Classical Conservative Properties of Five Difference Schemes for ...

CLASSICAL CONSERVATIVE PROPERTIES OF FIVE DIFFERENCE ¨ SCHEMES FOR COUPLED KLEIN-GORDON-SCHRODINGER EQUATIONS JIALIN HONG1 , SHANSHAN JIANG2 , AND LINGHUA KONG3

Abstract. In this paper we compare the classical conservative properties of five difference schemes applied to the coupled Klein-Gordon-Schr¨ odinger equations in the quantum physics, and investigate the numerical behaviors of schemes in implementation. Numerical results reveal merits and shortcomings of schemes, and show that all five schemes are stable in the classical conservation laws. In the sense of preservation of classical conservative properties, conservative schemes are better than others. Keywords Klein-Gordon-Schr¨ odinger equations; difference schemes; conservation laws

1. Introduction In this paper, we consider the standard coupled Klein-Gordon-Schr¨odinger (CKGS) equations   i∂ ϕ + 1 ∂ ϕ + uϕ = 0, xx t 2 (1.1)  ∂tt u − ∂xx u + u − |ϕ|2 = 0, where ϕ(x, t) denotes a complex scalar nucleon field, and u(x, t) denotes a real scalar √ meson field, respectively. Moreover, i = −1, and we denote the spatial and temporal direction by (x, t), and x ∈ R, t > 0. We supplement (1.1) by prescribing the initialboundary value conditions for ϕ(x, t) and u(x, t) with (1.2)

ϕ|t=0 = ϕ0 (x), lim |ϕ(x, t)| = 0,

|x|→∞

u|t=0 = u0 (x),

ut |t=0 = u1 (x),

lim u(x, t) = 0,

|x|→∞

where ϕ0 (x), u0 (x), and u1 (x) are given initial values. The CKGS equations (1.1)-(1.2) is a classical model to describe the interaction between conservative complex neutron field and neutral meson Yukawa in quantum field theory. Theoretical results concerning it can be found in the literatures [1] [2], [3]. Fukuda and Tsutsumi studied the CKGS equations (1.1)-(1.2) ([1]) firstly. Global existence and asymptotic behavior of the solutions are provided in [3]. While numerical methods for it are studied only in [6] and [7], with respect to the difference methods. In [6], authors investigate some multi-symplectic method and its numerical properties. In [7], authors considered an energy-preserving difference scheme and proved its convergence with order O(h2 + τ 2 ), where h, τ are the spatial and temporal step sizes, respectively. As for the spectral methods, Bao et.al have studied by time-splitting method in [8]. The purpose of this paper is to investigate numerically five difference J. Hong and S. Jiang are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, the NNSFC (No.19971089, No.10371128) and the Special Funds for Major State Basic Research Projects of China 2005CB321701; L. Kong is supported by 2005 postgraduate innovational foundation of USTC. 1

JIALIN HONG1 , SHANSHAN JIANG2 , AND LINGHUA KONG3

2

schemes applied to CKGS equations, especially to compare the preservation of classical conservation laws in numerical implementation. The outline of this paper is as follows: We review some properties of the CKGS equations in Section 2. Some numerical methods are also presented in the section. Then conservation laws are simply investigated in Section 3. Finally, some numerical experiments are shown in Section 4. We can also discover some interesting numerical phenomena in the section. Finally, we draw some conclusions to end the paper. 2. Invariants and Numerical Methods for CKGS Equations We review some properties of the CKGS equations (1.1) in the section, including charge conservation law, energy conservation law, momentum conservation law, mean value conservation law as well. Various difference schemes which are all of 2nd order both in time and space, are also constructed in the section. The invariants follow along with the CKGS equations: (I.) charge conservation law Z

(2.1)

Z

|ϕ(x, t)|2 dx =

C(t) := R

|ϕ0 (x)| := C(0),

t > 0.

R

(II.) energy conservation law Z

(u(x, t)2 + ut (x, t)2 + ux (x, t)2 + |ϕx (x, t)|2 ) − 2u(x, t)|ϕ(x, t)|2 dx

E(t) := R

Z

(2.2)

(u(x, 0)2 + ut (x, 0)2 + ux (x, 0)2 + |ϕx (x, 0)|2 ) − 2u(x, 0)|ϕ(x, 0)|2 dx := E(0).

