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ORTHOGONAL SPLINE COLLOCATION METHODS FOR  SCHRODINGER-TYPE EQUATIONS IN ONE SPACE VARIABLE

MARK P. ROBINSON3

AND

GRAEME FAIRWEATHERy

Abstract. We examine the use of orthogonal spline collocation for the semi-discretization of the cubic Schrodinger equation and the two-dimensional parabolic equation of Tappert. In each case, an optimal order L2 estimate of the error in the semidiscrete approximation is derived. For the cubic Schrodinger equation, we present the results of numerical experiments in which the integration in time is performed using a routine from a software library.

AMS(MOS) subject classi cations. 65M15, 65M20, 65M70

In this paper, we examine the use of orthogonal spline collocation, that is, spline collocation at Gauss points, for the semi-discretization of two problems of Schrodinger type. We rst consider the initial value problem for the cubic Schrodinger equation iut + uxx + qjuj2u = 0; (x; t) 2 (01; 1) 2 (0; T ]; (1.1) u(x; 0) = g (x); x 2 (01; 1); where i2 = 01, q is a given positive constant, the given function g (x) is complexvalued, and jg (x)j ! 0 as jxj ! 1: This equation describes many physical phenomena, including the behavior of a plane stationary light beam in a medium with a nonlinear refractive index [33] and the evolution of deep water waves (see [11] and references cited therein). For the numerical solution of (1.1), one restricts x to a nite interval [a; b] which is chosen so that the modulus of the solution u(x; t) is negligible for x outside [a; b]: By imposing homogeneous Dirichlet or homogeneous Neumann boundary conditions at x = a and x = b; the pure initial value problem (1.1) is converted into an initial boundary-value problem (IBVP). Since the choice of Neumann or Dirichlet boundary conditions is not signi cant, we consider the IBVP 1. Introduction.

(1.2a)

iut + uxx + qjuj2u = 0;

(x; t) 2 (a; b) 2 (0; T ];

3 Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101, U.S.A.

y Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, U.S.A. 1

(1.2b) (1.2c)

u(x; 0) = g (x); x 2 (a; b);

u(a; t) = u(b; t) = 0; t 2 [0; T ]:

Several numerical studies have examined the use of nite dierence methods and nite element methods based on the L2 -Galerkin semi-discretization for solving IBVPs of this form; see, for example, [1], [2], [6], [11], [12], [14], [23], [24], [25], [26], [27], [28], [30], and [31]. In [20], both the L2 -Galerkin method and the H 01 -Galerkin method were considered. The second Schrodinger-type equation which we consider is the two-dimensional parabolic equation (PE) of Tappert [29] (1.3)

i h i i 01(x)ut = k001 (x) n2 (x; t) 0 1 + i (x; t) u + (01(x)ux )x; 2k0 2

which has been used eectively in many problems involving long-range, low-frequency underwater acoustic wave propagation. In (1.3), x and t represent the depth and range, respectively, u(x; t) is the pressure eld, k0 is a given positive constant, (x) is the density, and n(x; t) and  (x; t) are prescribed functions. Previous studies of this equation have employed various numerical schemes including split-step Fourier methods, explicit and implicit nite dierence schemes, the method of lines with nite-dierence semi-discretizations with respect to depth, and nite element Galerkin methods; see, for example, [15] and references therein, [10], [13] and [19]. In the present study, we examine the use of orthogonal spline collocation for the discretization with respect to x of (1.2) and (1.3). Numerical methods based on spline collocation have proved to be exceedingly eective for the approximate solution of a broad class of problems, particularly boundary value problems for ordinary differential equations and in the method of lines solution of parabolic and hyperbolic initial-boundary value problems; see [9] for a comprehensive survey. The popularity of collocation methods is due in part to their conceptual simplicity, wide applicability, and ease of implementation. The obvious advantage of spline collocation methods over nite element Galerkin methods is that the calculation of the coecients in the equations determining the approximate solution is very fast since no integrals need be evaluated or approximated. In comparison with nite dierence methods, spline collocation provides approximations to the solution and its derivative with respect to x at all points of the domain of the problem. From the standpoint of solving a non2

