ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KLEIN-GORDON-SCHRÖDINGER EQUATION

Transcription

1 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KLEIN-GORDON-SCHRÖDINGER EQUATION JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, Abstract. In tis paper, we study te multi-symplectic metod for te Klein- Gordon-Scrödinger equation. Besides conserving te multi-symplectic structure of te equation, te multi-symplectic metod is also investigated for te conservation of carge, local energy, and total energy under corresponding discretization numerically. We deduce te transit law of te specific formal energy for te metod. Based on tese, we give te error estimations of te numerical solutions teoretically and present te results tat te numerical solutions converge to te exact solutions for proper norms. In order to testify te superiority of te multi-symplectic metod, it is compared wit some non-multi-symplectic metod. Numerical experiments sow tat te multi-symplectic metod preserves te carge conservation law precisely, and conserves oter conservation laws in better magnitude tan te non-multi-symplectic metod. Keywords: Error analysis; conservation laws; multi-symplectic sceme; Klein-Gordon- Scrödinger equation.. Introduction During te last few years, te multi-symplectic metods ave been proposed and developed for some famous Hamiltonian partial differential equations, suc as Dirac equations [], wave equations [], etc, for teir perfectly preserving te geometry structure of equations. In tis paper, we pay our attention to te Klein-Gordon-Scrödinger equation, wic describes a system of conserved scalar nucleons interacting wit neutral scalar mesons. Te Klein-Gordon-Scrödinger equation is one of important matematical models in quantum pysics, for it consists of one Klein-Gordon equation and one Scrödinger equation. Since Fukuda and Tsutsumi in [] gave te first teoretical studies for Klein- Gordon-Scrödinger system, a great amount of work as been done on existence of solutions, stability [9] and asymptotic beavior []. But up to now, few numerical metods and simulation proposed for te Klein-Gordon-Scrödinger system as we know. In [], Zang presented a conservative difference sceme for a class of Klein- Gordon-Scrödinger equations, and present te convergence of te metods. It as only teoretical analysis tere, but no numerical experiments. In Bao and Yang [], te autors present spectral metods for Klein-Gordon-Scrödinger equation. Kong in [8] gives te multi-symplectic metods for Klein-Gordon-Scrödinger equation, and discusses te carge conservation law teoretically. In tis paper, we investigate te Te first autor is supported by te Director Innovation Foundation of ICMSEC and AMSS, te Foundation of CAS, te NNSFC (No.99789, No.78) and te Special Funds for Maor State Basic Researc Proects of Cina 5CB7.

2 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, multi-symplectic metods for Klein-Gordon-Scrödinger system, wic is different from te angle of [, 8, ]. Te organization of tis paper is as follows. In te rest of tis section, we introduce te preliminary knowledge about Klein-Gordon-Scrödinger system and some pysical properties. In section, we present te teoretical analysis of te multi-symplectic integrators constructed by Runge-Kutta-type metods, and mention te preservation of te discrete multi-symplectic conservation law and te discrete carge conservation law under te mentioned discretization. Furtermore, we investigate te recursion reformulation of a specific formal energy for te multi-symplectic metod we present. In section, we focus on developing te error estimations of te numerical solutions for Klein-Gordon-Scrödinger equation and prove tat te numerical solutions converge to te teoretical solutions in some order. In section, we perform plentiful numerical experiments to testify te effectiveness of te multi-symplectic sceme listed in te previous section, and compare wit some non-multi-symplectic sceme, and give a detailed comparison for te numerical results obtained by different scemes. In te following we consider te standard Klein-Gordon-Scrödinger equation i t ϕ + (.) xxϕ + uϕ =, tt u xx u + u ϕ =, were ϕ(x, t) denotes a complex scalar nucleon field, and u(x, t) denotes a real scalar meson field, respectively. Moreover, i =, and we denote te spatial and temporal direction by (x, t), respectively, and x R, t. We supplement (.) by prescribing te initial-boundary value conditions for ϕ(x, t) and u(x, t) wit: (.) ϕ t= = ϕ (x), u t= = u (x), u t t= = u (x), lim x ϕ =, lim x u =, were, ϕ (x), u (x), and u (x) are known smoot functions. If we set te complex valued function ϕ(x, t) = q(x, t) + ip(x, t), and introduce new variables g = x q, f = x p, v = t u, w = x u, and state variable z = (q, p, g, f, u, v, w) T, te above equation (.) can be reformulated into te multi-symplectic Hamiltonian PDEs form (.) M t z + K x z = z S(z), were M and K are two skew-symmetric matrices (wic can be singular), M =, K =,

