2 Multidimensional ARKN Methods and the Corresponding Order Conditions
Tải bản đầy đủ - 0trang
10.2 Multidimensional ARKN Methods and the Corresponding Order Conditions
213
Definition 10.1 An s-stage multidimensional ARKN scheme for the numerical integration of the initial value problem (10.1) is defined by
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
s
a¯ i j f (Y j , Y j ) − MY j ,
Yi = yn + hci yn + h 2
i = 1, . . . , s,
j=1
s
ai j f (Y j , Y j ) − MY j ,
Yi = yn + h
i = 1, . . . , s,
j=1
(10.3)
s
⎪
⎪
⎪
2
⎪
y
=
φ
(V
)y
+
hφ
(V
)y
+
h
b¯i (V ) f (Yi , Yi ),
⎪
n+1
0
n
1
n
⎪
⎪
⎪
i=1
⎪
⎪
⎪
s
⎪
⎪
⎪
⎪
⎪
y
=
φ
(V
)y
−
h
Mφ
(V
)y
+
h
bi (V ) f (Yi , Yi ),
0
1
n
⎩ n+1
n
i=1
where a¯ i j and ai j are real constants, the weight functions bi (V ) : Rd×d → Rd×d and
b¯i (V ) : Rd×d → Rd×d , i = 1, . . . , s, in the updates are matrix-valued functions of
V = h 2 M. The scheme (10.3) can also be denoted by the following Butcher tableau:
c
A
A¯
b (V ) b¯ (V )
c1 a11 . . . a1s
..
..
..
..
.
.
.
.
cs as1 · · · ass
=
a¯ 11
..
.
a¯ s1
. . . a¯ 1s
..
..
.
.
· · · a¯ ss
b1 (V ) · · · bs (V ) b¯1 (V ) · · · b¯s (V )
In the special case where the right-hand-side function of (10.1) is independent of
y , the scheme (10.3) reduces to the corresponding ARKN method for the special
oscillatory system [17],
y + M y = f (y),
y(t0 ) = y0 ,
t ∈ [t0 , T ],
(10.4)
y (t0 ) = y0 ,
namely,
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
s
a¯ i j f (Y j ) − MY j ,
Yi = yn + hci yn + h 2
i = 1, . . . , s,
j=1
s
b¯i (V ) f (Yi ),
⎪
⎪
i=1
⎪
⎪
⎪
⎪
s
⎪
⎪
⎪
⎪
bi (V ) f (Yi ).
⎩ yn+1 = φ0 (V )yn − h Mφ1 (V )yn + h
yn+1 = φ0 (V )yn + hφ1 (V )yn + h 2
i=1
(10.5)
214
10 Multidimensional ARKN Methods for General Multi-frequency …
The scheme (10.5) can also be denoted by the following Butcher tableau:
c
c1 a¯ 11 . . . a¯ 1s
..
..
..
..
.
.
.
.
cs a¯ s1 · · · a¯ ss
A¯
b¯ (V )
b (V )
=
b¯1 (V ) · · · b¯s (V )
b1 (V ) · · · bs (V )
A multidimensional ARKN method (10.3) is said to be of order p if, for a sufficiently smooth problem (10.1), the conditions
en+1 := yn+1 − y(tn + h) = O(h p+1 )
en+1 := yn+1 − y (tn + h) = O(h p+1 )
(10.6)
are satisfied simultaneously, where y(tn + h) and y (tn + h) are the respective exact
solution and its derivative of (10.1) at tn + h, whereas yn+1 and yn+1 are the onestep numerical approximations obtained by the method from the local assumptions:
yn = y(tn ) and yn = y (tn ). The order conditions, established in [19] for the method
(10.3), are given below.
Theorem 10.1 (Wu et al. [19]) A necessary and sufficient condition for a multidimensional ARKN method (10.3) to be of order p is given by
⎧
⎪
¯ ) − ρ(τ )! φρ(τ )+1 (V ) = O(h p−ρ(τ ) ), ρ(τ ) = 1, . . . , p − 1,
⎨ Φ(τ ) ⊗ Im b(V
γ (τ )
ρ(τ )!
