the energy identity of continos model (.) at the discrete level. The implementation of (7.) in one dimension give s the desired soltion =.. Thesis Organization This paper is organized as follows, in section we will constrct operator L for linear eqations, gided by Von Nemann stability analysis, and we discss the spatial discretization for linear eqations. After spatial discretization the spatial derivative operator becomes matri, we pt some restriction on the matri to garantee that the nmerical scheme is stable for arbitrary large time step. In section 3, some nmerical eamples for linear eqations are given. In section 4, we develop a second order large time-stepping nmerical scheme for the Korteweg-de Vries eqation, preserving both momentm and energy at the discrete level. Both theoretical and nmerical reslts are given in this section. In section 5, we developed a nmerical scheme for Swift-Hohenberg eqation which preserves the nonincreasing energy fnctional at the discrete level. The pattern formation for D Swift-Hohenberg eqation is displayed in the nmerical test. In section 6, a stable nmerical scheme has been developed for

10 7 generalized Kramoto-Sivashinsky eqations. In chapter 7, we developed two stable nmerical schemes for a nonlinear model eqation t = ( ).

11 8 CHAPTER. Constrction of the operator L. Constrction of the operator L In this section, we discss the selection of the smoothing operator for linear PDEs with constant coefficients. For simplicity, we only consider one-dimensional case. Similar analysis can be well applied to mlti-d case. We consider the linear problem with constant coefficients where P ( ) = r l= A l l l, A l being constants. t = P ( ), (, ) = (), R, (.) We assme this problem is well-posed. To be self-contained, we recall the fndamental definition of well-posedness for problem. (.) in ([3]) Definition... The Cachy problem (.) is well posed if, for every () C (R):.There eists a niqe soltion (, t), and.there are constants c and K, independent of (), sch that (, t) Ke ct (., ), (.) where. denotes the norm in L (R) space and C (, t) denotes the infinitely differentiable fnction space. Or interest is to develop a large time step stable scheme to compte the soltion of (.). To do so we first propose an approimate model to (.), t U = φ, U = e ct, φ = (P ( ) c)u, (.3)

12 9 where (ɛ, ) = I ɛ(p ( ) c) and the choice of ɛ depends on the operator P ( ). We then discretize (.3) in time by forward Eler method to obtain U n+ U n t = φ n, φ n = (P ( ) c)u n, (.4) where U n = U n () U(t n, ). We now smmarize the choice of ɛ in the following theorem. Theorem... The scheme (.4) is stable in the sense of n+ e c t n (.5) for any t >, if ) (ɛ, ) = I t P ( ), c = for Re(P (iξ)) =, for all ξ R, ) (ɛ, ) = I ɛp ( ) with t 3) (ɛ, ) = I ɛ(p ( ) c) with t ɛ t, c = for Re(P (iξ) <, for all ξ R, ɛ t for Re(P (iξ)) c(c > ), for all ξ R. Proof. In order to prove (.5), it sffices to show U n+ U n. We apply the Forier transform to (.4) to obtain Û n+ (ξ) = Û n (ξ) + t ˆφ n (ξ), (ɛ, iξ) ˆφ n (ξ) = (P (iξ) c)û n (ξ), (.6) which gives Û n+ = Q(ξ)Û n (ξ), Q(ξ) = ( + (P (iξ) c) t ). (.7) (ɛ, iξ) Here Q(ξ) plays the role of the amplification factor. By taking (ɛ, ) = ɛ(p ( ) c) in (.7) we obtain Q = (ɛ t)(p (iξ) c). (.8) ɛ(p (iξ) c) We now estimate Q for different cases.

13 ) With the choice ɛ = t and c=, Q = + t P (iξ) t (.9) P (iξ). From (.9) and Re(P (iξ)) =, it follows Q =, which garantees the identity n+ = n. ) From (.8) together with the conditions Re[P (iξ)], c= and ɛ t, we have (ɛ t)p (iξ) Q = ɛp (iξ) ( ) (ɛ t)re(p (iξ)) + (Im(P (iξ))) = ( ɛre(p (iξ)) ) + (Im(P (iξ))), where Im(P (iξ)) denotes the imaginary part of P (iξ). The fact that Q implies n+ n. 3)By the assmption c >, Re(P (iξ)) we have Re(P (iξ)) c. We sbstitte P (iξ) c for P (iξ) in ) to obtain Q <, which implies U n+ U n. Then we sbstitte back = Ue ct to obtain n+ e c(n+) t n e cn t, n+ n e c t. We complete the proof of the theorem.. Spatial Discretization The semidiscrete scheme discssed in section (.) is nconditionally stable for any t >, in this section we discss the effect of spatial discretization on the stability of the flly discrete scheme. For simplicity we consider c=, then (.4) can be rewritten as (ɛ, ) n+ () n () t = P ( ) n (), (.)

