A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented

Mickens, Ronald E.

English

NASA Technical Reports Server

N91-28073
CONSTRUCTION OF STABLE EXPLICIT FINITE-DIFFERENCE
SCHEMES FOR SCHRODINGER TYPE DIFFERENTIAL EQUA_ONS
Ronald E. Mickens
Departments of Mathematics and Physics
Clark-Atlanta University
Atlanta, Georgia
Technical Monitor
John Shoosmith
Langley Research Center
ABSTRACT
A family of conditionally stable, forward Euler finite-difference equations can be constructed for
the simplest equation of Schr&tinger type, namely ut = iuxr. Generalization of this result to physically
realistic Schr&linger type equations is presented.
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https://ntrs.nasa.gov/search.jsp?R=19910018759 2020-03-19T16:43:27+00:00Z
CONS_UCTION OF STABLE EXPLICIT FINITE-DWFERENCE
SCHEMES FOR SCHRODINGER TYPE DIFFERENTIAL EQUATIONS
Introduction
Discrete finite-difference models of differential equations have been a traditional and popular
technique for obtaining numerical solutions of both ordinary and partial differential equations [1,2].
An essential component of this procedure is the replacement of the derivative by its discrete analogue
based on the definition of the derivative as a limit process; that is
dx x(t +h) - x(t)
_- = Lim h ' (1)
h-,,0
and
dx Xk+l --xk
• - -_ h ' (2)
where xk is an approximation to x(tk) with tk = hk. However, more general definitions of the derivative
can be defined. As an example consider
dx x(t +h) - x(t)
d--i-= Limh--O ¢(h) ' (3)
where _p(h) has the property
¢p(h) = h +0(h2). (4)
Note that for this case the discrete replacement for the derivative is
dx Xk+I -- Xk
• " '" ¢(h) ' (5)
which, in general, differs from Eq. (2) for finite values of the step-size h.
The major implication of the above remarks is that new discrete models of differential equations
can be constructed based on Eq. (3). These models have the potential for providing solutions that arise
when the conventional discrete derivation of Eq. (2) is used.
As an illustration, consider the differential equation
dx
_- = i3,x, x(t0) = xo, 2 = real parameter, (6)
with solution
48
x(t) = x0 exp [ i,t (t - to) ]. (7)
Application of conventional procedures [1] gives, for example, the finite-difference scheme
Xk+l -- Xk
h - i2Xk. (8)
In contrast, consider the following exact scheme [3]
Xk+l -- Xk
_(h) = iRxk, x0 = given, (9)
where
e/__l
_(h) = i_ (10)
An easy and direct calculation [4] shows that for Eqs. (6) and (8), we have for their solutions
x(tk) _ _, while for Eqs. (6) and (9), x(tk) = xk for all step-sizes h. Also, from Eq. (10), we obtain
_(h) = h + i2h2 + 0(h3). (11)
Consider the simplest Schr6dinger-type equation
Ut = iuxr. (12)
Conventional finite-differences, using a forward Euler replacement for the time-derivative, give a
scheme that is unconditionally unstable [5]. We now show how conditionally stable schemes can be
constructed.
To proceed, based on the discussion presented earlier, we construct the following finite-difference
scheme
u_ +I - u_ u,,,+l - 2u_ + u_-I
i_l(At,A) - _2 (Ax,A) (13)
where the "denominator functions" have the properties
¢1(At,2) =At + i2(At) 2 + 0[(At) 31 (14a)
_2 (Ax, t) = (Ax) 2 + 0[(Ax) 3] (14b)
and _., for the moment, is an unspecified parameter. Defining R(At, Ax, 2) as
 l(At, 4)
R - _pE(AX,2) = RI(At, Ax, 2) + iR2 (At, Ax, 2) (15)
we can rewrite Eq. (13) in the form
49
= +1 + (1 -2R)
The substitution of a typical Fourier mode
u_ = C(n) exp [ia,(Ax)m],
+ Ru_.
(17)
into Eq. (16) and requiring the C(n) be bounded allows the determination of the stability properties.
(This concept of stability is called "practical stability" [5,6].) A straight-forward calculation shows that
Eq. (16) is stable if the following condition is satisfied.
12 1
(R1 - _) + (R2) 2 _ i-6" (18)
This relation has the interesting geometric interpretation: In the (R1,R2) plane, Eq. (16) is stable for
points on and inside the circle of radius 0.25 centered at (0.25, 0). We will refer to the inequality of Eq.
(18) as the circle condition.
For a given application, the following procedure is to be followed:
a) Select denominator functions with the properties given in Eqs. (14) and calculate R in Eq. (15).
b) Choose a point (R1, R2) consistent with the circle condition of Eq. (18).
c) Determine RffAt, Ax, 2) and R2(At, Ax, 2) from Eq. (15) and set them equal, respectively to
if,1 and if,2, i.e.,
RI(At, Ax, it) = ffq, ff,2(At, Ax, it) = if,2. (19)
d) Select a value for the space-step, Ax, and solve the two relations of Eq. (19) for At and it in
terms of Ax. Doing this gives
At = f1(Ax), it = fE(AX). (20)
Thus, the selection of the point (RI, R2), satisfying the circle condition and the relations of Eq. (20),
defines completely the finite-difference scheme of Eq. (16).
As an example, consider the following denominator functions
_1 = At + i2(At) 2, _2 = (AX) 2, (21)
with
R=I 4 (22)
The following results are easily obtained
50
_ 1-i (23)At -- , 2 = 4 , Cpl = 4 "
and
Note that everything can be expressed in terms of Ax, the space step-size; its value can be selected as
desired.
A second, nontrivial example is the nonlinear Schr'6dinger equation
u, -- + lul2u. (25)
A finite-difference scheme that embodies the work of this paper and also uses the nonlocal modeling
of the nonlinear term [3] is
u_+l -u'qn_p_ -i -[unto+l- 2u'qn +unm-1]_ -
+ ]Un+l,2 (26)
where the star (*) denotes complex conjugation, and the denominator functions ¢1(At, 2) and
_2(Ax, 2) satisfy the conditions of Eqs. (14). Note that Eq. (20) is an explicit finite-difference scheme.
In summary, we have presented a new procedure to construct explicit finite-difference schemes
for SchrSdinger type partial differential equations. In general, we expect these schemes to be condi-
tionally stable. This paper summarizes research that has already been published in Physics Review A,
June 1989 and in the "Proceedings of the 2nd IMAC Conference on Computational Acoustics."
References
1. M.K. Jain, Numerical Solution of Differential Equations, Halstead Press, New York, 1984.
2. A.R. Mitchell, Computational Methods in Partial Differential Equations, Wiley,
New York, 1969.
3. R.E. Mickens, Mathematical modeling of differential equations by difference equations,
in ComputationalAcoustics, Vol. 1, D. Lee, R.L. Sternberg and M.H. Schultz, eds.,
North-Holland, Amsterdam, 1988.
4. R.E. Mickens, Difference Equations, Van Nostrand Reinhold, New York, 1987.
51
.6.
T.F. Chan, D. Lee and L Shen, Stable explicit schemes for equations for the Schr'6dinger
type,Siam JournalofNumericalAnalysis,23 (1986),274-281.
R.D. Richtmyer and K.W. Morton, Difference Methods for Initial-Value Problems,
Interscience, New York, 1967.
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Construction of stable explicit finite-difference schemes for Schroedinger type differential equations

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