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Chawla—Numerov method revisited
247
Journal of Computational and Applied Mathematics 36 (1991) 247-250 North-Holland
Letter Section
Chawla-Numerov
method revisited *
M. Meneguette Department of Mathematics, Faculty of Sciences and Technology, Srio Paul0 State University (UNESP), Presidente Prudente, SP, Brazil Received 26 September 1986 Revised 1 September 1990
Abstract Meneguette, M., Chawla-Numerov (1991) 247-250.
method revisited, Journal of Computational
and Applied Mathematics
36
The well-known two-step fourth-order Numerov method was shown to have better interval of periodicity when made explicit, see Chawla (1984). It is readily verifiable that the improved method still has phase-lag of order 4. We suggest a slight modification from which linear problems could benefit. Phase-lag of any order can be achieved, but only order 6 is derived. Keywork problems.
Chawla-Numerov
method, higher derivatives and phase-lag, periodic second-order
initial-value
The Numerov method Yn-+2
-
2Y,+i +Y, =
S2(f,+2
+
lOf,,l
+f,)
is the optimal two-step method of the Stormer-Cowell Y”=fk
Y),
Yo =Y(-%A
(1)
family for solving the initial problem
Yd ‘Y’(Xo)-
This method is fourth order and has (0, 6) as interval of periodic stability [4, Chapter 91. Its local truncation error LTE is given by LTE = &h6y,‘6!i + 0( h’). The stability and phase-lag can be derived by applying the method to the test equation y”=
-x2y,
x E R,.
* Work partially supported by CNPq (Brazil). 0377-0427/91/$03.50
Q 1991 - Elsevier Science Publishers B.V. (North-Holland)
(2)
248
M. Meneguette
/ Chawla-Numerov
method for ordinary differential equations
Solving (2) by (1) we obtain, for H2 = (Xh)2, Yn+2
2A(H2)
_
2Yn+l B(H
+Yn
(3)
= 0,
>
where and
A(H’)=l-&H2
= 1 + &H’.
B(H2)
Periodicity holds as long as the characteristic polynomial associated with (3) has zeros on the unit circle. That is so for H2 E (0, 6). The phase-lag analysis, for two-step methods, can be shown only to require the comparison of
4H2) BW)
...3
=l-;H2+$,H4-&H’+
with H2 cos H=l-2+41-6r+
H4
H6
... .
(4)
Clearly the order of phase-lag is 4, the actual phase-lag being H4/480. It is surprising that phase analysis has never been done in this way before; for example, in [2,5] a complicated tangent expansion is used. Chawla [l] improved (1) by making it explicit, that is, he replaced fn+2 by f,+2 =f(~,+~, j,,,), where u,+2
=
2Yn+, -Y,
+
h2fn+,,
(5)
whose LTE = &h4y,‘“+‘I + 0( h’). Hence the LTE of the new formula is h6 - ,,,Y,‘“!,
af
+ -(x,+2, aY
h6 Yn+,)~Yt$
+ O(h7).
The explicit method applied to (2) leads to (3) with and
B(H2)=1
A(H’)=l-q+$,
which again imposes order 4, but the phase-lag periodicity is enlarged to H2 E (0, 12) [l]. Although less attractive to nonlinear problems, provides phase-lag of any order; as an illustration however, imposed by linear problems as for instance problems. Let Y,,+~ be calculated from +
Yn+2 - 2Yn+I +Y,
= ikh2( fn+2 +
with fn+2 as before and dx f(
x
n+l,
Yn+l ).
However, the interval
of
incorporation of higher derivatives in (5) order 6 is derived. No major difficulty is, from semidiscretisation of some hyperbolic
h2fn+,+ 6h4f,‘:‘u
Yn+2 = 2Yn+I -Y,
f,‘:‘, := - d22
now is H4/6!.
lOfn+l+fn>,
(6)
M. Meneguette
/ Chawla-Numerov
method for ordinary differential equations
The error constant in our yn+z is l/20, therefore also reflected in the LTE of (6), which becomes _-
Applying
h6 24()yn(6!1+
af
and
which gives rise, by comparison
slightly smaller than that of Chawla’s. This is
h6
Y,+2)mY,‘%
-(x,+2, aY
(6) to (2) we obtain
B(H*)=l
249
+ O(h’)*
(3) with
A(I7*)=1--$+$-$, with (4), to phase-lag
H6/8!.