14 where (ɛ, ) depends on the spatial operator P ( ). We discretize the spatial operator P ( ) by finite difference operator, we will investigate what additional conditions need to be imposed in order to keep the flly discrete scheme nconditionally stable... Dissipative Case When P ( ) contains only even order derivatives and P (iξ) for any ξ R, the eqation is dissipative, from the discssion in section we know for this case = (I ɛp ( )) with t ɛ t, (.) becomes (I ɛp ( )) n+ () = (I (ɛ t)p ( )) n (). (.) We apply a spatial discretization on (.) to obtain (I ɛa) n+ = (I (ɛ t)a) n, (.) where n = ( n, n,, n N ) is an N dimensional vector, N is the nmber of grid points, A is a matri derived from the discretization of P ( ). Net we will show that the flly discrete scheme (.) remains stable for arbitrary time step for a class of A. For eample for P ( ) = we se central difference to approimate the operator, i = i+ 4 i+ + 6 i 4 i + i 4, where i is the i th grid point vale. If we se the periodic bondary condition then the matri A corresponding to is A =

15 Lemma... If A is symmetric and negative definite, and (I ɛa) is invertible, then scheme (.) is nconditionally stable in the sense of n+ n, provided t ɛ t. Proof. For given A, (I ɛa) is invertible, then n+ = (I ɛa) (I (ɛ t)a) n. Let B = (I ɛa) (I (ɛ t)a), and λ be an eigenvale of B, associated with the eigenvector, then B = λ. That is (I ɛa) (I (ɛ t)a) = λ. This leads to which implies A = λ ɛ t λɛ, λ(a) = λ(b) ɛ t λ(b)ɛ. A is symmetric, negative definite, so its eigenvales are negative. λ(b) ɛ t λ(b)ɛ <, i.e. Hence Note that t ɛ ((λ(b) )ɛ + t)(λ(b) ) <. t ɛ, becase of ɛ t. λ(b). Since A is symmetric, so is B. It follows from linear algebra that B is nitarily eqivalent to a diagonal matri Λ with each diagonal element λ i, i n where n is the dimension of Λ. Since we can write n+ = U T ΛU n,

16 3 it follows that U n+ = ΛU n. Hence which completes the proof. n+ = U n+ U n = n,.. Dispersive Case When P ( ) contains only odd order derivatives, the eqation is dispersive. In this case = (I t P ( )), and (.) becomes ( I t ) ( P ( ) n+ () = I + t ) P ( ) n (). (.3) Assme A is the matri corresponding to P ( ) and we discretize (.3) in space to obtain ( I t ) ( A n+ = I + t ) A n. (.4) For eample P ( ) =, we se central difference to approimate the operator, i = i+ i+ + i i 3, where i is the i th grid point vale. we se periodic bondary condition, then the matri A corresponding to is A = which is skew-hermitian ,

17 4 Lemma... If A is skew-hermitian, i.e. A = A, and (I ɛa) is invertible, then scheme (.4) is nconditionally stable in the sense that n+ L = n L, where n L = N i= (n i ). Proof. By the assmption that (I ɛa) is invertible, and from (.4) we have n+ = (I t A) (I + t A)n. Let B = (I t A), so B = (I + t A), then we have n+ L = (B B ) B B n L. We notice that (B B ) B B = B(B ) B B = B(B ) B B = B(BB ) B. From the assmption A is skew-hermitian, we have B is skew-hermitian, so B is normal. Then we have B(BB ) B = B(B B) B = BB (B ) B = I, i.e. n+ L = n L.

18 5 CHAPTER 3. Linear Eqations 3. Diffsion eqation We consider a forth order diffsion eqation t + =. (3.) according to or discssion in Section. The operator = ɛp ( ) with t ɛ t. For the spacial discretization of we se central difference. The damping is small when t < O( h 3 ) so we take ɛ > t which leads to ( + ɛ(d + D ) ) n+ = ( ( ɛ)(d + D ) ) n, where (D + D ) n = (n + 4n + + 6n 4n + n )/( 4 ). In or nmerical test we choose the initial condition to be (, ) = cos(π), [, ], and the eact soltion can be compted as eact (, t) = ep( 6π 4 t)cos(π). First we perform a large time step test, we take large time steps dt =.,.,.3,,, and fi the nmber of grid points to be 4, the final time tma =. From Figre 3. we see that, even when the time step is large, we still get a reasonable error, which shows that the scheme is stable for large time steps. Net we perform a test to see how the ratio θ = ɛ t effects the L error. We fi the grid cell nmber to be 4, becase from or nmerical eperiments we find the nmber of grid points does not effect the shape of the graph. From Figre 3. we can conclde when t is small, the optimal θ shold be close to /, and when t is large, the optimal θ shold be close to.

19 6.58 large time step test L error dt Figre 3.: Eample t =, 4 cells. We also perform a convergence test for the same eqation. We fi the ending time to be tma =. and take θ = 5/9, and calclate the error and order of convergence by the following formlae E k,l = ( N i eact ( i, t N )), (3.) (r) k,l = log() log(e k,l+ E k,l E k,l E k,l ), (3.3) (r) k,l = log() log(e k,l E k,l E k+,l E k,l ). (3.4) Table 3.: L error by compting eqation (3.) N t E-8.85E-8.885E-8.943E-8 8.3E-8.3E-8.63E-8.35E E-8.8E-8.4E-8.76E-8 3.9E-8.447E-8.77E-8.39E-8

20 7.4 3 dt=.,tma=. 3 dt=.5,tma=. L error L error theta theta 5 3 dt=.,tma= 5 3 dt=.5,tma= 4 4 L error 3 L error theta theta Figre 3.: Eample t =, 4 cells. These three tables show the nmerical scheme has convergence rate order in time direction and order in space direction. 3. Dispersive eqation We consider the dispersive eqation t + + =, in which P ( ) =. According to previos discssion or operator = I + t ( + ). The corresponding Table 3.: order of convergence in the t direction (r) kl of eqation (3.) N t