3 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION and S(z) is a real smoot function of te state variable z, wic for system (.) is in te form S(z) = u(q + p ) + (g + f u v + w ). It is well-known tat one of most interesting properties for (.) is tat it possesses te multi-symplectic conservation law (.) were ω = dz Mdz, t ω + x κ =, κ = dz Kdz, wic represents te geometric caracter on te pase. It can be easily seen tat (.) is a natural extension of conservation of symplecticity from Hamiltonian ODEs to Hamiltionian PDEs, wic indicates tat symplecticity can vary over te spatial domain and from to, and tis variation is not arbitrary as te canges in are exactly compensated by canges in (see [7]). Specially, (.) can be rewritten into te equivalent form (.5) t [dq dp + du dv] + x [dg dq + df dp + dw du] = for te Klein-Gordon-Scrödinger equation (.). A more spiriting fact is tat tere exist several local conservation laws along wit te multi-symplecticity of Hamiltonian PDEs, wic are local energy conservation law (.6) were t E(z) + x F (z) =, E(z) = S(z) zt K x z, F (z) = zt K t z, represent energy density and energy flux, respectively, and local momentum conservation law (.7) were t I(z) + x G(z) =, G(z) = S(z) zt M t z, I(z) = zt M x z. represent momentum density and momentum flux, respectively. And for te equation considered, E(z) = u(q + p ) + ( u v + qq xx + pp xx + uu xx ), F (z) = (gq t + fp t qg t pf t uu xt + wv), G(z) = (uq + up ) + (g + f u + w ) qp t + pq t uu xt, I(z) = (qf pg) + (uu xt vw).

4 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, Integrating te local energy conservation law (.6) over te spatial domain, and recalling te boundary conditions in (.), we derive te conservation of total energy d (.8) dt E(z) := d E(z)dx =, dt were E(z) = R ((u + v + w + f + g ) u(q + p ))dx. Tis is one of important global conservative quantities for Klein-Gordon-Scrödinger equation (.). Furtermore, by a straigt calculation, we ave te following proposition. It gives one intrinsic conservative quantity for Kilein-Gordon-Scrödinger equation itself. Proposition.. Te solution ϕ of te initial-boundary value problem (.)-(.) satisfies te carge conservation law (.9) ϕ(x, t) dx = ϕ (x) dx = C, were, C is a constant. R In te subsequent section, we will turn attention to applying some multi-symplectic sceme to te Klein-Gordon-Scrödinger equation and investigate some discrete conservative quantities under te numerical discretization. R R. Te special multi-symplectic sceme and its conservative laws In tis section, we focus on investigating te central box sceme, wic is multisymplectic. For convenience, te following notations are used: δ t V k+ = τ and k+ (V V k ), (U, V ) = U V, δt V k = τ k+ (V V k := V k +V k ), δxv k = (V + V k k +V ), k V k, V k := V k. Here, we coose as te spatial mes grid-size, and τ as te stepsize, respectively. And introduce a uniform grid (x, t k ) R, ten te approximation of te value of function V (x, t) at te mes grid (x, t k ) is denoted by V k. Wen we apply te central box sceme to te equation (.), ten ave te following numerical sceme u k+ + u k + = τv k+, + (.) u k+ + uk+ = w k+, + q k+ + qk+ = g k+, + p k+ + pk+ = f k+, +

5 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 5 (.) v k+ + v k + τ q k+ q k + + τ p k+ p k + + τ wk+ + wk+ + + u k+ + f k+ + f k+ g k+ + gk+ ((p k+ + + u k+ p k+ + + u k+ q k+ + + =, =, ) + (q k+ ) ) =, + in wic, u k + (vk+ = (uk + uk + ), wk + +v k), qk+ = (qk+ +q k), pk+ (gk + gk + ). Ten we give te following expression (.) E k+ + = = (wk + wk + ), uk+ E k+ + E k + τ = (pk+ +p k ), f k + + F k+ + F k+ = (uk+ + u k ), vk+ = = (f k +f k + ), gk + to denote te residual of local energy conservation law, toug te multi-symplectic sceme can not preserve te corresponding discretization of local energy conservation law (.6) exactly. In te numerical experiments, we will calculate suc residual numerically, for it is an important sign for te local property of te multi-symplectic metods. Similarly, we denote te discrete total energy (.8) by = (.) E k+ = E k+ + for use in te follows. Using te similar straigt calculations in reference [], we can obtain te following discrete multi-symplectic conservation law τ(dg k+ + dqk+ + + df k+ + dpk+ + + dwk+ + duk+ + ) (.5) τ(dg k+ dq k+ + df k+ dp k+ + dw k+ du k+ ) + (dq k+ dp k+ + du k+ dv k+ ) (dq k dp k + + du k + dv k ) + = wic is te corresponding discretization version of (.5). Ten we ave te following teorem (see [,, 5] and references terein): Teorem.. Te implicit midpoint discretization (.)-(.) is multi-symplectic, and it satisfies te discrete multi-symplectic conservation law (.5). It is well-known tat te discrete multi-symplectic conservation law is te discrete versions of te geometric property on te pase. However, tis can not be verified numerically.

