Absorbing boundary conditions on a hypersurface for the Schrödinger equation in a half-space

Appl. Math. Lett. Vol. 9, No. 4, pp. 55-59, 1996 CopyrightQ1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/96 $15.00 + 0.00

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A b s o r b i n g B o u n d a r y C o n d i t i o n s on a H y p e r s u r f a c e for the SchrSdinger E q u a t i o n in a H a l f - S p a c e L. DI MENZA Math@matiques Appliqu6es de Bordeaux, Universit6 Bordeaux I 351, cours de la Lib6ration, 33405 Talence, Prance

(Received and accepted November 1995) Abstract--Transparent boundary conditions for the Schr6dinger equation iOtu + Au = 0 are deduced from the solution of the Cauchy problem set in the whole space. They are established at the

hypersurface {xn = 0}. Approximating the symbol of these conditions by means of rational fractions, and introducing auxiliary functions at the boundary, local absorbing conditions are derived. Then, expressing them without surfacic functions, the link between the two kinds of conditions is made K e y w o r d s - - P a r t i a l differential equations, Transparent conditions, Square root approximations. 1. I N T R O D U C T I O N Transparent boundary conditions allow us to consider the restriction of the solution of a Cauchy problem set in ~ to a bounded set or a half-space without interfering with the behaviour of the corresponding solution. From a practical point of view, the purpose is to study a physical problem only in the region where the information turns out to be significant. Numerically, this enables us to compute the solution of the problem in a restricted domain. Unfortunately, these conditions are nonlocal both in time and space, which makes their numerical implementation difficult and costly in terms of CPU time. Another problem is then to find easier boundary conditions by means of approximation techniques, namely the absorbing boundary conditions. These conditions were first used for the wave equation [1-3], for the Maxwell system [4], and later for parabolic problems such as the heat equation or the linear advection-diffusion equation [5-7]. Implementation of these conditions for the SchrSdinger equation is more recent [8-10]. The aim of this paper is to introduce transparent and absorbing boundary conditions for the SchrSdinger equation for any space dimension, on the hypersurface F = {x E R~; xn = 0}. In the first part, we write the expression of transparent boundary conditions derived from the extension method, and another form coming from the pseudodifferential factorization of the Schr6dinger operator. Then, absorbing conditions are deduced with the use of approximation tools already employed for parabolic equations [5-7], or hyperbolic equations [2,3]. In the last part, we show the link between the two kinds of boundary conditions, expressing absorbing conditions without surfacic functions. The link shows the equivalence of the two different forms of transparent conditions seen above. 2. T R A N S P A R E N T

BOUNDARY

CONDITIONS

In the general case, two different ways of expressing transparent conditions can be found. The first involves the Fourier-Laplace transform and the behaviour of the solution of the Cauchy Typeset by AA4S-TEX 55

56

L. DI MENZA

problem at infinity, which means that we have to know the decreasing properties of the solution. The second relies on a factorization of the differential operator, making a distinction between an operator which propagates to the right and another to the left. Both methods are based on pseudodifferential operators, whose symbols are not necessarily polynomial with respect to dual variables (see [11]). Here, transparent conditions for the SchrSdinger equation are found by means of an extension method, making use of fundamental solutions. Let ~ = {x = (x ~,xn) = ( x l , . . . , x n ) E R~; xn < 0} and F = {x = (x ~, xn) = ( X l , . . . , X n ) E ]l~n; Xn = 0}, and let us introduce the Cauchy problem iOtu + A u = O, x E R ~, t > O, u l t = 0 = ?20,

(1)

X E ~n.

We take two different extensions ~ and ~ of the solution of (1) to ~t × R~ as follows:

~(t, x ~, x,~) =

u(t,x',xn), 0,

ifxn >0, otherwise,

u(t,x',Xn) = { u(t,x',Xn), u(t, x I, -x,~),

ifxn >0, otherwise.