= R

(III.) momentum conservation law Z

M (t) : =

Im(ϕ(x, t)ϕx (x, t)) − ut (x, t)ux (x, t)dx ZR

(2.3) =

Im(ϕ(x, 0)ϕx (x, 0)) − ut (x, 0)ux (x, 0)dx = M (0), R

where “Im” denotes the imaginary part, and ϕ denotes the complex conjugate of ϕ. (IV.) mean value conservation law (2.4)

e (u0 ) − C(0)) cos(t) + N e (u1 ) sin(t) = N (0), N (t) = C(0) + (N

R e (u(·, t)) = where N (t) := N R u(x, t)dx. These are formal invariants of CKGS equations. Naturally, the most interesting thing is to search for effective schemes to preserve these formal invariants as much as possible. Thus, it motivates us to investigate their various numerical schemes, and compare their numerical results. We list five difference schemes for CKGS equations and compare their numerical phenomena. We choose h as the spatial mesh grid size, and τ as the temporal step size, respectively. For convenience, we introduce a uniform grid (xj , tk ) ∈ R × R+ . The approximation of the value of function V (x, t) at the mesh grid (xj , tk ) is denoted by Vjk . The concrete schemes are the following: Scheme (M-M), which is constructed by applying implicit midpoint method to both directions of equations in [6]. n+1/2

n+1/2

n+1/2

+ (ϕj+1/2 uj+1/2 + ϕj−1/2 uj−1/2 ) = 0,

n+1/2

+ δx2 uj

i(δt ϕj+1/2 + δt ϕj−1/2 ) + δx2 ϕj (δt2 unj+1/2 + δt2 unj−1/2 ) − (δx2 uj n−1/2

n+1/2

n+1/2 n+1/2

n+1/2

n−1/2

n+1/2 n+1/2

n+1/2

n+1/2

n−1/2

) + 12 (uj+1/2 + uj−1/2 + uj+1/2

n−1/2

n−1/2

+uj−1/2 ) − 12 (|ϕj+1/2 |2 + |ϕj−1/2 |2 + |ϕj+1/2 |2 + |ϕj−1/2 |2 ) = 0,

CLASSICAL CONSERVATIVE PROPERTIES OF FIVE SCHEMES FOR KGS EQUATION

3

n+1/2 n+1/2 j n+1 1 1 n n n where ϕn+1/2 + ϕj ) = 41 (ϕn+1 j+1 + ϕj+1 + ϕn+1 + ϕj ), j+1/2 = 2 (ϕj+1/2 + ϕj+1/2 ) = 2 (ϕj+1 n+1/2

δt ϕj+1/2 =

ϕn+1 −ϕn j+1/2 j+1/2 , τ

δt2 unj+1/2 =

n−1 un+1 −2un j+1/2 +uj+1/2 j+1/2 , 2 τ

n+1/2

δx2 uj

n+1/2

=

uj+1

n+1/2

−2uj h2

n+1/2

+uj−1

,

etc., the same for the following. This scheme is proved to be multisymplectic. Scheme (M-C), which applies implicit midpoint method to t-direction and central difference method to x-direction: 1 n+1/2 n+1/2 + (δx2 ϕn+1 + δx2 ϕnj ) + uj ϕj = 0, j 4 1 1 n−1/2 1 n−1/2 2 n+1/2 n+1/2 2 δt2 unj − (δx2 un−1 + 2δx2 unj + δx2 un+1 ) + (uj + uj ) − (|ϕj | + |ϕj | ) = 0. j j 4 2 2 n+1/2

iδt ϕj

Scheme (T-C), which is constructed by applying trapezoidal method to t-direction and central difference method to x-direction: 1 1 + (δx2 ϕn+1 + δx2 ϕnj ) + (un+1 ϕn+1 + unj ϕnj ) = 0, j j 4 2 j 1 1 n−1/2 1 n−1/2 n+1/2 n+1/2 δt2 unj − (δx2 uj + δx2 uj ) + (uj + uj ) − (|ϕn+1 |2 + 2|ϕnj |2 + |ϕn−1 |2 ) = 0. j 2 2 4 j n+1/2

iδt ϕj

Scheme (T-T), which is constructed by applying trapezoidal method to both directions of equations. n+1/2