6 6 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, If te introduced variables v, w, f, g are eliminated, te multi-symplectic sceme (.)-(.) can be reformulated as (.6) (.7) τ (qk+ q k ) τ (qk+ q k ) + (pk+ + pk+ + p k+ ) + u k+ p k u k+ p k+ =, τ (pk+ p k ) τ (pk+ p k ) + (qk+ + qk+ + q k+ ) u k+ q k+ + + u k+ q k+ =, (.8) τ (uk+ u k + u k + u k+ u k u k + uk + u k ) + uk+ + u k + + u k+ + + u k + u k ) (uk+ + uk+ + u k+ ϕ k+ + ϕ k + (.9) ϕ k+ ϕ k =. Combining (.6) wit (.7), we obtain i τ (ϕk+ ϕ k + ϕ k+ ϕ k ) (ϕk+ + ϕk+ + ϕ k+ ) + u k+ ϕ k u k+ ϕ k+ =. Ten we derive te following numerical sceme (.) i(δ t ϕ k+ + + δ t ϕ k+ ) + δ xϕ k+ + u k+ + ϕ k+ + + u k+ ϕ k+ = (.) (δ t u k + ϕ k+ + + δt u k ) (δ xu k+ + δxu k ) + (u k+ + u k + + u k+ + ϕ k + ϕ k+ ϕ k =. by using te notations defined previously. Next, we turn to te discrete carge conservation law firstly. + u k ) Teorem.. Te multi-symplectic sceme (.)-(.) possesses te discrete carge conservation law, i.e. (.) ϕ k = Constant. Proof. We first multiply bot sides of (.) by ϕ k+, i.e. ϕ k+ + ϕ k, were ϕ denotes te complex conugate of ϕ. It follows from te first term tat (.) i τ (ϕk+ + ϕk ϕ k ϕ k+ + ϕ k+ ϕk ϕ k +ϕ k+ + ϕ k+ ϕ k ϕ k ϕ k+ + ϕ k+ + ϕ k+ + ϕ k+ ϕk+ + ϕ k+ ϕ k+ ϕ k +ϕ k ϕ k ϕ k ϕ k ϕ k ).

7 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 7 Te second term becomes (.) (ϕk+ + ϕk+ + ϕ k+ Similarly, te rest of left side yields (.5) u k+ ϕ k+ + + ϕk+ + u k+ ϕk+ ϕ k+ ϕk+ ϕ k+ + u k+ + ϕ k+ ). ϕ k+ ϕ k+ + u k+ ϕ k+ ϕ k+. summing over all spatial grid points, and taking te imaginary part, we can find tat (.) and (.5) bot become real functions, and it follows from (.) tat i τ [(ϕ k+ + ϕk+ + ϕ k+ ϕ k+ + ϕ k+ ϕk+ ) (ϕ k +ϕ k + ϕ k ϕ k + ϕ k ϕ k )], wit te zero boundary conditions. From te analysis above, we can furter deduce (.6) ϕ k+ = ϕ k+ ϕ k+ + + = ϕ k ϕ k + + = ϕ k = = ϕ. Tis completes te proof. It can be seen evidently tat Teorem. is consistent wit te carge conservation law (.9). And it means tat te carge conservation law of KGS equation can be preserved by our multi-symplectic sceme exactly. In general, multi-symplectic Runge- Kutta type metods ave te stability in te sense of te carge conservation law for multi-symplectic Hamiltonian PDEs, wic can be referred to []. As a quadratic invariant te carge conservation law plays a very important part in quantum pysics, owever, it can not be preserved by symplectic integrators in te case of Hamiltonian ODEs, and tis is ust one of interesting tings to introduce multi-symplectic metods. Te next result concerns te error estimation of te discrete total energy, and for te reason of te nonlinear terms in equation(.), te multi-symplectic metod can not preserve te discrete total energy conservation law exactly. Lemma.. Under te zero boundary conditions, te multi-symplectic sceme (.)- (.) satisfies te following two implicit discrete global conservation laws, i.e. [( ϕk+ + ϕk+ )( ϕk+ + ϕk+ ) ( ϕk + ϕk )( ϕk + ϕk )] (.7) (u k+ + ϕ k+ ϕ k+ + + u k + ϕ k + ϕ k ) + = R(u k + ϕ k ϕ k+ + + u k+ + ϕ k+ + ϕ k ). +

8 8 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, Here, R stands for real part. u k+ u k u k + [( + ) + ( τ u k + ) ] + τ [(u k+ + ) (u k ) ] + (.8) + = [( uk+ + uk+ (ϕ k+ ϕ k+ + + ) ( uk + uk ) ] + ϕ k + ϕ k )(u k+ + + u k ). + Proof. We can get from (.) tat i(δ t ϕ k+ + + u k+ ϕ k δ t ϕ k+ + δ t ϕ k + + u k+ ϕ k+ + δ t ϕ k ) + (δ xϕ k+ + δxϕ k ) + u k ϕ k u k ϕ k We multiply te above equation by ϕ k+ ϕ k, sum over all spatial grid points, and take te real part. Wit zero boundary conditions, te first term becomes imaginary functions. It follows from te second term tat (.9) = R((δ xϕ k+ + δ xϕ k )(ϕ k+ ϕ k )) + = R(ϕ k+ + ϕk+ ϕ k+ ϕ k+ + ϕ k+ ϕk+ ϕ k + ϕk + ϕ k ϕ k ϕ k ϕk ) R(ϕ k + ϕk+ ϕ k+ ϕ k + ϕ k ϕk+ ϕ k+ + ϕk + ϕ k+ ϕ k ϕ k+ ϕk ) [ ( ϕk+ + ϕk+ )( ϕk+ + ϕk+ Obtained from te rest terms in (.) tat (.) R((u k+ ϕ k u k+ ϕ k+ + u k ϕ k + + ) + ( ϕk + ϕk + u k =. )( ϕk + ϕk )]. ϕ k )(ϕ k+ ϕ k )) = R((u k+ ϕ k u k + ϕ k )(ϕ k+ + + ϕ k )) + J = = [(u k+ + ϕ k+ ϕ k+ + + u k + ϕ k + ϕ k + ) + R(u k + ϕ k ϕ k+ + + u k+ + ϕ k+ + ϕ k )]. +