Considering now the fundamental solution of the SchrSdinger equation with support in R + × R~ • e -inTr/4

E+(t, x) = - z (47~t)n/2 ei(llxH2)/4ty(t), the jump formula applied for ~ and ~ through F leads to (see [10]):

Ox.u (t, x', O) + 2 lot

u(T, y', O)E+(t - ~-, x' - y', O) dy'dT

+

/ot/

,,-~ u (T, y', O) A ± E + (t - T, x' - y', O) dy'dTyt

]

= O,

(2)

n--1 2 where the transverse Laplacian A± is defined by A± = Y~'~k=l 0~. Another way to write these conditions with the dual variables (w, ~P) E Nw x II~,-1 of (t, x r) E N + x R n--1 x, , based on the pseudodifferential faetorization of the SehrSdinger operator on F, is

O z ~ + e - ~ / 4 (iw + iN~'H2) l/2 it -= o,

Xn ----0.

(3)

REMARKS.

1. Transparent conditions (2) have been established using strongly the form of the fundamental solution of the Schr5dinger equation. It seems difficult to extend this process to more complex partial differential equations, whose fundamental solutions are not necessarily known. Moreover, we clearly see that conditions (2) are nonlocal conditions in time as well as in space. 2. Formulas (2) and (3) give two different forms of transparent conditions. In the case n --- 1, the orthogonal term in (2) vanishes. These two expressions are similar, and lead to the one given in 18]. Nevertheless, in the next section, approximation techniques will be applied for conditions (3).

3. A P P R O X I M A T I O N

OF TRANSPARENT

CONDITIONS

From equation (3), we get local boundary conditions, using approximation techniques described in [5,9,10], in order to implement them for the numerical simulation of dispersive waves.

Absorbing Boundary Conditions

57

First, we take an approximation of the complex square root in equation (3): R ( Z ) = Z 1/2 is replaced by ak Z Rm(Z)

= a~ +

m

=

Z + d~

k=l ke{0,...,m},

a~>0,

rn

am _ k

ak d k Z + d mk '

k=0 k---1 d~>0, ke{1,...,m},

the coefficients being computed as follows: among the set EP~ of all the rational fractions r ( Z ) = Pm(Z)/Qm(Z) (with deg. Pm = m, and deg. Q,~ = m) which interpolate R at a family of complex points {w0,(±iwk)k=l ...... },

wk E[0, p],

kE{0,...,m}

and

w k + l >_wk,

kE

{1,...,m-I},

we choose the one denoted by R m which minimizes I I r ( i w ) - v/~HL2([0,p]~). Numerically, this fully nonlinear problem is solved with successive one-dimensional minimizations (for more technical details, see [5,10]). We set am G ~ = Gin(p, ( k (P)), (d'~(p))) = min nr(iw) - R(iw)HL2([o,p]~)

I I R m ( i w ) - R(iw)l]L~([o,p].~ ).

rCEP~

A double subscript notation is kept, because modifying the value of m leads us to compute all the coefficients (a~ ¢ a ~ + l , k = 0 . . . . . m). Even if no theoretical proof confirms it, we numerically see that Glm decreases to 0, as m is large. Thus, in Fourier variables, condition (3) becomes Ox~ ~z + e -i7r/4

am

,

a~d'~

~

u-

k=l

= O.

i ~ +i11~'11 ~ +

In the second step, we introduce auxiliary functions (~k = ~ k ( t , x ' ) k = l .....m) at the boundary, which satisfy 1 mt%-- q$k, k = 1,...,m, (a) i ~ + i11~'112 + d k

and after inversion, in the usual space a t ~ k -- iA_L~k + d r ~ k

= u,

(5)

k -= 1 , . . . , m .

Taking as initial data (;klt=o = 0 (k -- 1 , . . . , rn), boundary conditions become ~nU q-

e-iZc/4

a rn k

u --

a km dkm ~ k

]

= O,

on F, t > 0,

iOt~k+/kA_~k+id~k=iu,

onF, t>0,

k=l,...,m,

(6)

~k It=0 -- 0,

on F,

k = 1 . . . . . m.