⎪
⎩ Φ(τ ) ⊗ Im b(V ) −
φρ(τ ) (V ) = O(h p+1−ρ(τ ) ), ρ(τ ) = 1, . . . , p,
γ (τ )
where τ is the Nyström tree associated with the elementary differential F (τ )(yn , yn )
of the function f˜(y, y ) = f (y, y ) − M y at (yn , yn ), the entries Φ j (τ ), j =
1, . . . , s, of Φ(τ ) are weight functions defined in [8], and φρ(τ ) (V ), ρ(τ ) =
1, . . . , p, are given by (4.7).
10.3 ARKN Methods for General Multi-frequency
and Multidimensional Oscillatory Systems
This section constructs three explicit multidimensional ARKN schemes for the general multi-frequency and multidimensional oscillatory system (10.1) and analyses
the stability and phase properties of the schemes.
10.3 ARKN Methods for General Multi-frequency …
215
10.3.1 Construction of Multidimensional ARKN Methods
First, consider an explicit three-stage ARKN method of order three. Since f depends
on both y and y in (10.1), at least three stages are required to satisfy all the order
conditions. The explicit three-stage ARKN methods for (10.1) can be expressed by
the following Butcher tableau:
c
A
A¯
b (V ) b¯ (V )
=
c1 0
c2 a21
c3 a31
0
0
a32
0
0
0
0
a¯ 21
a¯ 31
0
0
a¯ 32
0
0
0
b1 (V ) b2 (V ) b3 (V ) b¯1 (V ) b¯2 (V ) b¯3 (V )
From Theorem 10.1, a three-stage ARKN method is of order three if its coefficients
satisfy
⎧
(e ⊗ I )b(V ) = φ1 (V ) + O (h 3 ),
(c ⊗ I )b(V ) = φ2 (V ) + O (h 2 ),
⎪
⎪
⎪
2
⎪
¯ ) = φ2 (V ) + O (h 2 ),
(c ) ⊗ I b(V ) = 2φ3 (V ) + O (h), (e ⊗ I )b(V
⎪
⎪
⎨
¯
¯
( Ae) ⊗ I b(V ) = φ3 (V ) + O (h),
(c ⊗ I )b(V ) = φ3 (V ) + O (h),
¯ ) = φ3 (V ) + O (h), (Ae) ⊗ I b(V ) = φ2 (V ) + O (h 2 ),
(Ae) ⊗ I b(V
⎪
⎪
⎪
⎪
⎪
(Ac) ⊗ I b(V ) = φ3 (V ) + O (h), (c · Ae) ⊗ I b(V ) = 2φ3 (V ) + O (h),
⎪
⎩
(A2 e) ⊗ I b(V ) = φ3 (V ) + O (h), (Ae · Ae) ⊗ I b(V ) = 2φ3 (V ) + O (h),
(10.7)
where e = (1, 1, 1) .
1
Choosing c1 = 0, c2 = , c3 = 1, and solving all the equations in (10.7), we
2
obtain
⎧
1
⎪
⎨ a21 = , a31 = −1, a32 = 2,
2
(10.8)
⎪
⎩ a¯ 21 = 1 , a¯ 31 = 1 , a¯ 32 = 0,
8
2
and
⎧
⎪
⎪
⎪
⎨ b1 (V ) = φ1 (V ) − 3φ2 (V ) + 4φ3 (V ),
b2 (V ) = 4φ2 (V ) − 8φ3 (V ),
⎪
⎪
⎪
⎩ b (V ) = −φ (V ) + 4φ (V ),
3
2
3
3
b¯1 (V ) = φ2 (V ) − φ3 (V ),
2
b¯2 (V ) = φ3 (V ),
1
b¯3 (V ) = φ3 (V ).