9 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 9 Combining (.9) wit (.), we finally get J = [( ϕk+ + ϕk+ J = J = = (u k+ + R(u k + ϕ k+ ϕ k+ + + )( ϕk+ + ϕk+ ϕ k ϕ k+ + + u k + u k+ + ϕ k + ) ( ϕk + ϕk ϕ k+ + ϕ k ) + ϕ k ). + )( ϕk + ϕk )] Similarly, we multiply (.) by u k+ u k, sum over all spatial grid points. By making use of zero boundary conditions, it follows from te first term tat (δ t u k + + δ t u k )(u k+ u k ) (.) = τ = [(u k+ + + uk+ u k+ u k u k + [( + ) + ( τ u k + u k ) (u k + + u k u k + uk ) ] u k + ) ]. τ Te second term becomes (.) (δxu k+ + δxu k )(u k+ u k ) = [( uk+ + uk+ ) ( uk + uk ) ]. Te tird term yields (.) (u k+ + + u k + + u k+ + u k )(u k+ u k ) = [(u k+ + ) (u k ) ]. + Te rest terms read (.) = ( ϕ k+ + + ϕ k + (ϕ k+ ϕ k ϕ k+ + ϕ k + + ϕ k )(u k+ u k ) ϕ k )(u k+ + + u k ). +

10 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, Combining (.), (.), (.) and (.), we get u k+ u k u k + [( + ) + ( τ u k + ) ] + τ [(u k+ + ) (u k ) ] + + = [( uk+ + uk+ (ϕ k+ ϕ k+ + + ) ( uk + uk ) ] + ϕ k + ϕ k )(u k+ + + u k ). + Tis completes te proof. According to Lemma., if we set te corresponding discrete total energy of te multi-symplectic sceme (.)-(.) at t k+ as (.5) J E k+ = [( ϕk+ + ϕk+ )( ϕk+ + ϕk+ u k+ u k + ) + ( + ) + ( uk+ + uk+ ) τ = + (u k+ ) u k+ + ϕ k+ + ϕ k+ + ], + we ave te following result about te transit of te discrete total energy in temporal direction. Furtermore, (.5) can be considered as te discrete version of te total energy conservation law (.8) at t k+, wen applying te multi-symplectic sceme (.)-(.) to system (.). Teorem.. For te multi-symplectic sceme (.)-(.), te transit of te discrete total energy in te temporal direction obeys te following law J (.6) E k+ E k = (u k+ + = u k ) ϕ k+ + + ϕ k. + Proof. Due to te implicit discrete global conservation laws (.7)-(.8) in Lemma., we obtain E k+ E k = [R(u k ϕ k + ϕ k+ + u k+ + ϕ k+ + ϕ k + ) + + (ϕ k+ ϕ k ϕ k + ϕ k )(u k+ + + u k )] + = (u k+ + u k ) ϕ k+ + + ϕ k. + Tis completes te proof. Te discrete total energy at t k+ is te specific property for te multi-symplectic sceme we present, for its special geometry structure.

11 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION. Error estimations for te multi-symplectic sceme In tis section, we investigate te properties of te multi-symplectic sceme (.)- (.) furter, and calculate its convergence, wic is te most important sign of te effectiveness of te numerical metods. For te convenience of readers, We set truncation errors of sceme as (.) r k+ =i(δ t ϕ(x + + u(x +, t k+ ) + δ t ϕ(x )ϕ(x +, t k+ )) + δxϕ(x ) ) + u(x )ϕ(x ), (.) σ k = (δ t u(x + + u(x + ϕ(x +, t k, t k ) + δ t u(x ) + u(x ) ϕ(x +, t k )) (δ xu(x ) + u(x, t k Let global errors of solutions be denoted by, t k ) ) ϕ(x ) + δ xu(x, t k ) ϕ(x e k = ϕ ( x, t k ) ϕ k, η k = u ( x, t k ) u k. )) + u(x +, t k ), ) It can be verified by Taylor expansion tat te truncation errors of te multisymplectic sceme (.)-(.) are O(τ + ), i.e. (.) r k+ = O(τ + ), σ k = O(τ + ). According to te relationsip of recursion, we can obtain te following expression (.) r k+ = (rk+ + + r k+ ) + O(τ + ), σ k = (σk + σ k ) + O(τ + ). + Ten we ave te following global error estimations for te multi-symplectic scemes (.)-(.). Teorem.. For te te multi-symplectic sceme (.)-(.), te following estimates old: (.5) { δ t η N+, δ x η N+, η N+, e N } {( + Cτ)( δ t η + δ x η + τ N [ r k+ k= were, τ is sufficiently small, suc tat τ C. + η + e + e ) + r k + σk + O(τ + ) ]} exp NCτ, Proof. Firstly, it follows from subtracting (.) from (.) tat (.6) r k+ = i(δ t e k+ + + u(x + δ t e k+ ) + δ xe k+ )ϕ(x + u(x + ) u k+ ϕ k+ + + )ϕ(x + u k+ ϕ k+. )