linear equation such as the cubic Schrodinger equation, the ease of implementation of the collocation method makes it possible to achieve computational eciency without resorting to product approximation [5] as is usually done in nite element Galerkin schemes, and enables one to obtain approximations of high order accuracy. A brief outline of this paper is as follows. In Section 2, we introduce de nitions, notation, and other preliminaries. In Section 3, we describe the continuous-time orthogonal spline collocation method for the solution of (1.2), and derive an optimal order L2 error estimate. In Section 4, we present numerical results which demonstrate the ecacy of the collocation method for the solution of the cubic Schrodinger equation. In Section 5, an analysis similar to that of Section 3 is given for the collocation solution of the PE of Tappert in horizontally strati ed media, in which case the density function (x) is piecewise constant. It is shown that the interface conditions arising in such problems can be incorporated into the collocation procedure in a very natural way. 2. Preliminaries. Unless explicitly stated otherwise, functions considered in this paper are complex-valued. For a given function f , we denote by f1 and f2 its real and imaginary parts, respectively. Given any space S of functions, let 0; 4" for d; e 2 R, without explicit mention being made each time. Given a partition (2.1)

1 : 0 = x1 < x2 < ::: < xN +1 = 1

of I , let Ij = (xj ; xj+1), hj = xj+1 0xj ; j = 1; 2; :::; N , and h = max1jN hj . A family F of partitions is said to be quasi-uniform if there exists a nite positive number  such that max h 1jN hj



for every partition in F (cf. [8]). We assume that the partition 1 is a member of a quasi-uniform family. We de ne Mp(1) = fvjv 2 C 1 (I ); vjI 2 Pp ; j = 1; 2; :::; N g \ fvjv(0) = v(1) = 0g; j

where p  3 and Pp denotes the set of all polynomials of degree at most p. Let fk gpk0=11 denote the nodes for the (p 0 1)-point Gaussian quadrature rule on the interval I with corresponding weights fwkgpk0=11 , wk > 0, and set jk = xj + hj k ; j = 1; 2; :::; N; k = 1; 2; :::; p 0 1:

We de ne the quadratic form f1; 1g by ff; g g =

N X

ff; g gj ;

j =1

4

where ff; g gj = hj

and set

pX 01 k=1

jf j2D =

wk f (jk )g 3(jk);

N X

ff; f gj :

j =1

Since 1 belongs to a quasi-uniform family of partitions, it can be shown using results of [7] and [18] that there exist positive constants C1 = C1(p) and C2 = C2(p; ) such that, for any 2 Mp(1), (2.2)

C1 j

jD  k k  C2 j jD :

In the remainder of this section, we introduce notation and some basic results which are used in Section 5 when we consider the collocation solution of (1.3) with piecewise constant (x). The corresponding results required in Section 3 are easily obtained by setting  = 1. We assume that each partition 1 of I under consideration contains the points de ning the locations the media interfaces. To simplify the exposition, we de ne (x) by (x) = j ; xj < x < xj+1 ; j = 1; 2; :::; N;

where j > 0 for all j , noting, of course, that the only discontinuities in (x) occur at the media interfaces. Then we de ne M0p (1) = fvjv 2 C 0(I ); vjI 2 Pp ; j = 1; 2; :::; N g \fvj0j011 v0(x0j ) = 0j 1 v0(x+j ); j = 1; 2; :::; N g \ fvjv(0) = v(1) = 0g; where p  3. Note that if  = 1, then M0p(1) = Mp(1). Hence, any results derived for the space M0p(1) also apply to Mp(1). Minor extensions of results of [4] and [18] show that there exist positive constants C1 = C1(p) and C2 = C2(p; ; ^), where j

 ^ = maxj;k j ; k

such that (2.2) is satis ed for 2 M0p(1). 5

The following lemma is required in the analyses (cf. [4, 7]). For f; g 2