2
(10.9)
Equations (10.8)–(10.9) give a three-stage ARKN method of order three. The Taylor
expansions of the coefficients bi (V ) and b¯i (V ) of this method are given by
216
10 Multidimensional ARKN Methods for General Multi-frequency …
⎧
⎪
⎪
⎪ b1 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
b2 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ b3 (V ) =
3
5
1
7
1
I− V+
V2 −
V3 +
V4 + ··· ,
6
40
1008
51840
492800
1
1 2
2
1
1
I− V+
V −
V3 +
V4 + ··· ,
3
10
252
12960
1108800
1
1
1
1
1
I+
V−
V2 +
V3 −
V4 + ··· ,
6
120
1680
72576
5702400
1
1
19
7
11
⎪
⎪
⎪
b¯1 (V ) = I −
V+
V2 −
V3 +
V4 + ··· ,
⎪
⎪
4
240
10080
48348
79833600
⎪
⎪
⎪
⎪
1
1
1
1
1
⎪
⎪
V+
V2 −
V3 +
V4 + ··· ,
b¯2 (V ) = I −
⎪
⎪
6
120
5040
362880
39916800
⎪
⎪
⎪
⎪
1
1
⎪
⎩ b¯3 (V ) = 1 I − 1 V + 1 V 2 −
V3 +
V4 + ··· .
12
240
10080
725760
79833600
(10.10)
This method is denoted by ARKN3s3.
We next consider the explicit multidimensional ARKN methods of order four for
(10.1). In this case, four stages are needed. An explicit four-stage ARKN method
can be denoted by the following Butcher tableau:
c
A
A¯
b (V ) b¯ (V )
c1 0
c2 a21
c3 a31
c
= 4 a41
0
0
a32
a42
0
0
0
a43
0
0
0
0
0
a¯ 21
a¯ 31
a¯ 41
0
0
a¯ 32
a¯ 42
0
0
0
a¯ 43
0
0
0
0
b1 (V ) b2 (V ) b3 (V ) b4 (V ) b¯1 (V ) b¯2 (V ) b¯3 (V ) b¯4 (V )
The simplifying conditions Ae = c, A¯ = A2 are used to reduce the order to those
listed below,
⎧
(e ⊗ I )b(V ) = φ1 (V ) + O (h 4 ),
(c ⊗ I )b(V ) = φ2 (V ) + O (h 3 ),
⎪
⎪
⎪
2
2
2
⎪
⎪
⎪ (c 2) ⊗ I b(V ) = 2φ3 (V ) + O (h ), (Ac)2 ⊗ I b(V ) = φ3 (V ) + O (h ),
⎨
(A c) ⊗ I b(V ) = φ4 (V ) + O (h), (Ac ) ⊗ I b(V ) = 3φ4 (V ) + O (h),
⎪ (c3 ) ⊗ I b(V ) = 6φ4 (V ) + O (h), (c · Ac) ⊗ I b(V ) = 2φ4 (V ) + O (h),
⎪
⎪
⎪
¯ ) = φ2 (V ) + O (h 3 ),
¯ ) = φ3 (V ) + O (h 2 ),
(c ⊗ I )b(V
⎪ (e ⊗ I )b(V
⎪
⎩ 2
¯ ) = 2φ4 (V ) + O (h), (Ac) ⊗ I b(V
¯ ) = φ4 (V ) + O (h),
(c ) ⊗ I b(V
(10.11)
where e = (1, 1, 1, 1) .