12 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, Multiplying (.6) by ē k+, summing over all te spatial grid points and taking te imaginary part, we obtain τ ( ek+ e k ) = I (r k+ ē k+ ) I (u(x + )ϕ(x + ) u k+ + ϕ k+ + )ē k+. + Here, I stands for imaginary part. Using (.7) τ ( ek+ r k+ ē k+ = (r k+ + e k ) = I + O(τ + ))ē k+, we ave + (r k+ + + O(τ + ))ē k+ + I (η k+ + ϕ k+ + )ē k+. + It follows from Teorem., tere exists a constant suc tat C = max( ϕ k+ ), =,...; k =,..., N + (.8) e k+ e k τ[ r k+ τ[ r k+ + ek+ + C η k+ + C e k+ + O(τ + ) ] + C η k+ + (C + )( ek+ + e k ) + O(τ + ) ]. On te oter and, substituting (.) into (.), we ave (.9) σ k = (δ t η k + ϕ(x + + ϕ k+ + + δt η k ) (δ xη k+ + δxη k ) + η k+ + η k + + η k+ + + η k + ϕ k + ) ϕ(x + + ϕ k+, t k ) ϕ(x + ϕ k ) ϕ(x, t k ) Multiplying (.9) by τ (ηk+ η k ), and summing over, one as τ ( δ tη k+ δ t η k + δ x η k+ δ x η k + η k+ = σ k (δ tη k+ + δ t η k ) η k ) + ( ϕ(x + ) + ϕ(x +, t k ) ϕ k+ + ϕ k )(δ + t η k+ + + δ t η k ). +

13 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION From σ k(δ tη k+ + δ t η k ) = (σ k + + O(τ + ))(δ t η k+ + + δ t η k ), we obtain + (.) τ ( δ tη k+ δ t η k + δ x η k+ δ x η k + η k+ η k ) = (σ k + + O(τ + ))(δ t η k+ + + δ t η k ) + + (ϕ k+ ē k+ + + e k+ + + ϕ(x + ) + ϕ k ē k + + e k + + ϕ(x +, t k ))(δ t η k+ + + δ t η k ). + Recalling tat te exact solution ϕ(x, t) of te system (.) is bounded, i.e. tere exists a constant C suc tat Ten, we derive C = max( ϕ(x, t) ). δ t η k+ δ t η k + δ x η k+ δ x η k + η k+ η k (.) τ[ σk + 8 (C + C )( e k+ + e k + e k ) + (C + C + )( δ tη k+ + δ t η k ) + O(τ + ) ]. Combining (.8) wit (.), we get te inequality (.) and Define δ t η k+ δ t η k + δ x η k+ δ x η k + η k+ η k + e k+ e k + e k e k τ[ r k+ + r k + 8 (5C + C + )( e k+ +(C + C + )( δ tη k+ Te above becomes + σk + e k W k = δ t η k+ + δ x η k+ + δ t η k + C ( η k+ ) + e k + η k ) ) + O(τ + ) ]. + η k+ + e k+ C = max{c, 8 (5C + C + ), (C + C + )}. W k W k Cτ(W k + W k ) + τ[ r k+ Terefore, N W N Cτ W k +{(+Cτ)W +τ k= N [ r k+ k= By Gronwall s inequality [6], we obtain tat + e k, + r k + σk + O(τ + ) ]. + r k + σk +O(τ + ) ]}.

14 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, W N {( + Cτ)W + τ N [ r k+ k= + r k + σk + O(τ + ) ]} exp NCτ, were, τ is sufficiently small, suc tat τ C. Recalling tat te definition of W k, we ave te following estimations: { δ t η N+, δ x η N+, η N+, e N } {( + Cτ)( δ t η + δ x η + τ N [ r k+ k= Tis completes te proof. + η + e + e ) + r k + σk + O(τ + ) ]} exp NCτ. We d like to empasize ere tat te estimates of te solutions are bounded by te initial values and te local truncation errors of solutions. Corollary.. Te numerical solutions of te multi-symplectic sceme (.)-(.) converge to te solutions of system (.) wit order O(τ + ), i.e. e N O(τ + ), η N+ O(τ + ). Proof. Recalling tat te initial values e, e, η, η is at least two order precisely, (.) and (.), it yields (.) δ t η N+ + δ x η N+ Here, C is a bounded constant. Ten we derive (.) δ t η N+ + η N+ + e N+ + e N O(τ + ), δ x η N+ η N+ O(τ + ), e N Tis completes te proof. O(τ + ), O(τ + ). C O(τ + ). Lemma.. (Discrete Sobolev inequality) For any discrete function u = {u =,,...} in te real axis and for any given ε >, tere exists a constant K dependent on ε suc tat u ε δu + K u. Utilizing Teorem., corollary. and Lemma., we can obtain te following corollary. Corollary.. Te numerical solution of te multi-symplectic sceme (.)-(.) converges to te solution of te system (.) wit order O(τ + ) in te L norm for u k+. + Te results of te numerical experiments are listed in te following section.