n+1/2

n+1/2

i(δt ϕj+1 + 2δt ϕj + δt ϕj−1 ) + (δx2 ϕn+1 + δx2 ϕnj )+ j 1 n+1 n+1 n+1 n n n n n n (u ϕ + 2un+1 ϕn+1 + un+1 j j j−1 ϕj−1 + uj+1 ϕj+1 + 2uj ϕj + uj−1 ϕj−1 ) = 0, 2 j+1 j+1 n+1/2 n−1/2 (δt2 unj+1 + δt2 unj + δt2 unj−1 ) − (δx2 un+1 + δx2 unj + δx2 un−1 ) + (uj+1/2 + uj+1/2 + j j 1 n+1/2 n−1/2 n+1 2 uj−1/2 + uj−1/2 ) − [|ϕn+1 |2 + 2|ϕn+1 |2 + |ϕj−1 | + 2(|ϕnj+1 |2 + 2|ϕnj |2 + |ϕnj−1 |2 ) j 4 j+1 n−1 2 2 2 + |ϕn−1 | + |ϕn−1 j+1 | + 2|ϕj j−1 | ] = 0.

Conservative scheme (C-C), which was proposed by Zhang [7]: 1 1 + (δx2 ϕn+1 + δx2 ϕnj ) + (unj + un+1 )(ϕnj + ϕn+1 ) = 0, j j j 4 4 1 δx2 unj − δx2 unj + (un+1 + un−1 ) − |ϕnj |2 = 0. j 2 j n+1/2

iδϕj

We would like to emphasize here that all the five schemes are implicit since all of them, applied to the nonlinear equations, need iterative process from time tk to tk+1 , and the differences among schemes (M-M) and (T-T), (M-C) and (T-C) are in the discretization of the nonlinear terms. Many researchers have investigated the above five difference schemes, for Schr¨ odinger equation ([5] and references therein), Dirac equation ([4] and references therein), etc, however, hardly for the CKGS equations. In the subsequent section, we investigate the discrete conservative quantities of the five difference schemes. 3. Discrete Conservation Laws of the Difference Schemes In this section, we study the discrete conservation laws of the difference schemes presented in the previous section. Lemma 3.1. Both scheme (C-C) and scheme (T-C) possesses the discrete charge conservation law, i.e., X X (3.1) h |ϕkj |2 = h |ϕ0j |2 , j

j


4

where ϕ0j = ϕ0 (xj ) is the initial value. We refer to [7] for the proof of this Lemma. Lemma 3.2. ([6]) The scheme (M-M) admits the discrete charge conservation law, i.e. (3.2)

h

X ϕ0j + ϕ0j+1 X ϕkj + ϕkj+1 | |2 = h | |2 . 2 2 j

j

Lemma 3.3. ([7]) The scheme (C-C) satisfies the discrete energy conservation law in the following sense, E n = E n−1 = · · · = E(0),

(3.3)

where En = h

P

|

j

n+1 ϕn+1 j+1 −ϕj |2 h

+ 12 h

+h

P

|

un+1 −un j 2 j | τ

+h

n+1 P unj+1 −unj un+1 j+1 −uj

h

j j P n+1 2 P (|uj | + |unj |2 ) − h (un+1 + unj )|ϕn+1 |2 , j j j

h

j

which is an approximation of the energy conservation law (2.2). The other four schemes can not preserve the discrete energy conservation laws exactly. But we can investigate their residuals numerically. Furthermore, the truncation errors of the above five schemes reach O(τ 2 +h2 ), which can be easily verified by Taylor expansion. 4. Numerical Experiments In this section, we carry out some numerical experiments to observe the the numerical phenomena of the five difference schemes presented in the previous section. For the convenience of comparison, we give some solitary-wave solutions of the CKGS equations (1.1),

(4.1)