10.3 ARKN Methods for General Multi-frequency …
217
1
1
Choosing c1 = 0, c2 = , c3 = , c4 = 1, and solving the above equations, we
2
2
obtain
⎧
1
1
⎪
⎨ a21 = , a31 = 0, a32 = , a41 = 0, a42 = 0, a43 = 1,
2
2
(10.12)
1
1
⎪
⎩ a¯ 21 = 0, a¯ 31 = , a¯ 32 = 0, a¯ 41 = 0, a¯ 42 = , a¯ 43 = 0,
4
2
and
⎧
b1 (V ) = φ1 (V ) − 3φ2 (V ) + 4φ3 (V ),
⎪
⎪
⎨
b2 (V ) = 2φ2 (V ) − 4φ3 (V ),
b (V ) = 2φ2 (V ) − 4φ3 (V ),
⎪
⎪
⎩ 3
b4 (V ) = −φ2 (V ) + 4φ3 (V ),
b¯1 (V ) = φ2 (V ) − 3φ3 (V ) + 4φ4 (V ),
b¯2 (V ) = 2φ3 (V ) − 4φ4 (V ),
b¯3 (V ) = 2φ3 (V ) − 4φ4 (V ),
b¯4 (V ) = −φ3 (V ) + 4φ4 (V ).
(10.13)
Equations (10.12)–(10.13) give a four-stage ARKN method of order four. The Taylor
expansions of the coefficients bi (V ) and b¯i (V ) of this method are
⎧
⎪
⎪
b1 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ b2 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
b3 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ b4 (V ) =
⎪
⎪
⎪
b¯1 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
b¯2 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ b¯3 (V ) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ b¯4 (V ) =
3
5
1
7
I− V+
V2 −
V3 + ··· ,
6
40
1008
51840
1
1 2
1
1
I− V+
V −
V3 + ··· ,
3
20
504
25920
1
1 2
1
1
I− V+
V −
V3 + ··· ,
3
20
504
25920
1
1
1
1
I+
V−
V2 +
V3 + ··· ,
6
120
1680
72576
1
1
1
1
I− V+
V2 −
V3 + ··· ,
6
45
1120
56700
1
1
1
1
I− V+
V2 −
V3 + ··· ,
6
90
3360
226800
1
1
1
1
I− V+
V2 −
V3 + ··· ,
6
90
3360
226800
1
1
1
V−
V2 +
V3 + ··· .
360
10080
604800
(10.14)
This method is denoted by ARKN4s4.
In what follows, the construction of the ARKN methods of order five for (10.1) is
considered. In this case, at least six stages are required. Thus, consider the following
Butcher tableau:
218
c1
c2
c3
c4
c5
c6
10 Multidimensional ARKN Methods for General Multi-frequency …
0
a21
a31
a41
a51
a61
0
0
a32
a42
a52
a62
0
0
0
a43
a53
a63
0
0
0
0
a54
a64
0
0
0
0
0
a65
0
0
0
0
0
0
0
a¯ 21
a¯ 31
a¯ 41
a¯ 51
a¯ 61
0
0
a¯ 32
a¯ 42
a¯ 52
a¯ 62
0
0
0
a¯ 43
a¯ 53
a¯ 63
0
0
0
0
a¯ 54
a¯ 64
0
0
0
0
0
a¯ 65
0
0
0
0
0
0
b1 (V ) b2 (V ) b3 (V ) b4 (V ) b5 (V ) b6 (V ) b¯1 (V ) b¯2 (V ) b¯3 (V ) b¯4 (V ) b¯5 (V ) b¯6 (V )