21 8 Table 3.3: order of convergence in the direction (r) k,l of eqation (3.) N t semi-discrete scheme is (I + t ( + )) n+ n =, t (I + t ( + )) n+ = (I t ( + )) n. We se central difference for spatial discretization to obtain the nmerical scheme ( + t (D + D D + D )) n+ = ( t (D + D D + D )) n. (3.5) We now carry nmerical test for eqation (3.5) with the initial condition (, t = ) = cos(π), [, ]. The eact soltion is, eact (, t) = cos(π (π (π) 3 )t). We display the nmerical reslt at final time tma =. in the following tables. Table 3.4: L Errors to compte dispersive eqation N t These three tables show the scheme is second order in both time and space. Net we condct large time step test, with the same initial data and eact soltion, we notice

22 9 Table 3.5: order of convergence in the t direction of dispersive eqation (r) kl N t Table 3.6: order of convergence in the direction of dispersive eqation N t the speed of the traveling wave is π (π) 3, so one period is /(π (π) 3 ). We se large time step t = compte p to final time t = /(π (π) 3 ). We display the nmerical soltion and the eact soltion in Fig.3.3, which shows the wave is resolved very well. 3.3 Linear KS eqation We consider the linear eqation t = P ( ). (3.6) This corresponds to the case P ( ) =, which is the combination of second and forth order derivatives. Forier analysis shows the problem is well-posed in the sense that (, t) L L e t 4, t >. According to Case 3 in Theorem.. the nmerical scheme will be n+ = ( + tp ( )) n e t 4, where (ɛ, ) = I ɛ( /4I),

23 Figre 3.3: large time step test for dispersive eqation with t ɛ t. We se central difference to approimate second and forth order derivative and apply the periodic bondary condition. (D + D ) n = n + 4 n n 4 n + n, (D + D ) n = n + n + n.

24 The corresponding matri for periodic bondary condition and forth order derivative is X = , and for the second order derivative is X = First we perform a convergence test on or scheme we choose the initial condition to be (, ) = cos(π), and the eact soltion is (, t) = ep( 6π 4 t + 4π t)cos(π), The L error and order of convergence which is compted by formlae (3.-3.4) is displayed in the following tables. From this nmerical test we see the scheme have nmerical convergence order in time and order in space.

25 Table 3.7: L Errors to compte Eqation 3.6 by scheme 3.3 N t E-8.8E-8.849E E-8.3E-8.76E E-8.9E-8.37E-8.5.4E-8.6E-8.3E-8.9 Table 3.8: order of convergence in the t direction (r) kl N t The combination of second order and forth order derivative term in the KS eqation, describe a balance between long-wave(small freqency ξ) instability and short wave(large freqency ξ) stability. We notice the freqency ξ = / is an optimal one at which the wave grow fastest. Net we perform a nmerical test to see how the freqency ξ and time step effect the error between nmerical reslt and the eact soltion. We take the initial vale to be (, ) = e ξi, and the eact soltion is We take the ξ to be (, t) = e ξ4 t+ξ t e ξi.,, π, 5π, π separately, and for some mode we perform both large time step test and small step test. For large time test we choose t =.,.3,.9, Table 3.9: order of convergence in the direction (r) k,l N t

26 3.7,.8,.43,.79,.87 and for small step test we se of the t for large time test. And the final time we take for large time test is 5, and.5 for small time step test, the nmerical reslt is displayed in Fig From the reslt we see, for instable wave ξ there is no big difference as we change θ and ξ, however for short waves which is stable the error growth as θ getting closer to.5.

27 4 large time step test ξ=sqrt()/ 5 L error dt large time step test ξ= theta=.6 theta=.7 theta=.8 theta=.9 theta=.5 theta= 5 L error dt theta=.6 theta=.7 theta=.8 theta=.9 theta=.5 theta= Figre 3.4: large time step and small time step error comparison for linear KS eqation

28 5 large time step test ξ=π 5 L error dt small time step test mode π theta=.6 theta=.7 theta=.8 theta=.9 theta=.5 theta= 5 L error dt theta=.6 theta=.7 theta=.8 theta=.9 theta=.5 theta= Figre 3.5: large time step and small time step error comparison for linear KS eqation

29 6 large time step test ξ=5π 5 L error dt large time step test ξ=π theta=.6 theta=.7 theta=.8 theta=.9 theta=.5 theta= 5 L error dt theta=.6 theta=.7 theta=.8 theta=.9 theta=.5 theta= Figre 3.6: large time step and small time step error comparison for linear KS eqation

30 7 CHAPTER 4. The KdV eqation 4. Backgrond In this chapter we consider the generalized Korteweg-de Vries(KdV) eqation of the form t + f() + =, (4.) Here f() is the given fl fnction, say f = /. This eqation describes the evoltion of long one-dimensional waves in many physical settings, in which the dispersion conterbalances the nonlinear term f(). The KdV eqation is well-known to satisfy infinitely many conservation laws, the most fndamental properties are the preservation of the mass and energy R (, t) = ()d (mass), (4.) R R (, t)d = ()d (energy). (4.3) R where () is a given initial data decaying at far field. De to the higher order spatial derivatives in (4.) any eplicit time discretization wold restrict the time step to order of ( ) 3. This is not sitable for long time simlation. On the other hand it is believed that the nmerical method preserving both mass and energy is more stable and sitable for long-time integration. Or goal here is to design a second order finite difference method for (4.): (i) to preserve both the mass and energy relations (4.) and (4.3) respectively, at the discrete level; and (ii) to be stable for large time step. We note that, in a recent work [], a finite volme method was introdced in order to preserve both mass and energy; bt in their method two eqations for and are evolved.