15 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 5. Numerical experiments Te purpose of tis section is to testify te numerical performance of te multisymplectic sceme presented in Section. (.) We first concentrate on te solitary-wave solutions of system (.) ϕ(x, t, q) = q sec q (x qt) exp(i(qx + q + q ( q ) t)), u(x, t, q) = ( q ) sec (x qt), q were, q indicates te propagating velocity of wave. Te initial values ϕ (x) = ϕ(x,, q) u (x) = u(x,, q), v (x) = u t (x, t, q) t= are obtained from (.) as t =. In te following for te sake of calculation, we set te spatial domain to be considered is te interval [ L, L], wit L =, and assume zero boundary conditions, i.e., ϕ( L, t) = ϕ(l, t) =, u( L, t) = u(l, t) =. Furtermore, we set a series of notations as (.) (error ϕ ) k = ( (error u ) k+ = max e k ) +, (erroru ) k+ = ( η k+, + to sow L errors and L error of solutions. And as stated in section, we make use of η k+ ) +, (N ) k = ϕ k to represent te discrete carge conservation law, and let (.) (error c ) k = (N ) k (N ) to denote te global error of discrete carge conservation law (.9). And we also denote te error propagation of discrete total energy as (.) (Error e ) k = E k E. In order to illustrate te perfect beavior of te multi-symplectic sceme (in sort, SMID), we introduce anoter implicit non-multi-symplectic metods of order (in sort TRAP), i.e trapezoidal metod in bot directions of and for Klein-Gordon- Scrödinger equation. Ten we can also define all te above notations of errors at te point (x, t k ) for TRAP.

16 6 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, Fig presents te evolution of te numerical solutions ϕ, u for solitary-wave, wen applying SMID to simulate te solutions ϕ(x, t,.), u(x, t,.) in te interval[,8]. If we don t empasize in te following, te computing conditions are all as tis. Table - sow te errors and te experimental order of accuracy for SMID and TRAP respectively, in L and L norms at T =. Looking at te table, it can be observed tat te numerical results are consistent to te teoretical results of Teorem. and its corollaries. Te numerical solution u is of order O(τ + ) in bot L norm and L norm. And te numerical solution ϕ is also of order O(τ + ) in te L norm. Table indicates tat te errors of numerical solutions for TRAP are in te same order wit SMID. In Fig - we sow te long beavior of te numerical solutions ϕ(x, t,.), u(x, t,.) in te interval [, ] wit te temporal stepsize τ =.5 and spatial mesgrid-size =.. Fig presents te errors of carge conservation law. As we ave proved in te Teorem. teoretically, SMID preserves te carge conservation law exactly in te scale of, and TRAP simulates te te carge conservation law in te scale of 6. Ten, we set max E k+ to denote te maximum error of te + residual of local energy conservation law (.), and its variation is exibited in Fig. Te two scemes present te similar numerical beavior and te variation does not ave evident increment for a long. Te global errors of total energy presented in Fig are similarly to. However, SMID is superior to TRAP in one or two magnitude for bot of local energy conservation law and te total energy. Fig 5 presents errors of discrete carge conservation law wit te cange of temporal stepsize τ and spatial mesgrid-size. And wen tey cange, tere is no evident difference in te errors for SMID. We can read tat SMID preserves te discrete carge conservation law accurately in long beavior, wic can be seen in te diagram are in te scale of. Toug te graps exibit te trend of accumulation of error, te amplifying velocity is extremely slower tan te magnitude of te error. But for TRAP it can not conserve te carge conservation law, and te error is reduced to about 6 rougly, as bot te mesgrid-size and te stepsize are reduced to. In Fig 6, we list te maximum errors of local energy conservation law wit different temporal stepsize and spatial mesgrid-size. We d like to point out ere from te data in te numerical experiments tat te maximum errors approximately reac te magnitude of O( + τ ) for bot SMID and TRAP. However, te results of SMID are better tan TRAP. In Fig 7, from te cange process it can be observed tat te global error of total energy for SMID rougly reaces te magnitude of O( τ ) at least. If we reduce spatial mesgrid-size or temporal stepsize to alf, te error is reduced to about also, similarly, if we reduce bot of spatial mesgrid-size and temporal stepsize to alf, te error is reduced to about 6 correspondingly. And for TRAP, te global error of total