The order conditions up to order five are imposed using the simplifying assumptions Ae = c, A¯ = A2 and the coefficients must then satisfy
⎧
(e ⊗ I )b(V ) = φ1 (V ) + O (h 5 ),
(c ⊗ I )b(V ) = φ2 (V ) + O (h 4 ),
⎪
⎪
⎪
⎪
2
3
⎪
(Ac) ⊗ I b(V ) = φ3 (V ) + O (h 3 ),
(c ) ⊗ I b(V ) = 2φ3 (V ) + O (h ),
⎪
⎪
⎪
⎪
⎪
((A2 c) ⊗ I )b(V ) = φ4 (V ) + O (h 2 ),
((Ac2 ) ⊗ I )b(V ) = 2φ4 (V ) + O (h 2 ),
⎪
⎪
⎪
3
2
⎪
(c · Ac) ⊗ I b(V ) = 3φ4 (V ) + O (h 2 ),
⎪
⎨ (c ) ⊗ I b(V ) = 6φ4 (V ) + O (h ),
3
(c · A2 c) ⊗ I b(V ) = 4φ5 (V ) + O (h),
(A c) ⊗ I b(V ) = 6φ5 (V ) + O (h),
⎪
⎪
4
⎪ (c ) ⊗ I b(V ) = 24φ5 (V ) + O (h),
(c2 · Ac) ⊗ I b(V ) = 12φ5 (V ) + O (h),
⎪
⎪
⎪
⎪
⎪
(Ac · Ac) ⊗ I b(V ) = 6φ5 (V ) + O (h), (c · Ac2 ) ⊗ I b(V ) = 8φ5 (V ) + O (h),
⎪
⎪
⎪
⎪
⎪
(A(c · Ac)) ⊗ I b(V ) = 3φ5 (V ) + O (h),
(A2 c2 ) ⊗ I b(V ) = 2φ5 (V ) + O (h),
⎪
⎪
⎩
3
(Ac ) ⊗ I b(V ) = 6φ5 (V ) + O (h),
(10.15)
and
⎧
¯ ) = φ2 (V ) + O (h 4 ),
(e ⊗ I )b(V
⎪
⎪
⎪
⎨ (c2 ) ⊗ I b(V
¯ ) = 2φ4 (V ) + O (h 2 ),
¯ ) = φ5 (V ) + O (h),
⎪
(A2 c) ⊗ I b(V
⎪
⎪
⎩ 3
¯ ) = 6φ5 (V ) + O (h),
(c ) ⊗ I b(V
¯ ) = φ3 (V ) + O (h 3 ),
(c ⊗ I )b(V
¯ ) = φ4 (V ) + O (h 2 ),
(Ac) ⊗ I b(V
¯ ) = 2φ5 (V ) + O (h),
(Ac2 ) ⊗ I b(V
¯ ) = 3φ5 (V ) + O (h),
(c · Ac) ⊗ I b(V
(10.16)
where e = (1, 1, 1, 1, 1, 1) .
1
1
1
2
Choosing c1 = 0, c2 = , c3 = , c4 = , c5 = and c6 = 1, and solving the
6
3
2
3
above equations, we obtain
⎧
⎪
⎪
a21
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
a43
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨ a61
⎪
⎪
⎪
⎪
a¯ 21
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
a¯ 43
⎪
⎪
⎪
⎪
⎪
⎩ a¯ 61
1
1
, a41 = − ,
3
4
1
2
1
= 0,
a51 = − , a52 = , a53 = ,
27
9
3
2
3
27
, a63 =
, a64 = −4,
= − , a62 =
11
11
11
1
1
, a¯ 32 = 0, a¯ 41 = ,
= 0,
a¯ 31 =
18
8
2
= 0,
a¯ 51 = 0,
a¯ 52 = , a¯ 53 = 0,
9
21
18
9
4
=
, a¯ 62 = − , a¯ 63 =
, a¯ 64 =
,
22
11
11
11
=
1
,
6
a31 = 0,
a32 =
3
,
4
4
a54 =
,
27
27
,
a65 =
11
a42 =
a¯ 42 = 0,
a¯ 54 = 0,
a¯ 65 = 0,
(10.17)
10.3 ARKN Methods for General Multi-frequency …
and
⎧
15
⎪
φ2 (V ) + 40φ3 (V ) − 135φ4 (V ) + 216φ5 (V ),
b1 (V ) = φ1 (V ) −
⎪
⎪
⎪
2
⎪
⎪
⎪ b (V ) = 0,
⎪
⎪
2
⎪
⎪
⎪
⎪
b
⎪