31 8 Or method is more efficient since only one eqation is sed. For other nmerical methods developed for the KdV eqation ranging from finite difference method, finite element methods to spectral methods, see e.g. [],[3],[6],[35]. 4. Strang splitting and time discretization In (4.) we take f() = ( /). The initial vale problem (4.) becomes t + + =, (, ) = (). (4.4) First we describe the operator splitting strategy, see [7]. The strategy is alternately to solve the eqation t + =, (, ) = (), (4.5) and the linear dispersive eqation t + =, (, ) = (). (4.6) Let S t be the soltion operator associated with the conservation law (4.5), we write the niqe weak soltion to (4.5) as (, t) = S t (). Similarly we denote the soltion operator associated with the linear dispersive eqation (4.6) by A t. Then we approimate the soltion of (4.4) by t (, n t) = [A t S t ] n (). (4.7) When implementing this approach, both A t and S t mst be replaced by nmerical methods. In time direction we se Crank-Nicolson n+ n t + n+/ n+/ + n+/ =. In practice, for each time step evoltion we se the Strang splitting method n+ = [S t A t S t ] n, to obtain second order accracy in time direction.

32 9 4.3 Spacial discretization and Energy preserving scheme First we define We propose a nmerical scheme D = +, h D + = +, h D =. h n+ n+ n t n t + D D + D ( n+/ ) =, (4.8) + 6h (n+/ + + n+/ + n+/ )( n+/ + n+/ ) =. (4.9) Where we se D D + D to approimate, and in (4.6) and for (4.5) we apply the semidiscrete scheme d dt n + 3 ( n+ + n + n )( + ) =, h which is refered to as /3 dispersive scheme, and has been stdied by Levermore and Li [4]. Theorem Assme (n ) < and Σ < for Z, then scheme (4.8) and (4.9) satisfies (n+ soltion has compact spport. ) = (n ), and (n+ ) = (n ) if we assme the nmerical Proof. Mltiplying both sides of (4.8) by the factor n+/ ( n+ = n+ + n, we obtain ) ( n ) + t n+/ D D + D n+/ =. Smming this over the vales of, Z, and omitting the sbscript on the norm. we obtain n+ n = tσ Z n+/ D D + D n+/. (4.)

33 3 Net we prove the right hand side of (4.) is. For simplicity we omit the sperscript for n+/. Σ Z D (D + D ), = Σ Z ( D + D + D + D ), = Σ Z ( ( ) ( + ), (4.) = Σ Z ( + ) + ( + ), (4.) =. (4.3) Where we have assmed = for large. So, (4.8) conserves the L norm at discrete level. For (4.9) we also mltiply both sides by n+/ vales of to obtain = n+ + n and smming this over the n+ n = t/(3h)σ J = n+/ ( n+/ + + n+/ We still omit the sperscript to prove the right hand side of (4.4) is. + n+/ )( n+/ + n+/ ). (4.4) Σ Z ( )( + ) = Σ Z ( ( + ) ( ) + ( ) + ( ) ), (4.5) = (4.6) This completes the proof of L norm conservation (n+ ) = (n ). We can prove the conservation of mass n+ = n in the same way. 4.4 Nmerical implementation First we make a N partition of the comptational domain [a, b], so = b a. In order to implement (4.9) each time step we need an internal iteration. We rewrite (4.9) as n+ + n n + t h (n+/ + + n+/ + n+/ )( n+/ + n+/ ) =,

34 3 which can be written as where n+/ + t h (n+/ + + n+/ n+/ + n+/ )( n+/ + n+/ ) n =, (4.7) = (n+ + n ). (4.7) is a system of nonlinear eqations when goes from to N. For simplicity we se V to denote n+/, and λ = t h, the system becomes V + λ(v V N + V (V V N )) =, V + λ(v + V + V (V + V )) =, =, 3,, n, V N + λ(v V N + V N (V V N )) N =. In order to solve this system of nonlinear eqations F (V ) =, where V = (v,, v N ). We apply the Newton s method J(V k )s k = F (V k ), V k+ = V k + s k, where J(V) is the Jacobian (matri) of F(V). The iterations are condcted ntil the following criterion is satisfied: F (V k ) < δ, (4.8) where δ is a pre-given tolerance nmber. 4.5 Nmerical eperiments Eample Consider the one-soliton soltion to eqation (4.)with f() = and (, t) = Asech (k wt ), (4.9) where A = k,w = 4k 3 We take k =.3 and =, and take (, ) as the initial condition.

35 3 Table 4.: Eample 5., nmerical errors and convergence rate of at t=. N L order L order L order 5 5.E E E-4 -.3E E E E E-6.95.E E E E-6.9 Table 4.: Eample 5., nmerical errors and convergence rate of at t=. N L order L order L order 5 7.9E-5-5.E-5 -.5E-4 -.E-5.9.4E E E E-6.96.E E E-7.9.7E-6.9 Firstly we check the accracy and convergence rates of the method. The soltion region is taken to be (-,). We assign the bondary conditions with the eact soltion to make the nmerical simlation accrate. We take small t =., and perform the comptation on the grid of 5,,5, cells, respectively, p to t =., and the L, L, L errors and the corresponding convergence rates are presented in Table 5. and Table 5.. We can see from the tables that or method is second-order accrate for both and. Secondly, we se this soltion to test the stability and robstness of the method in longtime integration. We take the comptational domain to be (-,), with periodic bondary conditions. We notice the soliton has an advection speed of.36, a period of the soliton is We ths se this observation to make comparison between the nmerical soltion and the eact soltion. We take 4 cells t =., p to 55.5 ( period), ( periods) and 66.5 (3 periods) (4 periods), the soltions are displayed in Fig.4. we see this method is very stable in long-time simlation. We notice that the phase error becomes severe at longer time, becase the nmerical error accmlates as the comptational time becomes longer, we can redce the phase error by making finer grid cell or taking smaller time step t.