√  3 2 1 1 − q2 + q4  2  p p  ϕ(x, t, q) = sech (x − qt − x ) exp(i(qx + t)), 0  2(1 − q 2 ) 4 1 − q2 2 1 − q2  1 3   sech2 p (x − qt − x0 ),  u(x, t, q) = 2 4(1 − q ) 2 1 − q2

where q(0 6 |q| < 1) indicates the propagating velocity of wave, x0 is the initial phase. The initial values ϕ0 (x) = ϕ(x, 0, q), u0 (x) = u(x, 0, q), v0 (x) = ut (x, t, q)|t=0 are obtained from (4.1) as t = 0. 4.1. Solitary Wave Tests. In this subsection, we test the accuracy of the five schemes. We also want to know how well the schemes preserve the invariants. In the following calculations, we choose x0 = −20, q = 0.3, and set the spatial domain [−L, L] large enough, such that the zero boundary conditions are satisfied, i.e., ϕ(−L, t) = ϕ(L, t) = 0, u(−L, t) = u(L, t) = 0. In the following practical computation, we choose L = 40. Furthermore, errors between numerical solutions and analytic solutions in sense of 2 L and L∞ norms are, respectively, defined as: (4.2)

||errorϕn ||2 = (h

X j

1

|ϕ(xj , tn ) − ϕnj |2 ) 2 , ||errorϕn ||∞ = max |ϕ(xj , tn ) − ϕnj |. j

The dicretization of (2.1),(2.2) and (2.3) at t = tn are denoted C n , E n , M n .


5

Table 1 shows errors for the five difference schemes at T = 1, and Table 2 lists the residuals of conservative quantities for the five schemes. From the two tables, we can observe that the numerical results are consistent to the theoretical ones that the numerical solutions u, ϕ are of order O(τ 2 + h2 ). The charge is conserved exactly by schemes (M-M),(M-C) and (C-C) within the roundoff error, while the other two not. Only scheme (C-C) conserves the energy exactly. An interesting phenomenon can be observed from table 2, is that the residuals of the conservative quantities which the scheme preserves increase with the thickened mesh grid. However, the residuals of the conservative quantities which the scheme does not preserve decrease with the thickened mesh grid. This demonstrates that, for the conservative quantities which the scheme preserve, the truncation errors are negligible and roundoff errors increase with the thickened mesh grid for the increase of computation. Table 1. Errors between numerical solutions and analytic solutions τ \h 0.1\0.2

method M-M M-C T-C T-T C-C

L∞ error of u 3.404209e-4 8.853035e-4 1.196352e-3 1.477323e-3 1.258607e-3

L2 error of u 3.341671e-3 1.271846e-3 1.366835e-3 2.750816e-3 1.424779e-3

L∞ error of ϕ 5.032287e-3 1.964584e-3 2.059972e-3 4.483451e-3 2.078082e-3

L2 error of ϕ 6.402466e-3 3.053696e-3 3.174964e-3 6.540302e-3 3.134800e-3

0.05\0.1

M-M M-C T-C T-T C-C

9.515770e-5 2.350719e-4 3.163652e-4 3.911607e-4 3.325115e-4

8.840607e-4 3.387470e-4 3.636545e-4 7.221917e-4 3.784952e-4

1.335464e-3 5.078567e-4 5.341095e-4 1.178744e-3 5.392754e-4

1.693839e-3 8.004568e-4 8.328130e-4 1.701153e-3 8.222223e-4

0.025\0.05

M-M M-C T-C T-T C-C

2.500168e-5 6.063758e-5 8.134725e-5 1.011365e-4 8.563031e-5

2.269768e-4 8.729277e-5 9.368392e-5 1.854060e-4 9.742859e-5

3.420941e-4 1.292030e-4 1.358407e-4 3.002562e-4 1.371329e-4

4.347211e-4 2.045860e-4 2.129327e-4 4.341851e-4 2.102248e-4

Table 2. Residuals of conservative quantities for different schemes τ \h 0.1\0.2

method M-M M-C T-C T-T C-C

charge error -4.440892e-15 3.552713e-15 -2.44645e-6 5.011719e-6 5.773159e-15

energy error 1.891756e-6 4.819882e-8 1.245640e-6 1.404062e-4 -3.44169e-15

mean meson error -4.751145e-4 -8.991595e-4 -2.019303e-7 4.0304846e-7 -2.846999e-9