3 (V ) = 27 φ2 (V ) − 9φ3 (V ) + 39φ4 (V ) − 72φ5 (V ) ,
⎪
⎪
⎪
⎪
⎪
b
(V ) = −32 φ2 (V ) − 11φ3 (V ) + 54φ4 (V ) − 108φ5 (V ) ,
⎪
⎪ 4
⎪
⎪
⎪
27
⎪
⎪
b5 (V ) =
φ2 (V ) − 12φ3 (V ) + 66φ4 (V ) − 144φ5 (V ) ,
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
b6 (V ) = −φ2 (V ) + 13φ3 (V ) − 81φ4 (V ) + 216φ5 (V ),
⎪
⎪
⎪
⎨
64
b¯1 (V ) = φ2 (V ) − 5φ3 (V ) +
φ4 (V ) − 13φ5 (V ),
⎪
5
⎪
⎪
⎪
⎪
b¯2 (V ) = 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎪ b¯3 (V ) = 9φ3 (V ) − 171 φ4 (V ) + 45φ5 (V ),
⎪
⎪
⎪
5
⎪
⎪
⎪
⎪
64
⎪
¯
⎪
b
(V
)
=
−4φ
(V
)
+
φ4 (V ) − 16φ5 (V ),
4
3
⎪
⎪
5
⎪
⎪
⎪
⎪
54
⎪
⎪
b¯5 (V ) =
φ4 (V ) − 27φ5 (V ),
⎪
⎪
⎪
5
⎪
⎪
⎪
⎪
⎩ b¯ (V ) = − 11 φ (V ) + 11φ (V ).
6
4
5
5
219
(10.18)
Equations (10.17)–(10.18) give a six-stage ARKN method of order five. The Taylor
expansions of the coefficients (10.18) are given by
⎧
11
259
3
25 2
3
⎪
⎪
⎪ b1 (V ) = 120 I − 70 V + 8064 V − 2851200 V + · · · ,
⎪
⎪
⎪
⎪
⎪
⎪
b2 (V ) = 0,
⎪
⎪
⎪
⎪
⎪
27
17
99
9 2
⎪
⎪ b3 (V ) =
I−
V+
V −
V3 + ··· ,
⎪
⎪
40
560
896
70400
⎪
⎪
⎪
⎪
8
19
4
1 2
⎪
⎪
b (V ) = − I +
V−
V +
V3 + ··· ,
⎪
⎪
⎪ 4
15
35
126
89100
⎪
⎪
⎪
27
3
9
3 2
⎪
⎪
⎪
b (V ) =
I−
V+
V −
V3 + ··· ,
⎪
⎪ 5
40
140
896
35200
⎪
⎪
⎪
⎪
11
47
1
1
⎪
⎪
I+
V−
V2 +
V3 + ··· ,
⎨ b6 (V ) =
120
336
4480
7983360
2839
⎪
⎪ b¯ (V ) = 11 I − 383 V + 1231 V 2 −
⎪
V3 + ··· ,
1
⎪
⎪
120
25200
1814400
199584000
⎪
⎪
⎪
⎪
⎪ b¯2 (V ) = 0,
⎪
⎪
⎪
⎪
⎪
61
⎪ b¯ (V ) = 9 I − 51 V + 107 V 2 −
⎪
V3 + ··· ,
⎪
3
⎪
20
1400
100800
3696000
⎪
⎪
⎪
⎪
4
197
59
59
⎪¯
⎪
⎪
b4 (V ) = − I +
V−
V2 +
V3 + ··· ,
⎪
⎪
15
3150
113400
24948000
⎪
⎪
⎪
⎪
9
17
27
13
⎪
⎪
b¯5 (V ) =
I−
V+
V2 −
V3 + ··· ,
⎪
⎪
40
2280
67200
7392000
⎪
⎪
⎪
⎪
1
⎪
⎩ b¯6 (V ) = 11 V − 11 V 2 +
V3 + ··· .
12600
453600
3024000
This method is denoted by ARKN6s5.
(10.19)
220
10 Multidimensional ARKN Methods for General Multi-frequency …
10.3.2 Stability and Phase Properties of Multidimensional
ARKN Methods
This section is concerned with the stability and phase properties of multidimensional ARKN methods (10.3). In the case of the classical RKN methods, the stability
properties are analysed using the second-order homogeneous linear test equation
y (t) = −λ2 y(t), with λ > 0.