Higher
orders can be achieved
2.0
1.5
1.0
/ 0. 5
-0.5
-1.0
-1.5
-2. 0
Fig. 1. 1: Numerov; 2: Chawla-Numerov;
3: phase-lag 6; 4: phase-lag 8.
by
250
M. Meneguette / Chawla-Numerov
using higher derivatives with coefficients value y,, +z is given by
method for ordinary differentiaI equations
to match the cos H expansion, that is, the predicted
L
j,,+z
= ~Y,,+I
-Yn
d2’ f(x n+l, Yn+A +h2fn+l+ C h*j+*YjdX2j ;=1
where y, = 4!/(2j
+ 4)! and L is a positive integer.
Remark. A fourth-order explicit method with phase-lag of order 6 and similar periodicity characteristics as in (6) was presented in [3]. The interval of periodicity is calculated from 1A( H2)/B( H2) 1 -c 1, therefore a straightforward search followed by a bisection computation provides (0, 7.57) as the interval for (6). Although this is a smaller interval, it is still reasonably large. However, if the fourth total derivative of f is also incorporated in (6), the new method has the same LTE (7); the phase-lag becomes H’/lO! and the interval of periodicity is (0, 21.48121). Figure 1 gives the behaviour of A( H2)/B( H2) against H2 for the various methods.
Acknowledgements The author is grateful to Professor S. McKee and the referee for improvements in this note.
References 111M.M. Chawla, Numerov made explicit has better stability, BIT 24 (1984) 117-118. 121M.M. Chawla and P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of second order
periodic initial-value problems, J. Comput. Appl. Math. 11 (3) (1984) 277-281. [31 M.M. Chawla and P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: Explicit method, J. Comput. Appl. Math. 15 (3) (1986) 329-337. [41 J.D. Lambert, Computational Methods in Ordinav Differential Equations (Wiley, London, 1973). [51 E.H. Twizell, Phase-lag analysis for a family of two-step methods for second order periodic initial value problems, J. Comput. Appl. Math. 15 (2) (1986) 261-263.
Journal of Computational and Applied Mathematics 36 (1991) 247-250 North-Holland
Letter Section
Chawla-Numerov
method revisited *
M. Meneguette Department of Mathematics, Faculty of Sciences and Technology, Srio Paul0 State University (UNESP), Presidente Prudente, SP, Brazil Received 26 September 1986 Revised 1 September 1990
Abstract Meneguette, M., Chawla-Numerov (1991) 247-250.
method revisited, Journal of Computational
and Applied Mathematics
36
The well-known two-step fourth-order Numerov method was shown to have better interval of periodicity when made explicit, see Chawla (1984). It is readily verifiable that the improved method still has phase-lag of order 4. We suggest a slight modification from which linear problems could benefit. Phase-lag of any order can be achieved, but only order 6 is derived. Keywork problems.
Chawla-Numerov
method, higher derivatives and phase-lag, periodic second-order
initial-value
The Numerov method Yn-+2
-
2Y,+i +Y, =
S2(f,+2
+
lOf,,l
+f,)
is the optimal two-step method of the Stormer-Cowell Y”=fk
Y),
Yo =Y(-%A
(1)
family for solving the initial problem
Yd ‘Y’(Xo)-
This method is fourth order and has (0, 6) as interval of periodic stability [4, Chapter 91. Its local truncation error LTE is given by LTE = &h6y,‘6!i + 0( h’). The stability and phase-lag can be derived by applying the method to the test equation y”=
-x2y,
x E R,.
* Work partially supported by CNPq (Brazil). 0377-0427/91/$03.50
Q 1991 - Elsevier Science Publishers B.V. (North-Holland)
(2)
248
M. Meneguette
/ Chawla-Numerov
method for ordinary differential equations
Solving (2) by (1) we obtain, for H2 = (Xh)2, Yn+2
2A(H2)
_
2Yn+l B(H
+Yn
(3)
= 0,
>
where and
A(H’)=l-&H2
= 1 + &H’.