36 33 Eample We consider the two-soliton soltion of (5.) with the same f() ( (, t) = kep(θ ) + k ep(θ ) + (k k ) ep(θ + θ ) + a (k ep(θ ))+ ) kep(θ ) ep(θ + θ ) )/( + ep θ + ep θ + a ep θ + θ, where k =.4, k =.6, a = ( (k k ) (k + k ) ) = 5, (4.) θ = k k 3 t +, θ = k k 3 t +, (4.) = 4, = 5 (4.) We set p the comptational domain to be (-4,4), and apply the periodic bondary condition. Initially we have two solitons the tall one is to the left of the short one. Both solitons travel from left to right and speed of the tall one is bigger than the speed of the short one. So after some time the tall one catches p the short one. After they goes ot from the right end, they reappears from left de to the periodic bondary condition. Then the tall one will catch p the short one again, this process will keep going on. To show the stability and robstness of or scheme in long-time simlation and to display the interactions of solitons, we display the (,t)-contor of the soltion in the time interval (,5), see Fig. 4.. In this pictre we see the nmerical soltion preserve a stable and smooth shape for the solitons Net we compare the nmerical reslt with the eact soltion. The comptational domain is still taken to be (-4,4), and t =., we plot both the nmerical soltion and eact soltion at t =, 4, in Fig.4.3. The solid line represent the eact soltion and the * line is the nmerical soltion, we see they agree each other very well. This eample has been sed in [9]. Eample Zabsky-Krskal s problem [39] Consider the KdV eqation t + + (. ) =, (4.3)

37 34 with periodic bondary condition. The soltion is described by Zabsky and Krskal in 965. t = T B = /π is the breakdown time, and at t = 3.6T B 8 soltions is generated. Using scheme (4.8), (4.9), we perform comptations for t =./π, = 4, and we stop at the critical time t = 3.6/π. These three waves displayed in Fig.4.4 agree with the reslt given by Zabsky and Krskal in 965 see [39]. Net we compare or scheme with the leap frog scheme sed by Zabsky and Krskal n+ = n t 3 (n + + n + n )( n + n ) (4.4). t 3 ( i+ i+ + i i ). The nmerical reslt by (4.4) with small time step t = 5 5 π, = 4 is displayed in [39]. The waveform will blow p for πt >. By comparison we perform the comptation by or scheme (4.8), (4.9) ntil the final time t = /π, we display the nmerical reslt at time t = 9.9/π and t = /π in Fig.4.5. The nmerical reslt at t = 9.9 agree with the waveform given by the twelve-points scheme see [39], and at the break down time /π the waveform is still smooth by or scheme. And we notice the time step we se is t =./π while in twelve-point scheme t =.5/π and for the leap frog scheme t =.5/π, or time step is mch larger than the other two schemes.

38 35 (b)nmerical soltion at t=55.5 ( period) (d)nmerical soltion at t= ( period) (d)nmerical soltion at t=66.5 (3 period) (d)nmerical soltion at t= (4 period) Figre 4.: Nmerical eample 4.5., soltion plots at different times.

39 t Figre 4.: Nmerical eample 4.5., (,t)-contor of the soltion in the time interval (,5).4 (b)nmerical soltion at t= (c)nmerical soltion at t= (d)nmerical soltion at t= Figre 4.3: Nmerical eample 4.5., soltion plots at different times.

40 Figre 4.4: t =./π, = /4. we denote the initial wave by dotted line, denote the soltion for t = /π by dash-dot line, denote the soltion for t = 3.6/π by solid line Figre 4.5: t =./π, = /4, nmerical reslt displayed at t = 9.9/π and t = /π

41 38 CHAPTER 5. The Swift-Hohenberg eqation In this chapter we consider a class of diffsion eqations of the form t = α ( + ) + f(), D R, (5.) where f() = df () d. This eqation has been etensively stdied in literatres,see e.g.[8], [7]. One eample is the Swift-Hohenberg eqation t = α ( + ) 3, where α is a bifrcation parameter and the potential fnction F () = (α ) 4 4, then f() = df () d = (α ) 3. The Swift-Hohenberg eqation is first proposed in 977 by Swift and Hohenberg [3] and later has been applied to the problem of laser [3] and Taylor-Coette flow [5]. The swift- Hohenberg eqation is interesting in pattern formation; with qadratic and cbic nonlinearities, it ehibits many stable eqilibrim states. The pattern of the eqilibrim states depends on both the parameters and the spatial domain, see [38],[5],[7],[9],[8].