(10.20)
Since the ARKN methods integrate y + M y = 0 exactly, it is pointless to consider
the stability and phase properties of ARKN methods on the basis of the conventional
linear test equation (10.20).
For the stability analysis of multidimensional ARKN methods, we use the following revised test equation [13, 16]:
y (t) + ω2 y(t) = −εy(t), with ε + ω2 > 0,
(10.21)
where ω represents an estimate of the dominant frequency λ and ε = λ2 − ω2 is the
error of that estimation. Applying an ARKN integrator to (10.21), we obtain
⎧
z = εh 2 ,
V = h 2 ω2 ,
⎨ Y = eyn + chyn − (V + z)AY,
y
= φ0 (V )yn + φ1 (V )hyn − z b¯ (V )Y,
⎩ n+1
hyn+1 = −V φ1 (V )yn + φ0 (V )hyn − zb (V )Y.
(10.22)
It follows from (10.22) that
yn+1
hyn+1
= R(V, z)
yn
hyn
,
where the stability matrix R(V, z) is given by
R(V, z) =
φ0 (V ) − z b¯ (V )N −1 e φ1 (V ) − z b¯ (V )N −1 c
−V φ1 (V ) − zb (V )N −1 e φ0 (V ) − zb (V )N −1 c
,
with N = I + (V + z)A and e = (1, 1, . . . , 1) .
The spectral radius ρ(R(V, z)) represents the stability of an ARKN method. Since
the stability matrix R(V, z) depends on the variables V and z, the characterization of
stability is determined by two-dimensional regions in the (V, z)-plane. Accordingly,
we have the following definitions of stability for an ARKN method:
(i) Rs = {(V, z)| V > 0, z > 0 and ρ(R) < 1} is called the stability region of an
ARKN method.
(ii) R p = {(V, z)| V > 0, z > 0, ρ(R) = 1 and tr(R)2 < 4det(R)} is called the
periodicity region of an ARKN method.
10.3 ARKN Methods for General Multi-frequency …
Stability Region of Method ARKN3s3
10
8
6
4
Stability Region of Method ARKN6s5
Stability Region of Method ARKN4s4
10
5
0
5
10
V
15
20
z
z
z
2
0
−2
−4
−6
−8
−10
10
8
6
4
2
0
−2
−4
−6
−8
−10
221
0
−5
−10
0
5
10
V
15
20
0
5
10
V
15
20
Fig. 10.1 Stability regions of the methods ARKN3s3 (left), ARKN4s4 (middle), and ARKN6s5
(right)
(iii) If Rs = (0, ∞) × (0, ∞), the ARKN method is called A-stable.
(iv) If R p = (0, ∞) × (0, ∞), the ARKN method is called P-stable.
The stability regions based on the test equation (10.21) for the methods derived
in this section are depicted in Fig. 10.1.
For the integration of oscillatory problems, it is common practice to consider the
phase properties (dispersion order and dissipation order) of the numerical methods.
Definition 10.2 The quantities
tr(R)
, d(H ) = 1 −
√
2 det(R)
φ(H ) = H − arccos
det(R)
are, respectively, called
√ the dispersion error and the dissipation error of ARKN methods, where H = V + z. Then, a method is said to be dispersive of order q and
dissipative of order r , if φ(H ) = O(H q+1 ) and d(H ) = O(H r +1 ). If φ(H ) = 0
and d(H ) = 0, then the method is said to be zero dispersive and zero dissipative.
The dissipation errors and the dispersion errors for the methods derived in
Sect. 10.3.1 are
• ARKN3s3:
d(H ) =
ε
H 4 + O(H 6 ),
96(ε + ω2 )
φ(H ) = −
• ARKN4s4:
ε
H 5 + O(H 7 );
480(ε + ω2 )
d(H ) =
ε(4ε + 3ω2 ) 6
H + O(H 8 ),
576(ε + ω2 )2
φ(H ) =
ε
H 5 + O(H 7 );
120(ε + ω2 )
222
10 Multidimensional ARKN Methods for General Multi-frequency …
• ARKN6s5:
d(H ) = −
φ(H ) =
ε(154ε + 129ω2 ) 6
H + O(H 8 ),
110880(ε + ω2 )2
ε(44ε + 27ω2 ) 7
H + O(H 9 ).