B(H2)
Periodicity holds as long as the characteristic polynomial associated with (3) has zeros on the unit circle. That is so for H2 E (0, 6). The phase-lag analysis, for two-step methods, can be shown only to require the comparison of
4H2) BW)
...3
=l-;H2+$,H4-&H’+
with H2 cos H=l-2+41-6r+
H4
H6
... .
(4)
Clearly the order of phase-lag is 4, the actual phase-lag being H4/480. It is surprising that phase analysis has never been done in this way before; for example, in [2,5] a complicated tangent expansion is used. Chawla [l] improved (1) by making it explicit, that is, he replaced fn+2 by f,+2 =f(~,+~, j,,,), where u,+2
=
2Yn+, -Y,
+
h2fn+,,
(5)
whose LTE = &h4y,‘“+‘I + 0( h’). Hence the LTE of the new formula is h6 - ,,,Y,‘“!,
af
+ -(x,+2, aY
h6 Yn+,)~Yt$
+ O(h7).
The explicit method applied to (2) leads to (3) with and
B(H2)=1
A(H’)=l-q+$,
which again imposes order 4, but the phase-lag periodicity is enlarged to H2 E (0, 12) [l]. Although less attractive to nonlinear problems, provides phase-lag of any order; as an illustration however, imposed by linear problems as for instance problems. Let Y,,+~ be calculated from +
Yn+2 - 2Yn+I +Y,
= ikh2( fn+2 +
with fn+2 as before and dx f(
x
n+l,
Yn+l ).
However, the interval
of
incorporation of higher derivatives in (5) order 6 is derived. No major difficulty is, from semidiscretisation of some hyperbolic
h2fn+,+ 6h4f,‘:‘u
Yn+2 = 2Yn+I -Y,
f,‘:‘, := - d22
now is H4/6!.
lOfn+l+fn>,
(6)
M. Meneguette
/ Chawla-Numerov
method for ordinary differential equations
The error constant in our yn+z is l/20, therefore also reflected in the LTE of (6), which becomes _-
Applying
h6 24()yn(6!1+
af
and
which gives rise, by comparison
slightly smaller than that of Chawla’s. This is
h6
Y,+2)mY,‘%
-(x,+2, aY
(6) to (2) we obtain
B(H*)=l
249
+ O(h’)*
(3) with
A(I7*)=1--$+$-$, with (4), to phase-lag
H6/8!.
Higher
orders can be achieved
2.0
1.5
1.0
/ 0. 5
-0.5
-1.0
-1.5
-2. 0
Fig. 1. 1: Numerov; 2: Chawla-Numerov;
3: phase-lag 6; 4: phase-lag 8.
by
250
M. Meneguette / Chawla-Numerov
using higher derivatives with coefficients value y,, +z is given by
method for ordinary differentiaI equations
to match the cos H expansion, that is, the predicted
L
j,,+z
= ~Y,,+I
-Yn
d2’ f(x n+l, Yn+A +h2fn+l+ C h*j+*YjdX2j ;=1
where y, = 4!/(2j
+ 4)! and L is a positive integer.
Remark. A fourth-order explicit method with phase-lag of order 6 and similar periodicity characteristics as in (6) was presented in [3]. The interval of periodicity is calculated from 1A( H2)/B( H2) 1 -c 1, therefore a straightforward search followed by a bisection computation provides (0, 7.57) as the interval for (6). Although this is a smaller interval, it is still reasonably large. However, if the fourth total derivative of f is also incorporated in (6), the new method has the same LTE (7); the phase-lag becomes H’/lO! and the interval of periodicity is (0, 21.48121). Figure 1 gives the behaviour of A( H2)/B( H2) against H2 for the various methods.
Acknowledgements The author is grateful to Professor S. McKee and the referee for improvements in this note.
References 111M.M. Chawla, Numerov made explicit has better stability, BIT 24 (1984) 117-118. 121M.M. Chawla and P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of second order
periodic initial-value problems, J. Comput. Appl. Math. 11 (3) (1984) 277-281. [31 M.M. Chawla and P.S. Rao, A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: Explicit method, J. Comput. Appl. Math. 15 (3) (1986) 329-337. [41 J.D. Lambert, Computational Methods in Ordinav Differential Equations (Wiley, London, 1973). [51 E.H. Twizell, Phase-lag analysis for a family of two-step methods for second order periodic initial value problems, J. Comput. Appl. Math. 15 (2) (1986) 261-263.