42 39 5. Algorithm eqation as design and energy analysis where We rewrite 5. as t = L + df d, (5.) L = = yyyy yy yy. Let (, t) be a smooth soltion of (5.), then sbect to the Dirichlet bondary condition = =, (, y) Ω. (5.3) The eqation (5.) admits a nonincreasing energy fnctional Ψ = { ( ) ( ) F ()}ddy, (5.4) dψ ( ) ddy dt = <. (5.5) t Ω Ω We epect the nmerical scheme preserve (5.5) at discrete level. We discretize (5.) in time to obtain n+ n t ( = L θ n+ + ( θ) n) + F (n+ ) F ( n ) n+ n, (5.6) where n = (t n,, y), t n = n t, θ is a parameter in (, ]. The se of parameter θ servers to smoothen the nmerical soltion n (). Second term of right hand side is the divided difference of df d in (5.6). We now show that for certain θ, the semi-discrete scheme (5.6) is stable. We smmarize the reslt in the following: Theorem 5... Consider the semi-discrete scheme (5.6) with any given potential fnction F(). ).If θ =, the scheme (5.6) is stable in the sense of Ψ[ n+ ] = Ψ[ n ( n+ n ) ] ddy Ψ[ n ]. (5.7) Ω t for arbitrary t. ). If θ >, then the scheme (5.6) is stable in the sense of Ψ[ n+ ] Ψ[ n ],

43 4 for all t θ. Proof. ) (5.6) can be rewritten as n+ n t + ( + + yyyy + yy + yy ) (θ n+ +( θ) n ) F (n+ ) F ( n ) n+ n =. (5.8) We mltiply n+ n on both sides and integrate over Ω, sing integration by parts, together with the admissible bondary condition = =, = yy =, to obtain Ω { ( n+ n ) t (θ n+ + ( θ) n )( n+ n ) + (θ n+ + ( θ) n )( n+ n ) (θ n+ y + ( θ) n y )( n+ y n y ) + (θ n+ yy + ( θ) n yy)( n+ yy n yy) } + (θ n+ y + ( θ) n y)( n+ y n y) (F ( n+ ) F ( n )) ddy =. We regrop the terms to obtain Ω { (n+ n ) t (( n+ ) ( n ) ) + ((n+ ) ( n ) ) + (θ ( ) ( n+ n ) ( n+ n ) ) (( n+ y ( + ( n+ y ) ( n y ) ) + ((n+ yy ) ( n yy) ) + (θ ( ) ( n+ yy n yy) ( n+ y n y ) ) ) ( n y) ) + (θ )(n+ y n y) [F ( n+ ) F ( n )]}ddy =. (5.9) By the definition of the energy fnctional Ψ we have { ( Ψ[ n+ ] Ψ[ n n+ n ) ] + Ω t + (θ ( ) ( n+ n ) ( n+ n ) ), + (θ ( ) ( n+ yy n yy) ( n+ y n y ) ), + (θ ) ( ( n+ y n y) } ddy =. (5.)

44 4 We see when θ = / the scheme satisfy the energy identity (5.7) at the discrete level. ) From (5.), we have { ( Ψ[ n+ ] Ψ[ n n+ n ) ] + Ω t + (θ ( ) ) ( n+ n ) + ( n+ n )( n+ n ) + (θ ( ) ) ( n+ yy n yy) + ( n+ yy n yy)( n+ n ) + (θ )(n+ y } n y) ddy =. We add and sbstract (θ )(n+ n ) to obtain { Ψ[ n+ ] Ψ[ n ] + ( Ω t θ + )(n+ n ) + (θ ( ) ) ( n+ n ) + ( n+ n ) + (θ ( ) ) ( n+ yy n yy) + ( n+ n ) We see when θ > / + (θ )(n+ y n y) } ddy. Ψ[ n+ ] Ψ[ n ] (θ t ) Ω ( n+ n ) ddy. Then we can conclde that when t θ, the energy fnctional Ψ is nonincreasing i.e. The proof is now complete. Ψ[ n+ ] Ψ[ n ]. 5. Nmerical Algorithms We consider the Swift-Hohenberg eqation t = α ( + ) 3, < < L, t >. (5.) We take θ = / as an eample. To implement (5.6) in each time step we need to apply an internal iteration and discretization in space.

45 4 ).Internal iteration [ + t ( α)] n,k+ + t n,k+ + t 4 ( n,k + n,k n + n) n,k+ = n t (( α) + + ) n t 4 3 n t n,k, (5.) where n,k+, n,k denote the k+ and k iteration step, respectively. ).Spatial discretization To discretize and we se central difference sch that n i n i+ 4n i+ + 6n i 4n i + n i 4, i n i+ n i + n i ( ). 5.3 Nmerical test We still consider (5.) with α =.3 and = = at =, = L. We se the initial data = Asin( π L ), A =. Since no eact soltion is available, we compare the nmerical soltion n with a reference soltion n, which is obtained by the same scheme taking smaller t =.. The L and L error is evalated by e L = e L =,. ) Accracy test. We perform this test for the optimal choice θ =, and the initial fnction we choose is = Asin( π L ), A =, and we take α =.3. We take small t =. and large grid cell nmber N = 6, and se the corresponding nmerical soltion as a reference soltion.

46 43 Table 5.: order of convergence of scheme (5.) N h L error order L error order L error order.3.88e E-3 -.8E E E E E-4.953E E E E E-4 - From this test we see the scheme is first order in space, which shows the nmerical convergence of the scheme. The second test we do is to show how the change of θ effects the stability of the scheme. We fi N =, tma = 4 (From nmerical test we see that the nmerical soltion reach steady state soltion from this time). We change θ and t then compare the soltion with the eact soltion in L norm. We consider θ =, t =. as the reference soltion. Table 5.: stability test for scheme (5.) t =. t =. t = θ = 6 nstable nstable nstable θ = 3 nstable nstable 8.8E-3 θ = 5 nstable nstable 9.77E-3 θ = 4.9E E-3 θ = E E E-3 θ = E-4.33E-3.5E- θ =.46E-4.594E-3.5E- This nmerical test is consistent with the energy analysis that when θ the nmerical scheme is stable. 5.4 Pattern formation In the following nmerical eperiment we will show how the parameters and the spatial domain effect the eqilibrim states. We consider the D Swift-Hohenberg eqation in the form of t = α ( + a) + β 3, [, L] (5.3)