17 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 7 energy rougly reaces te magnitude of O( ), since tat te cange of error seems not depend on te cange of temporal stepsize. In all, te errors of SMID are about tree magnitude better tan tose of TRAP. All te graps in Fig 6-7 exibit pretty performance, and wit oscillation of little amplitude around some equilibrium state, and ave no any evident amplification for a long, wic indicate tat te multi-symplectic sceme preserves te conservation laws perfectly in long- beavior, toug it can not preserve tem exactly. Now, we turn to te collision of two solitons. Te corresponding initial values are given as ϕ (x) = ϕ (x x,, q ) + ϕ (x x,, q ), u (x) = u (x x,, q ) + u (x x,, q ), v (x) = {u t (x x, t, q ) + u t (x x, t, q )} t=, were, x, x are initial pases, q, q are propagating velocities of two solitons, respectively. Case: Collision of two solitons wit same speed and opposite direction: x = 5, x = 5 and q =.7, q =.7, respectively, and te inverval [,]. In tis case, we take τ =. and =.. Fig 8 sows collision of two solitons in case by applying SMID to te equation at various s. Left column represents te evolution of ϕ, and rigt column represents te evolution of u. Te two solutions stay at te same position at around t =. (tird row), and te solutions keep in tis state after te and result in fusion, and are accompanied by a series of emission of waves (below row). In Fig 9, we give te corresponding global errors of carge conservation law, total energy and local energy conservation law for collision of two solitons in case. It can be observed tat te error of conservation law is in te scale of for SMID, wile for TRAP in te scale of. And te errors of total energy and local energy conservation law cange larger during collision, and exibit pretty performance wit oscillation of little amplitude around some equilibrium state, and ave no any evident amplification for a long after collision for bot SMID and TRAP. In general, SMID is better tan TRAP. In Fig, te evolutions of numerical solutions in case, wen applying te multisymplectic sceme, is exibited. Case: Collision of two solitons wit different speed and opposite direction, for we cose x = 5, x = 5 and q =.8, q =.6, respectively, and te interval [,]. In tis case, we set τ =. and =.. Fig sows collision of two solitons in case at various s. Left column represents te evolution of ϕ, and rigt column represents te evolution of u. From te

18 8 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, graps, we can see tat two soliton propagate in te opposite directions, and te two solutions stay at te same position at t =.6 (tird row), after collision tey propagate in teir original directions again. Te amplitude is becoming bigger for te soliton wit larger amplitude originally, and it becomes smaller for te oter. It means tat te soliton wit larger amplitude absorbs part of te oter during te collision. In Fig, we list te corresponding global errors of carge conservation law, total energy and local energy conservation law for collision of two solitons in case. It can be seen tat te error of conservation law is in te scale of for SMID. And oters exibit analogus performance as in Fig 9. In Fig, te evolutions of numerical solutions in case is revealed wen applying te multi-symplectic sceme. Table. Numerical results for SMID τ \ L error of ϕ L order of ϕ L error of u L order of u L error of u L order of u.\..9777e e e \ e e e-.68.5\..98e e-.87.6e \ e-.5569e-.7e- Table. Numerical results for TRAP τ \ L error of ϕ L order of ϕ L error of u L order of u L error of u L order of u.\..8775e e e \..898e e e-.99.5\..78e e-.75.8e \.5.5e-.7766e-.8e- 5. Conclusion We detailed te properties of te central box sceme, wen applied it to te Klein- Gordon-Scrödinger equation. Including te multi-symplectic geometry structure, te preservation of carge, total energy, local energy conservation law and formal energy of te Klein-Gordon-Scrödinger equation itself. Te results are as follows.. Teorem. indicates tat SMID possesses te discrete carge conservation law under zero boundary conditions.. Teorem. sows tat SMID as specific formal energy wic can be seen as a discrete version of energy at t k+.. Teorem. gives several estimate of te solutions wic is bounded by te initial values of solutions and te local truncation errors. Furtermore, Corollary. sows tat te numerical solutions of SMID approximate te real solutions wit order O(τ + ) teoretically.. Numerical experiments implies tat SMID preserve te carge conservation law exactly, wic is consistent to te teoretical result. And from te graps numerically, it follows tat SMID preserves te total energy and te local energy conservation law in te magnitude of O( τ ) and O(τ + ), respectively.

19 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 9 Figure. Te evolution of numerical solutions: Real part (Top), imaginary part (Mid) of ϕ, and u (Below), respectively, wit τ =. and =.. 5. From te numerical comparisons, toug SMID as te same order as TRAP wen approximating te solutions of te Klein-Grodon-Scrödinger equation, it is superior to te oter not only on preserving te carge conservation law exactly, but also on preserving te total energy and local energy conservation law in te iger magnitude.

20 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, 6 x x Figure. Global errors of carge conservation law, i.e., (Error c ) k wit temporal stepsize τ =.5 and spatial mes-grid =., respectively. SMID(Left column); TRAP(Rigt column). 7.5 x x Figure. Te maximum errors of local energy conservation law, i.e., max E k+ wit temporal stepsize τ =.5 and spatial mes-grid + =., respectively. SMID(Left column); TRAP(Rigt column). 7 x x Figure. Global errors of total energy, i.e., (Error e ) k wit temporal stepsize τ =.5 and spatial mes-grid =., respectively. SMID(Left column); TRAP(Rigt column). All tese are te most interesting tings tat motivate us to discuss te multisymplectic metods ere. Acknowledgement: Autors would like to tank Dr. Kong for useful discussions.