momentum error -5.040177e-6 -3.583390e-5 -4.905929e-5 1.011148e-5 -5.187411e-5

0.05\0.1

M-M M-C T-C T-T C-C

7.549516e-15 -1.110223e-14 -1.611851e-7 3.293989e-7 -1.110223e-14

1.329776e-7 1.641491e-8 1.047686e-7 1.042627e-5 4.551914e-15

-1.841521e-4 -2.363798e-4 -1.686126e-8 2.627202e-8 -2.996891e-9

-4.048984e-7 -4.340682e-6 -6.010579e-6 7.475347e-7 6.487119e-6

0.025\0.05

M-M M-C T-C T-T C-C

5.728750e-14 -6.661338e-14 -1.033481e-8 2.111651e-8 -7.416289e-14

8.790105e-9 1.392436e-9 7.349691e-9 7.029029e-7 3.408384e-14

-5.406053e-5 -6.053030e-5 -3.987706e-9 -7.189601e-11 -3.072619e-9

-2.813102e-8 -5.270855e-7 -7.359422e-7 5.058114e-8 -8.019507e-7

In order to illuminate the specialities of the five schemes, we give their numerical behaviors in the time interval [0, 100] with various temporal step sizes and spatial mesh grid sizes. Fig. 1 presents the residuals of energy E n . Except for scheme (C-C) preserving the energy conservation law exactly in the scale of 10−12 , all the other four schemes


6

−6

−5

2

x 10

x 10

1

1.8 0

1.6 1.4

−1

1.2

−2

1 0.8

−3

0.6

−4

0.4 −5

0.2 0 0

20

40

60

80

100

time

(a)

−6 0

20

40

60

80

100

time

(b) −7

−6

x 10

x 10

0.5

1.2

0

1 −0.5

0.8

−1 −1.5

0.6

−2

0.4

−2.5 −3

0.2

−3.5

0 0

25

50

75

100

time

(a)

0

50

75

100

time −8

−8

8

25

(b)

x 10

1

7

x 10

0.5

6

0

5 −0.5

4 −1

3 −1.5

2

−2

1 0 0 (a)

12.5

25

37.5

50 time

62.5

75

87.5

100

−2.5 0 (b)

12.5

25

37.5

50

62.5

75

87.5

100

time

Figure 1. Residuals of energy conservation law with various spatial and temporal step sizes: Above for τ = 0.1 and h = 0.2; Mid for τ = 0.05 and h = 0.1; Below for τ = 0.025 and h = 0.05; for (a) scheme M-M, (b) scheme M-C, (c) scheme T-C, (d) scheme T-T, (e) scheme C-C.

1 exhibit the similar changes: the residuals are reduced to about 16 roughly, as both the 1 spatial mesh grid size and the temporal step size are reduced to 2 simultaneity. And all graphs stay oscillation of small amplitude, and no evident fluctuation for a long time. But scheme (T-T) is inferior to the others. Fig. 2 shows the residuals of momentum M n . All the five schemes can not preserve the momentum conservation law exactly. Schemes (M-M) and (T-T) exhibit analogous 1 performance along with the increase of time, and the errors are reduced to about 16 roughly, as both of the spatial mesh grid size and the temporal step size are reduced to 12 . Schemes (M-C), (T-C) and (C-C) exhibit analogous performance likewise, they have the similar scope of errors.

CLASSICAL CONSERVATIVE PROPERTIES OF FIVE SCHEMES FOR KGS EQUATION −6

12

7

−4

x 10

8

x 10

7

10

6

8

5 6

4 4

3

2

2

0

1

−2 0

20

40

60

80

1000

time

(c)

0 0

20

−7

7

40

60

80

100

time

(d) −5

x 10

5

x 10

4.5

6

4 5

3.5 4

3

3

2.5 2

2

1.5 1

1

0

0.5

−1 0

25

50

(c)

75

100

0 0

3

4

2.5

3

2

2

1.5

1

1

0

0.5

(c)

12.5

50

75

100

time −6

−8

x 10

−1 0

25

(d)

time

25

37.5

50 time

62.5

75

87.5

100

x 10

0 0

12.5

(d)

25

37.5

50

62.5

75

87.5

100

time

Figure 1. continued.

Fig 3 displays the residuals of charge C n . We can observe that schemes (M-M), (M-C) and (C-C) all preserve the charge conservation law exactly, which are all in the magnitude of 10−12 . This is consistent to the theoretical results in Lemma 3.1. Though schemes (T-C) and (T-T) can not preserve the charge conservation law exactly, they remain the errors in better magnitude and have no evident increase for a long time with different mesh grid sizes and time step sizes. Moreover, the residuals are in accordance with the rule that they are increased to about 16 times roughly, as both the mesh grid size and the time step size are densified 2 times. It can be seen from the experiments that in the case of solitary-wave solutions, all the five schemes are stable in the sense of the energy, charge, and momentum conservation laws.

