zakharov equations - American Mathematical Society

mathematics of computation volume 64,number 210

april

1995,pages 537-553

FINITE DIFFERENCE METHOD FOR GENERALIZED

ZAKHAROVEQUATIONS QIANSHUNCHANG, BOLINGGUO, AND HONG JIANG Abstract. A conservative difference scheme is presented for the initialboundary value problem for generalized Zakharov equations. The scheme can be implicit or semiexplicit depending on the choice of a parameter. On the basis of a priori estimates and an inequality about norms, convergence of the difference solution is proved in order 0(h2 + t2) , which is better than previous results.

Introduction

The Zakharov equations [20] (1.1)

iEt + Exx-NE

(1.2)

^Ntt-{N+\E\2)xx

= 0, =0

describe the propagation of Langmuir waves in plasmas. Here the complex unknown function E is the slowly varying envelope of the highly oscillatory electric field, and the unknown real function N denotes the fluctuation of the ion density about its equilibrium value. The global existence of a weak solution for the Zakharov equations in one dimension is proved in [19], and existence and uniqueness of a smooth solution for the equations are obtained provided smooth initial data are prescribed. Numerical methods for the Zakharov equations are studied only in [5, 9, 10, and 15]. A spectral method is used to compute solitary waves and the collision of two solitary waves in [15]. In [9, 10], Glassey considered an energy-preserving implicit difference scheme for the equations and proved its convergence in order 0(h + t) . In [5], we propose a new conservative difference scheme which involves a parameter 6, 0 < 6 < j ; when 6 = j , the new scheme is identical to Glassey's scheme. For 0 = 0, the new scheme is semiexplicit, explicit in TV, but implicit in E. Numerical experiments demonstrate that the new scheme with 6 = 0 is more accurate and efficient compared to 6 = \ . Convergence of these schemes is proved in order 0(h + t) in [5, 9, and 10], while the order of the truncation errors is 0(h2 + x2). Received by the editor September 16, 1993 and, in revised form, December 28, 1993 and April

7, 1994. 1991 Mathematics Subject Classification.Primary 65M06, 65M12. Key words and phrases. Difference scheme, Zakharov equation, convergence. ©1995 American Mathematical Society 0025-5718/95 $1.00+ $.25 per page

537

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538

QIANSHUNCHANG, BOLINGGUO, AND HONG JIANG

For suitable initial data, the solution of the initial value problem for (1.1)(1.2) converges as X -> oo to a solution of the cubic nonlinear Schrödinger equation

(1.3)

Œ, + Exx + \E\2E = 0

(see [1, 17]). The generalized nonlinear Schrödinger equation

(1.4)

iEt + Exx + f(\E\2)-E = 0

has been considered in physics (see, for example, [2, 3, and 11]). Here, f(s) = sP(p > 0), f(s) = 1 - e~s, f(s) = f-Sj or f(s) = ln(l + s). Existence and uniqueness of the generalized solution for the equation ( 1.4) have been obtained and numerical methods for (1.4) have been studied (see [4, 6, and 18]). The generalized Zakharov equations are also considered in [21]. In the present paper, we consider the following initial-boundary value problem of generalized Zakharov equations in one dimension:

(1.5) (1-6)

iEt + Exx = Nf(\E\2)E,

xL0,

Ntt-NXX = ^(F(\E\2)),

where

(1.7) (1.8) (1.9)

f£C°°(R+), E(x, 0) = E°(x), E\x=XL = E\x=XR = 0,

F(s)= fSf(r)dr; Jo

N(x,0)

= N°(x),

Nt(x, 0) = Nx(x),

N\x=XL = N\x=XR = 0,

u\x=XL = u\x=XR = 0,

and the potential function u is given by (1.10)

uxx = Nt.

Moreover, we supplement (1.5)—(1.10) by imposing the compatibility condition

(1.11)

[RNx{x)dx

= 0.