36960(ε + ω2 )2
10.4 Numerical Experiments
In order to show the remarkable efficiency of the ARKN methods derived in Sect. 10.3
in comparison with some existing methods in the scientific literature, four problems
are considered. The methods we select for comparison are
•
•
•
•
ARKN3s3: the three-stage ARKN method of order three;
ARKN4s4: the four-stage ARKN method of order four;
ARKN6s5: the six-stage ARKN method of order five;
RKN4s4: the classical four-stage RKN method of order four (see, e.g. II.14 of
[8]);
• RKN6s5: the six-stage RKN method of order five obtained from ARKN6s5 with
V = 0.
For each experiment, we display the efficiency curves of accuracy versus computational cost as measured by the number of function evaluations required by each
method.
Problem 10.1 The linear problem:
⎧
2
⎪
⎨ y (t) + ω y(t) = −δy (t),
⎪
⎩ y(0) = 1, y (0) = − δ .
2
The analytic solution of this initial value problem is given by
⎛
⎞
2
δ
δ
t⎠ .
y(t) = exp − t cos ⎝ ω2 −
2
4
In this test, the parameter values ω = 1, δ = 10−3 are chosen. The problem is
integrated on the interval [0, 100] with the stepsizes h = 1/2 j , j = 1, 2, 3, 4. The
numerical results are displayed in Fig. 10.2 (left).
10.4 Numerical Experiments
Problem 1: The efficiency curves
−1
ARKN3s3
ARKN4s4
ARKN6s5
RKN4s4
RKN6s5
−2
−1
−2
−3
−4
−5
−6
−7
−8
−9
−10
−11
2.6 2.8
Problem 2: The efficiency curves
ARKN3s3
ARKN4s4
ARKN6s5
RKN4s4
RKN6s5
log10(GE)
−3
log10(GE)
223
−4
−5
−6
−7
−8
−9
−10
2.6
2.8
3.0 3.2 3.4 3.6 3.8 4.0
log10(Function evaluations)
3.0 3.2 3.4 3.6 3.8 4.0
log10(Function evaluations)
Fig. 10.2 Results for Problems 10.1 and 10.2: The log–log plot of maximum global error against
number of function evaluations
Problem 10.2 The van de Pol equation:
⎧
2
⎪
⎨ y (t) + y(t) = δ(1 − y(t) )y (t),
⎪
⎩ y(0) = 2 + 1 δ 2 + 1033 δ 4 + 1019689 δ 6 , y (0) = 0,
96
552960
55738368000
with δ = 0.8 × 10−4 . Integrate the problem on the interval [0, 100] with the stepsizes
h = 1/2 j , j = 1, 2, 3, 4. In order to evaluate the error for each method, a reference
numerical solution is obtained by the method RKN4s4 in II.14 of [8] with a very
small stepsize. The numerical results are displayed in Fig. 10.2 (right).
Problem 10.3 The oscillatory initial value problem
⎧
⎪
⎪
⎨ y (t) +
⎪
⎪
⎩ y(0) =
12ε
13 −12
y(t) =
−12 13
5
−4
ε
,
, y (0) =
6
ε
with
f 1 (t) =
3 2
−2 −3
y (t) + ε2
f 1 (t)
, 0 < t ≤ tend ,
f 2 (t)
36
24
sin(t) + 24 sin(5t), f 2 (t) = − sin(t) − 36 sin(5t).
5
5
The analytic solution to this problem is given by
y(t) =
sin(t) − sin(5t) + ε cos(t)
.
sin(t) + sin(5t) + ε cos(5t)