8

JIALIN HONG1 , SHANSHAN JIANG2 , AND LINGHUA KONG3 −12

1

x 10

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

20

40

60

80

100

time

(e) −12

1

x 10

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

25

50

75

100

time

(e) −12

8

x 10

7 6 5 4 3 2 1 0 −1 −2 0 (e)

12.5

25

37.5

50

62.5

75

87.5

100

time

Figure 1. continued. 4.2. Collision of Solitons. Now, we turn our attentions to the collision of two solitons. The corresponding initial values are chosen ϕ0 (x) = ϕ0 (x − x1 , 0, q1 ) + ϕ0 (x − x2 , 0, q2 ), u0 (x) = u0 (x − x1 , 0, q1 ) + u0 (x − x2 , 0, q2 ), v0 (x) = {ut (x − x1 , t, q1 ) + ut (x − x2 , t, q2 )}|t=0 , where x1 , x2 are initial phases, q1 , q2 are propagating velocities of two solitons, respectively. We choose collision of two solitons with same speed and opposite direction (symmetric collision): x1 = −15, x2 = 15 and q1 = 0.6, q2 = −0.6, respectively. We solve the problem with the five schemes in the spatial domain [−40, 40] till T = 50. In this case, we take τ = 0.01 and h = 0.2. Fig. 4 plots the evolution of the solitons calculated by scheme (M-M) as an example. We can find that the two solitons stay at the same position at time around t = 22.5, and


−5

1

9

x 10

2

x 10

0 0

−1 −2

−2 −3

−4

−4 −6

−5 −6

−8

−7 0

20

40

60

80

100

time

(a)

0

−6

1

4

0

2

−1

0

−2

−2

−3

−4

−4

−6

−5 0

25

50

75

100

time

(a)

80

100

25

50

75

100

time −7

6

0

4

−0.5

2

−1

0

−1.5

−2

−2

−4

−2.5

−6

12.5

x 10

−8 0

−7

(a)

60 time

(b)

x 10

−3 0

40

−6

x 10

0.5

20

(b)

25

37.5

50 time

62.5

75

87.5

100

x 10

−8 0 (b)

12.5

25

37.5

50

62.5

75

87.5

100

time

Figure 2. residuals of momentum conservation law, i.e., M n with various spatial and temporal step-sizes : Above for τ = 0.1 and h = 0.2; Mid for τ = 0.05 and h = 0.1; Below for τ = 0.025 and h = 0.05; for (a) M-M scheme, (b) M-C scheme, (c) T-C scheme, (d) T-T scheme, (e) C-C scheme.

the solitons keep in this state. After that, the collision results in fusion accompanied by a series of emission of waves. In Fig. 5, we give the corresponding residuals of energy for collision of two solitons. It can be observed that the error of energy conservation law is in the scale of 10−11 for scheme (C-C) because it is an energy-preserving scheme, while for schemes (M-M), (M-C) and (T-C) in the scale of 10−4 , for scheme (T-T) even in the scale of 10−1 . And the errors change larger during collision than others, and exhibit pretty performance with oscillation of little amplitude around some equilibrium state, and have no evident amplification for a long time before and after collision for all the schemes.


10

−5

−5

2

x 10

10

0

8

−2

6

−4

4

−6

2

0

−8 0

20

40

(c)

60

80

100

0

20

40

(d)

time −6

4

x 10

60

80

100

time −6

x 10

7

x 10

6

2

5 0

4 −2

3 2

−4

1 −6

0 −8 0

25

50

75

100

time

(c)

−1 0

25

75

100

−7

−7

6

50 time

(d)

x 10

4

x 10

3.5

4

3

2 2.5

0

2

−2

1.5 1

−4

0.5

−6

0

−8 0 (c)

12.5

25

37.5

50 time

62.5

75

87.5

100

−0.5 0

12.5

(d)