JxL

We propose a conservative difference scheme with parameter 6 of the generalized Zakharov equations. The difference scheme conserves two conservation laws that the differential equations possess. For 9 = 0, the scheme is semiexplicit. We will prove the convergence of the difference solution for all 6 £ [0, \] in order 0(h2 + t2) , which is consistent with the order of the truncation error of the difference scheme. This improves the results of order 0(h + t) which were given in [5, 9]. In §2, we describe the difference scheme and its basic properties. Some a priori estimates and the proof of convergence of the difference scheme are given


FINITE DIFFERENCEMETHOD FOR GENERALIZEDZAKHAROVEQUATIONS

539

in §3. Long and highly technical proofs of two lemmas in §3 are placed in the Supplement section at the end of this issue.

2. Finite difference

scheme

We consider a finite difference method for the problem (1.5)—(1.11). As usual, the following notations are used:

xj = xl + jh,

t" = n • x,

Xr —Xl

0"+1\\\ - ||el2) = lm(RE, e]+x +e]) + Pi,


FINITE DIFFERENCEMETHODFOR GENERALIZEDZAKHAROVEQUATIONS

549

where

n) + N(j n + l))E(\E(j,n + l)\2)-F(\EU,n)\2) =ImfV(; 4(7VU ,n) + JM(j,n + i)) mL n + 1)|2 _ IEUtn)x2 ■(E(j,n + l) + E(j,n)) 1 (Nn

, F(\E1+X\2)-F(\E"\2) J Aiyj+iVjI Nn+X)) ,F„+X,2 ,F„,2 J

' tpn+l K^j

,

, i pn\ pn+l,pn +£'j)>ej +ej

N

K„n , „n+\yF(\EJ+ I ^ ~ F^EJ^ , rvn „n+l =im ^;+o ¿,+i,L.^|2 f^n+l ^r+^)»gr+g;

+ Imi^(/V(;,«)

+ /V(;,/j+l))

/F(|£Q, « + 1)|2) - F(|£(;, ^ \E(j,n + l)\2-\E(j,n)\2

n)|2)

F(|g;+112) - F(|E;|2)\ \E»+X\2-\EJ\2 j

. (EJ+X + EJ), e]+x+ eA. Using the inequalities (4.2), (4.3) in the Supplement, and Lemma 3, we obtain (3.15)

|P3| < C\\nn+X\\2+ \\r,»\\2 + \\en+x\\\ + \\en\\22).

It is easy to obtain the estimate \lm(RE, ej+x + ej)\ = \lm(0(h2 + x2), e]+x + ej)\

"+1n2 \\ex Il2-t+ \\nn+h\2 llu* II24+ \\nn+x\\2 II" II2-t+ Hm"ii2i II" \w-

Thus, combining (3.25) and (3.22) yields

lkB+1llHlkxB+1ll2 + ll'/"+1lll+llalli + ll^+ill2

+ 1)|a_|£;(j.n)|2

rn+uF(\Ep\*)-F(\E»\>)iM

. (£n+i _ E?) - (n? + /v; ■(E(j,n+l)-EU,n))

\En+112

(E(j,n +1) + E(j,n))

-(Ep

_ |£"|2

+ (N? + Np)(F(\Ep\')

+E?)

- F(\E?\2))

- hJ2\F{\E{j,n+ 1)|2)- F(\E(j,n)\2) - F(\Ep\>) + F(\E?)\>)} J=i ■[N(j,n + 1) + N(j,n) - Np =Re | /i¿((/V;n+1 y=i

(£n+l - Ef)\

- N?}

+ N?)(E(j,n+l)-E(j,n))

- (N(j,n)

+ N(j,n + 1))

F(\E(J,n + l)\2-F(\E(j,n)\>)

\E{j,n + l)\>-\EU,n)\>

IEU,"+1) + E(j,n))

F{\Ep\*)-F(\Et\*) \ßn + l\2 _ |£»n|2

=Re | ^¿[(Ar(i,n)

^1

^ *} I

+ N(j,n + l))(ep

- e») - (E(j,n + 1) - E(j,n)

>=1

(vp+n?)}

F(\Ep\>)

- F(\E»\>)

IRV.+ 1I2 _ |£n+l|2_|£J

|Fn|2

^3

+ ej'J

© 1995 American Mathematical Society

0025-5718/95$1.00+ $.25 per page

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