21 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION x x x x x x Figure 5. Global errors of carge conservation law, i.e., (Error c ) k wit different spatial and temporal stepsize : above for τ =. and =.; Mid for τ =. and =.; Below for τ =. and =.. SMID(Left column); TRAP(Rigt column). References [] W. Bao, L. Yang, Efficient and accurate numerical metods for te Klein-Gordon-Scrödinger equations, preprint 5. [] T.J. Bridges, S. Reic, Multi-symplectic integrators: numerical scemes for Hamiltonian PDEs tat conserve symplecticity, Pys. Lett. A 8 (), 8-9. [] I. Fukuda, M. Tsutsumi, On te Uukawa-coupled Klein-Gordon-Scrödinger equations in tree dimensions, Prof. Japan Acad., 5 (975), -5. [] J. Hong, C. Li, Multi-symplectic RungeCKutta metods for nonlinear Dirac equations, J. Comput. Pys., (6), 8-7. [5] J. Hong, Y. Liu, H Munte-Kaas and A Zanna, Globally conservative properties and error estimation of a multi-symplectic sceme for Scrödinger equations wit variable coefficients, Appl. Numer. Mat., 56 (6), 8-8.

22 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI,.5 x x. x x x x x x x x Figure 6. Maximum errors of local energy conservation law, i.e., max E k+ wit different spatial and temporal stepsizes: Above for + τ =. and =.; Second for τ =. and =.; Mid for τ =. and =.; Fourt for τ =. and =.; Below for τ =. and =.. SMID(Left column); TRAP(Rigt column).

23 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 7 x x x x 6 7 x x x x x x Figure 7. Global error of total energy, i.e., (Error e ) k wit different spatial and temporal stepsizes: Above for τ =. and =.; Second for τ =. and =.; Mid for τ =. and =.; Fourt for τ =. and =.; Below for τ =. and =.. SMID(Left column); TRAP(Rigt column).

24 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI,.5 t=.5 t= t=7 t= t=. t= t=5.5 t= t= t= t=8 t= Figure 8. Te evolution of numerical solutions for soliton-soliton collison wit q =.7, q =.7, x = 5, x = 5 at various s. Left column: ϕ ; Rigt column: u.

25 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 5 x x x Figure 9. Global error of carge conservation, total energy and local energy conservation law for soliton-soliton collison wit q =.7, q =.7, x = 5, x = 5, wit spatial stepsize τ =. and temporal stepsize =.. SMID(Left column); TRAP(Rigt column). [6] A. Iserles, A First Course in te Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge 996. [7] A. L. Islas and C. M. Scober, On te preservation of pase structure under multisymplectic discretization, J. Comput. Pys. 97 () [8] L. Kong, R. Liu and Z. Xu, Numerical simulation of interaction between Scrödinger field and Klein-Gordon field by multisymplectic metod, Appl. Mat. Comput., 6 to appear. [9] M. Ota, Stability of stationary states for te coupled Klein-Gordon-Scrödinger equations, Non. Anal., 7 (996), [] T. Ozawa, Y. Tsutsumi, Asymptotic beaviour of solutions for te coupled Klein-Gordon- Scrödinger equations, Adv. Stud. Pure Mat. (99), [] S. Reic, Multi-symplectic Runge-Kutta collocation metods for Hamiltonian wave equation, J. Comput. Pys., 57 (), [] L. Zang, Convergence of a conservative difference sceme for a class of Klein-Gordon-Scrödinger equations in one dimension, Appl. Mat. Comput., 6 (5), -55.

26 6 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, Figure. Graps of tree dimensions of numerical solutions for soliton-soliton collison wit q =.7, q =.7, x = 5, x = 5, Left: ϕ ; Rigt: u. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Matematics and Scientific/Engineering Computing, Academy of Matematics and System Sciences, Cinese Academy of Sciences, P.O.Box 79, Beiing 8, P. R. Cina address: l@lsec.cc.ac.cn State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Matematics and Scientific/Engineering Computing, Academy of Matematics and System Sciences, Cinese Academy of Sciences, P.O.Box 79, Beiing 8, P. R. Cina Graduate Scool of te Cinese Academy of Sciences, Beiing 8, P. R. Cina address: iangss@lsec.cc.ac.cn State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Matematics and Scientific/Engineering Computing, Academy of Matematics and System Sciences, Cinese Academy of Sciences, P.O.Box 79, Beiing 8, P. R. Cina Graduate Scool of te Cinese Academy of Sciences, Beiing 8, P. R. Cina address: licun@lsec.cc.ac.cn

27 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION t= t= t=5.5 t= t=.6 t= t= t= t= t= t= t= Figure. Te evolution of numerical solutions for soliton-soliton collison wit q =.8, q =.6, x =, x = 5 at various s. Left column: ϕ ; Rigt column: u.

28 8 JIALIN HONG, SHANSHAN JIANG,, AND CHUN LI, x x x Figure. Global error of carge conservation, total energy and local energy conservation law for soliton-soliton collison wit, q =.8, q =.6, x =, x = 5, wit spatial stepsize τ =. and temporal stepsize =.. SMID(Left column); TRAP(Rigt column).

29 ERROR ANALYSIS OF SOME MULTI-SYMPLECTIC SCHEME FOR KGS EQUATION 9 Figure. Graps of tree dimensions of numerical solutions for soliton-soliton collison wit q =.8, q =.6, x =, x = 5, Left: ϕ ; Rigt: u.