25

37.5

50

62.5

75

87.5

100

time

Figure 2. continued. Fig. 6 lists the corresponding errors of momentum for collision of two solitons. We can see that schemes (M-C), (T-C) and (C-C) present the similar performance: the errors change larger during collision, and exhibit pretty performance with oscillation of little amplitude around some equilibrium state, and have no any evident amplification for a long time after collision. But, as a whole, the errors are in the scale of 10−1 , and much larger than in the case of solitary wave. Surprisingly, the results of schemes (M-M) and (T-T) much better than they are expected, which are in the scale of 10−10 . In Fig. 7, we illuminate the corresponding errors of charge conservation law for collision of two solitons. As in the case of solitary wave, schemes (M-M), (M-C) and (C-C) preserve the charge conservation law exactly, which are in the magnitude of 10−12 . But schemes (T-C) and (T-T) in the sale of 10−4 , and the errors change larger during collision, and exhibit pretty performance with oscillation of little amplitude around some equilibrium state, and have no any evident amplification for a long time after collision as other invariant quantities.


11

−5

2

x 10

0

−2

−4

−6

−8 0

20

40

60

80

100

time

(e) −6

4

x 10

2 0 −2 −4 −6 −8 0

25

50

75

100

time

(e) −7

6

x 10

4 2 0 −2 −4 −6 −8 0 (e)

12.5

25

37.5

50

62.5

75

87.5

100

time

Figure 2. continued. In all, though in the case of collision of solitons, all the five schemes remain the stability for the energy, charge, and momentum conservation laws. 5. Conclusions We present five difference schemes to simulate the CKGS equations, and investigate their properties numerically, including the discrete energy, charge and momentum conservation laws. The five difference schemes are all of order O(τ 2 + h2 ). Scheme (C-C) can preserve both the discrete charge and energy conservation laws exactly, and schemes (M-M),(M-C) preserve the discrete charge conservation law exactly, whereas none of them can preserve the discrete momentum conservation law exactly. In sum, scheme (C-C) is superior to schemes (M-M) and (M-C), and the latter two are exceed to schemes (T-C) and (T-T). But sometimes schemes (M-M) and (T-T) can simulate the discrete momentum conservation law precisely.


12 −12

1

−12

x 10

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 0

20

40

60

80

100

−1 0

time

(a)

20

40

60

80

100

time

(b) −12

−12

1

x 10

x 10

1

x 10

0.8 0.5

0.6 0.4

0

0.2 −0.5

0 −0.2

−1

−0.4 −0.6

−1.5

−0.8 −1 0

25

50

(a)

75

100

−2 0

−12

10

25

50

(b)

time

75

100

time −12

x 10

6

x 10

4 8

2 6

0 −2

4

−4 2

−6 −8

0

−10 −2 −4 0 (a)

−12 −14 12.5

25

37.5

50 time

62.5

75

87.5

100

0 (b)

12.5

25

37.5

50

62.5

75

87.5

100

time

Figure 3. residuals of charge conservation law, i.e., C n with various spatial and temporal step-sizes : Above for τ = 0.1 and h = 0.2; Mid for τ = 0.05 and h = 0.1; Below for τ = 0.025 and h = 0.05; for (a) M-M scheme, (b) M-C scheme, (c) M-C scheme, (d) T-C scheme, (e) C-C scheme.

References [1] I. Fukuda, M. Tsutsumi, On the Uukawa-coupled Klein-Gordon-Schr¨ odinger equations in three space dimensions, Prof. Japan Acad., 51 (1975), 402-405. [2] M. Ohta, Stability of stationary states for the coupled Klein-Gordon-Schr¨ odinger equations, Non. Anal., 27 (1996), 455-461. [3] T. Ozawa, Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-GordonSchr¨ odinger equations, Adv. Stud. Pure Math. 23 (1994), 295-305. [4] J. Hong, C. Li, Multi-symplectic RungeCKutta methods for nonlinear Dirac equations, J. Comput. Phys., 211 (2006), 448-472.