47 44 where α, a, β and L are parameters. We slightly modify (5.) to obtain the nmerical scheme for (5.3) as following n+ n t = ( + α a ) n k+ + n a n k + n + 3 β(n k + n ) n k+ + 3 β(n ) 4 ((n k ) + n k n + ( n ) ) n k+ 4 (n ) 3. We se the same initial fnction = Asin( π L ), A =. We have the following nmerical reslt for one-dimensional Swift-Hohenberg eqation pattern formation. Case :First we fi a >, β > and L > and change α. From Fig.5. we see when α a trivial soltion appears. For α > the sitation is different. When < α <.5 a stable sine-like pattern appears and as α increases the period becomes less. The nmerical test tells s for < α <.55, 3.5 periods appear in [, ] and for.55 < α <.5,.5 periods appear. When α >.5, the pattern changes as shown in Fig.5.. Case : Keep other parameter vales constant: e.g. α =, β =, L = and vary smoothly the constant a. See Fig.5.3 and Fig.5.4. As a grows, sine-like periodic pattern is observed, which is symmetric and the period becomes smaller as a increases. Case 3: In this case we take random data as the initial vale to perform nmerical tests. We fi α =, a =, and vary coefficient β to see how the pattern changes. See Fig.5.5 and Fig.5.6..When < β < 6.55, still sine-like pattern appears, bt the symmetric property breaks as β increases..when 6.55 < β < 6.7. There is no stable pattern for β in this region. We take β = 6.58 as an eample, we observe for different patterns among tests performed starting for random initial data. 3. When 6.8 < β, the pattern retains stable shape, st the amplitde increases as β increases.

48 45 Finally, we fi all the parameters in eqation (5.3) to be α =, β =, a = to see how the pattern changes when we increase the comptational domain L. See Fig5.7, when L is small enogh, the soltion tends to trivial soltion; For L = 6 the final state is nontrivial and it does change sign on the interval (, L); When L become bigger, the soltion tends to trivial soltion again, after L = 9 the final profile has the stable sine-like pattern, the nmber of periods increases as L grows.

49 46 α=,β=,a=,l=,t=. α=.5,β=,a=,l=,t= α=,β=,a=,l=,t= α=.5,β=,a=,l=,t= α=,β=,a=,l=,t= α=.5,β=,a=,l=,t= Figre 5.: Swift-Hohenberg eqation pattern formation

50 47.4 α=,β=,a=,l=,t=.8 α=3,β=,a=,l=,t= α=3.5,β=,a=,l=,t= α=4,β=,a=,l=,t= Figre 5.: Swift-Hohenberg eqation pattern formation

51 48.4 α=,β=,a=,l=,t=.4 α=,β=,a=.3,l=,t= α=,β=,a=.5,l=,t= α=,β=,a=.7,l=,t= Figre 5.3: Swift-Hohenberg eqation pattern formation 3

52 49.5 α=,β=,a=.9,l=,t=4.5 α=,β=,a=.,l=,t= α=,β=,a=.5,l=,t= α=,β=,a=3.5,l=,t= Figre 5.4: Swift-Hohenberg eqation pattern formation 4

53 5 α=,β=,a=,l=,t=4 5 α=,β=4,a=,l=,t= α=,β=6.58,a=,l=,t= α=,β=6.58,a=,l=,t= α=,β=6.58,a=,l=,t= α=,β=6.58,a=,l=,t= Figre 5.5: Swift-Hohenberg eqation pattern formation 5

54 5 7 α=,β=6.8,a=,l=,t=4 8 α=,β=7,a=,l=,t= α=,β=8,a=,l=,t= α=,β=5,a=,l=,t= Figre 5.6: Swift-Hohenberg eqation pattern formation 6

55 5. α=.3,β=,a=,l=5,t=.. α=.3,β=,a=,l=6,t= α=.3,β=,a=,l=8,t= α=.3,β=,a=,l=9,t= α=.3,β=,a=,l=,t= α=.3,β=,a=,l=5,t= Figre 5.7: Swift-Hohenberg eqation pattern formation 7

56 53 CHAPTER 6. The Kramoto-Sivashinsky type eqation 6. Backgrond The Kramoto-Sivashinsky eqation t + + a + b =, (6.) has attracted considerable attention over the last decades. The well-poseness of eqation (6.) is proved in [3]. When the coefficients a and b are both positive, its second order term has a destabilizing effect and the forth order term has a stabilizing effect, with the nonlinear term providing a mechanism for energy transfer from low to high wavenmbers. It arises in varios contets and admits different types of soltions like traveling wave soltion, see [8],[9],[5],[37]. From a dynamical system point of view, it is an interesting PDE that can ehibit chaotic soltions. 6. Nmerical scheme for generalized Kramoto-Sivashinsky eqation We consider the general form of the Kramoto-Sivashinsky type eqation t + + (a() ) + (b ()(b() ) ) + (c() ) =, R, (6.) with an initial condition (, ) = (), R, (6.3) where a,b,c are continos fnctions of. We consider the problem on [,] with periodic bondary condition, mltiply both sides of (6.) by and integrate by parts over [,], obtain the identity d dt (, t)d a()( ) d + c()( ) d =. (6.4)