These conditions can be seen as local conditions, since the nonlocal aspect in equation (4) is overcome by solving m SchrSdinger equations at the boundary. Numerically, these functions can be easily computed with finite-differences schemes (see [10]), or finite elements schemes (see [5]). 4. L I N K

BETWEEN

ABSORBING

TRANSPARENT CONDITIONS

AND

We now express boundary conditions (6) without auxiliary quantities. A change of variable cancelling the linear contribution in (5), and a convolution product of the source term with the fundamental solution E +_ 1 yields ~k(t,x')

= i

edr('-t)u(r,y',O)E+~_~(t

- r,z'

- y')dy'dr,

k = 1 .....

m.

58

L. DI MENZA

Replacing each ~0k in conditions (6), and using equation (5), the absorbing boundary conditions become Ox. u + e -i~/4

+

f0 t

lot

a~' + / _ . , k ~ k=l

(

a~ +

m ar~edr(r-t) k=l

)

)/.

.., u-

n-1 u'

u(r, y', 0)A±Zn+_l(t - T, X' -- y ' ) d y ' d r

= O.

(7)

From a practical point of view, conditions (7) are not useful, because of time and orthogonal space integrals. Nevertheless, they enable us to show the link between the two kinds of boundary conditions. We start from the relations for the coefficients a t ( p ) = v ~ a r ( 1 ) (k = 0 , . . . ,m), d r ( p ) = pdr(1 ) (k = 1 , . . . , m ) , and a m ( p , ( ak m( p ) ) , ( d r ( p ) ) ) = p a m ( 1 , ( a r ( 1 ) ) , ( d r ( 1 ) ) ) (see [5,10]), and we consider the coefficients still noted a r and d r which correspond to the L=-minimization on the interval [0, (G1)-1/2]. We have then , ( am k ), ( d r )

)=

(Glm) -1/2 G 1 ---- (([71) 1/2

Thus, these coefficients are chosen so that the approximation domain becomes larger, and IlRm(i~o) - R(iw)llL~ decreases to 0, as m tends to infinity. Denoting by Y the Heaviside distribution, we have the convergence property Ve > 0, 1/((1 + D t ) a / 2 + e ) . ~ - I ( Y P ~ ) converges to 1/((1 + D t ) a / 2 + e ) . ~ - I ( y R ) in L2(N), m --* oo. The proof relies on the following estimate on Rm: 3 C > O,

gw > O,

V m > 1,

IRm(iW)l <

C(1 + w),

and on the convergence of Rm(iW) to R(iw) a.e. in ]R+. Lebesgue's theorem achieves the proof. Let us now introduce the sequence (fro)m>1 defined with fm(t) = Y(t)(ar~ "" A-.~k=l X--'m a km e -drt~] (m _> 1, t _> 0) and f ( t ) = Y ( t ) / v / - ~ . We formally have . ~ - l ( R m ) ( t ) = Orfro(t)

and

~'-l(R)(t) = 0 t f ( t ) .

Therefore, the induction formula satisfied by each fundamental solution of the SchrSdinger equation e -i~r/4

n-1 , Vn _> 1, V ( t , x ' ) e R + x R x,

E+(t, x', O) -

-~--~E+_,(t,x ') =

1

e - i = / 4 f ( t ) E + _ l ( t , x')

allows us to see the link between conditions (2) and conditions (7), comparing e -i'~/4

/0

f m ( t - r)

~-, U(T, y ' ) E + _ l ( t - T, X' -- y') dy'dr yl

to e -'~/4

f ( t - r) .10

u(r, y')E+~_l(t - r, z' - y') dy'dr ~.~

yt

and e- ~ / 4

f m ( t - r)

~-~ u ( r , y ' ) A ± E + _ l ( t - r , x ' - y ' ) d y ' d r yt

to

e -'~/4

'u(r, y')AzE~+_l(t - r, x' - y') dy'dr

f ( t - T) aO

~7 ~

u(r, y', 0)A±E+(t - T, X' -- y', 0) dy'dr.

= 2 yl

Absorbing Boundary Conditions

59

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