0.5

13

−5

x 10

5

0

x 10

4

−0.5

3

−1

2 −1.5

1 −2

0 −2.5 0

20

40

60

80

100

time

(c)

0

20

40

60

80

100

time

(d) −6

−7

x 10

3

x 10

0 2.5

2

−5 1.5

1

−10 0.5

0

−15 0

25

50

(c)

75

100

0

25

50

−8

75

100

time

(d)

time

−8

x 10

x 10

0

18

−1

16

−2

14

−3

12

−4

10 8

−5

6 −6

4 −7

2

−8 −9 0 (c)

0 12.5

25

37.5

50 time

62.5

75

87.5

100

−2 0

12.5

25

(d)

37.5

50

62.5

75

87.5

100

time


[5] J. Hong, Y. Liu, H Munthe-Kaas and A Zanna, Globally conservative properties and error estimation of a multi-symplectic scheme for Schr¨ odinger equations with variable coefficients, Appl. Numer. Math., 56 (2006), 814-843. [6] L. Kong, R. Liu and Z. Xu, Numerical simulation of interaction between Schr¨ odinger field and Klein-Gordon field by multisymplectic method, Appl. Math. Comput., 2006 to appear. [7] L. Zhang, Convergence of a conservative difference scheme for a class of Klein-Gordon-Schr¨ odinger equations in one space dimension, Appl. Math. Comput., 163 (2005), 343-355. [8] W. Bao, L. Yang, Efficient and accurate numerical methods for the Klein-Gordon-Schr¨ odinger equations, preprint 2005.

14

JIALIN HONG1 , SHANSHAN JIANG2 , AND LINGHUA KONG3 −12

1

x 10

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

20

40

60

80

100

time

(e) −12

1

x 10

0.5

0

−0.5

−1

−1.5

−2 0

25

50

75

100

time

(e) −12

4

x 10

2 0 −2 −4 −6 −8 −10 −12 −14 0 (e)

12.5

25

37.5

50

62.5

75

87.5

100

time


Figure 4. numerical solutions of symmetric collision for scheme (MM); Left for u, right for |ϕ|.


2

x 10

15

−4

2

1

x 10

1

0

0

−1

−1

−2

−2

−3

−3

−4

−4 −5

−5

−6

−6 0

10

20

30

40

50

time

(a)

−7 0

10

20

30

40

50

40

50

time

(b)

−4

0.1

x 10 10

0

−0.1 5

−0.2

0

−0.3

−0.4 −5 0 (c)

10

20

30

40

50

−0.5 0

time

10

20

30 time

(d) −11

1

x 10

0.5

0

−0.5

−1

−1.5 0 (e)

10

20

30

40

50

time

Figure 5. Residuals of energy conservation law for collision: (a) scheme M-M, (b) scheme M-C, (c) scheme T-C, (d) scheme T-T, (e) scheme C-C. 1,2

State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O.Box 2719, Beijing 100080, P. R. China E-mail address: [email protected] 2

Graduate School of the Chinese Academy of Sciences, Beijing 100080, P. R. China E-mail address: [email protected] 3

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China E-mail address: [email protected]


16 −10

0.3

x 10 6

0.2

5

0.1

4 3

0

2 1

−0.1

0

−0.2

−1 −2 −3 0

10

20

30

40

50

−0.3 0

time

(a)

10

20

30

40

50

40

50

time

(b) −10

0.3

x 10 6

0.2

5

0.1

4 3

0

2 1

−0.1

0 −1

−0.2

−2

−0.3 0 (c)

10

20

30

40

50

−3 0

time

10

20

30 time

(d)

0.3

0.2

0.1

0

−0.1

−0.2

−0.3 0 (e)

10

20

30

40

50

time

Figure 6. residuals of momentum conservation law for collision: for (a) scheme M-M, (b) scheme M-C, (c) scheme T-C, (d) scheme T-T, (e) scheme C-C.


3.5

17

−12

x 10

x 10

1

3

0.5

2.5

0

2

−0.5

1.5

−1

1

−1.5 0.5

−2 0

−2.5 −0.5 0

10

20

30

40

−3 0

50

time

(a) −4

30

40

50

40

50

time

x 10

2

2

0

0

−2

−2

−4

−4

−6

−6

(c)

20

−4

x 10

−8 0

10

(b)

10

20

30

40

50

−8 0

time

10

20

30 time

(d) −12

x 10

1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0

(e)

10

20

30

40

50

time

Figure 7. residuals of charge conservation law for collision: for (a) scheme M-M, (b) scheme M-C, (c) scheme T-C, (d) scheme T-T, (e) scheme C-C.