57 54 Net we will give a lemma to show that the identity (6.4) implies the L stability of soltion of (6.). Lemma 6... Assme a() M and c() r. Let be a smoooth periodic soltion to (6.), then the identity (6.4) implies (, t) e M r, (6.5) where denotes the L norm in L (, ). Proof. From (6.4), with the lower bond of c() and pper bond of a(), we have d dt = M r. (6.6) For periodic and smooth we apply the Cachy s ineqality with ɛ to obtain M 4r + r M. Ths (6.6) yields d dt M 4r, from which (6.5) follows. We propose the following scheme in order to preserve (6.4) at discrete level n+ n t + 6h (v + + v + v )(v + v ) + D + (a(v )D v ) +D (b (v )D + D b(v )) + D + D (c(v )D + D v ) =. (6.7) where v = n+ + n. This scheme for a =, b =, c = has been stdied in ([]). Net we will prove the nmerical scheme (6.7) indeed keep the identity (6.4) at discrete level which also implies the L stability of scheme (6.7). Lemma 6... Assme, =,,..., N is the soltion of eqation (6.7), and it is periodical, i.e. v +N = v then scheme (6.7) satisfies (( n+ ) ( n ) ) = t a(v )(D + v ) c(v )(D + D v ). (6.8)

58 55 Proof. Mltiplying both sides of (6.7) by the factor v = n+ + n, and sm this over the vales of we obtain n+ n = t/(3h) v (v + + v + v )(v + v ), + v D + (a(v )D v ) + v D (b (v )D + D b(v )) (v )D + D (c(v )D + D v ). First we compte v (v + + v + v )(v + v ) = v (v + ) v (v ) + (v ) v + (v ) v = N N+ + N N+. By applying the periodicity of i, we have = n, = n+, which give s N N+ + N N+ =. And by smmation by parts formla [6] and the periodicity assmption we have (v D + (a(v )D v )) = a(v )(D + v ), v D + D c(v )D + D v = c(v )(D + D v ). and we notice Σ v D (b (v )D + D b(v )) = Σ ( D v )b (v )D + D b(v ) = Σ ( D b(v ))D + D b(v ) = Σ (b(v +) b(v ))(b(v + ) b(v ) + b(v )) =. This completes the proof.

59 Nmerical tests (6.7). In this section we se Newton iteration to perform several nmerical tests for the scheme Eample : Accracy test. We compte the Kramoto-Sivashinsky eqation t =, (6.9) with the initial condition (, ) = c ( 9tanh(k( )) + tanh 3 (k( ))). The eact soltion is (, t) = c ( 9tanh(k( ct )) + tanh 3 (k( ct ))). This nmerical eample has been sed in [36]. From (6.7) we have the nmerical scheme for (6.9) n+ n t + 6h (n+/ + + n+/ +D + D n+/ + n+/ )( n+/ + n+/ ) (6.) + D + D D + D n+/ =. Table 6.: C = 5, k = t =. 9, =. Newmann bondary condition. Uniform meshes with N cells at time N L order L order E E-.69.7E-..3E E-..79E-. We display the nmerical reslt in Fig.6.. From the graph we see or nmerical scheme sccessflly captres the traveling wave soltion of the KS eqation, and from Table (6.) we see we have a clean second order scheme

60 57 7 t= 7 t= t=4 7 t= Figre 6.: Plots of the comptational reslt for the KS eqation by scheme 6.9 on [-3,3], N=3, T=6 in space. Eample : Long time simlation We perform the nmerical test for the eqation t σ + =. (6.) The eact soltion is (, t) = c+9 5(tanh(k( ct ))+tanh (k( ct )) tanh 3 (k( ct ))), (6.)

61 58 and initial condition is (, t) = c + 9 5(tanh(k( )) + tanh (k( )) tanh 3 (k( ))), (6.3) where c =, k = /, σ = 4 From (6.7) we have the nmerical scheme for (6.) n+ n t + 6h (n+/ + + n+/ + n+/ )( n+/ σd D + D n+/ + n+/ ) + D + D n+/ + + D + D D + D n+/ =. (6.4) Table 6.: Newmann bondary condition. Uniform meshes with N cells at time t =.. N L order L order E E- -.4E-.98.7E E-..89E E E-3.99 We notice the soliton has an advection speed of, we take the comptational domain to be [, ], a period of the soliton is =, we take cells t =. and display the soliton p to 4( periods), (5 periods) and ( periods) in (6.), we see the scheme (6.4) is statble in long-time simlation with large time step. There is also phase error when the simlation time is long de to the nmerical error of the second order scheme, we can redce the error by making finer grid cell or taking smaller time step t. Eample 3: Another type K.S. eqation We perform a nmerical test for the eqation t + + =, (6.5) with the initial condition The eact soltion (, ) = c ( 3tanh(k( )) + tanh 3 (k( ))). (6.6) (, t) = c ( 3tanh(k( ct )) + tanh 3 (k( ct ))), (6.7)

62 59 5 (b)nmerical soltion at t= 5 (b)nmerical soltion at periods t= (c)nmerical soltion at 5 periods t= 5 (d)nmerical soltion at periods t= Figre 6.: Plots of the comptational reslt for KS eqation (6.) where c =.8, k = 9, =. n+ From (6.7) we have the nmerical scheme for (6.5) n t + 6h (n+/ + + n+/ + n+/ )( n+/ + n+/ ) D + D n+/ +D + D D + D n+/ =. (6.8) And we take Newmann bondary condition. The shock profile wave propagation for eqation is displayed in (6.3), we see the moving shock profile is resolved very well.