Order conditions for general linear methods
A. Cardone,∗ Z. Jackiewicz,† James H. Verner,‡ and Bruno Welfert§
October 6, 2013
Abstract
We describe the derivation of order conditions, without restrictions on stage order, for general linear methods for ordinary differential equations. This derivation is based on the extension of Albrecht
approach proposed in the context of Runge-Kutta and composite and
linear cyclic methods. This approach was generalized by Jackiewicz
and Tracogna to two-step Runge-Kutta methods, by Jackiewicz and
Vermiglio to general linear methods with external stages of different
orders, and by Garrappa to some classes of Runge-Kutta methods for
Volterra integral equations with weakly singular kernels. This leads
to general order conditions for many special cases of general linear
methods such as diagonally implicit multistage integration methods,
Nordsieck methods, and general linear methods with inherent RungeKutta stability. Examples of high order methods with some desirable
stability properties are also presented.
Keywords: General linear methods; order conditions; Nordsieck methods; two-step Runge-Kutta formulas
2010 MSC : 65L05, 65L20
∗
Dipartimento di Matematica, Università di Salerno, Fisciano (Sa), 84084 Italy, e-mail:
[email protected]. The work of this author was supported by travel fellowship from the
Department of Mathematics, University of Salerno.
†
Department of Mathematics, Arizona State University, Tempe, Arizona 85287, e-mail:
[email protected], and AGH University of Science and Technology, Kraków, Poland.
‡
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, B.C. V5A 1S6, Canada e-mail: [email protected].
§
Department of Mathematics, Arizona State University, Tempe, Arizona 85287, e-mail:
[email protected]
1
1
Introduction
Consider the initial value problem for a system of ordinary differential equations (ODEs)
{
(
)
y ′ (x) = f y(x) , x ∈ [x0 , X],
(1.1)
y(x0 ) = y0 ,
where y0 ∈ Rm , and the function f : Rm → Rm is assumed to be sufficiently
smooth. Let N be a positive integer and define the stepsize h = (X − x0 )/N ,
and the uniform grid xn = x0 + nh, n = 0, 1, . . . , N . To approximate the
solution y = y(x) to (1.1) we consider the class of general linear methods
(GLMs) defined by

s
r
∑
∑

[n+1]
[n+1]
[n]


=h
aij f (Yj
)+
uij yj , i = 1, 2 . . . , s,

 Yi
j=1
j=1
(1.2)
s
r
∑
∑

[n+1]
[n+1]
[n]


=h
bij f (Yj
)+
vij yj , i = 1, 2 . . . , r,

 yi
j=1
j=1
[n+1]
n = 0, 1, . . . , N − 1. Here, Yi
order q to y(xn + ci h), i.e.,
[n+1]
Yi
are approximations, possibly of low stage
= y(xn + ci h) + O(hq+1 ),
i = 1, 2, . . . , s,
(1.3)
[n]
where q is the minimum of all stage orders, and yi are approximations of
order p to the linear combinations of the derivatives of the solution y at the
point xn , i.e.,
[n]
yi
=
p
∑
qik hk y (k) (xn ) + O(hp+1 ),
i = 1, 2, . . . , r.
(1.4)
k=0
These methods can be characterized by the abscissa vector c = [c1 , . . . , cs ]T ,
the coefficient matrices
A = [aij ] ∈ Rs×s , U = [aij ] ∈ Rs×r , B = [aij ] ∈ Rr×s , V = [aij ] ∈ Rr×r ,
the vectors q0 , q1 , . . . , qp defined by




q1,1
q1,0


 . 
..  , q1 =  ...  ,
q0 = 




qr,1
qr,0
2

q1,p
 . 
. 
qp = 
 . ,
qr,p

...,
and the four integers: the order p, the stage order q, the number of external
approximations r, and the number of stages or internal approximations s.
Introducing the notation


 (



[n+1] )
[n+1]
[n]
f Y1
Y1
y1





(
) 
..
..
 , f Y [n+1] = 
 , y [n] =  ...  ,
Y [n+1] = 
.
.






( [n+1] )
[n+1]
[n]
f Ys
Ys
ys
the GLM (1.2) can be written in a more compact vector form

(
)
 Y [n+1] = h(A ⊗ I)f Y [n+1] + (U ⊗ I)y [n] ,
 y [n+1] = h(B ⊗ I)f (Y [n+1] ) + (V ⊗ I)y [n] ,
(1.5)
n = 0, 1, . . . , N − 1, where I is the identity matrix of dimension m and ‘⊗’
stands for the Kronecker product of matrices. Moreover, the relations (1.4)
and (1.5) take the form
Y [n+1] = y(xn + ch) + O(hq+1 ),
(1.6)
y [n] = z(xn ) + O(hp+1 ),
(1.7)
and

y(x + c1 h)


..
 ∈ Rs×m ,
y(x + ch) = 
.


y(x + cs h)

where
and the so-called exact value function z(x) ∈ Rs×m is defined by
z(x) =
p
∑
qk hk y (k) (x) = q0 y(x) + q1 hy ′ (x) + · · · + qp hp y (p) (x).
k=0
To guarantee zero-stability we assume that the coefficient matrix V is
power bounded. This is equivalent to the condition that the minimal polynomial of V has no zeros with magnitude greater than 1 and all zeros with
magnitude equal to 1 are simple. If V has only one eigenvalue on the unit
circle equal to 1 and all other eigenvalues inside of the unit circle the GLM
(1.5) is called strictly zero-stable.
3
Applying (1.5) to the linear test equation
y ′ = ξ y,
t ≥ 0,
(1.8)
where ξ is a complex parameter, leads to the matrix recurrence relation
y [n+1] = M(z)y [n] ,
n = 0, 1, . . . ,
(1.9)
where z = hξ and the so-called stability matrix M(z) is defined by
M(z) = V + zB(I − zA)−1 U.
We also define the stability function p(w, z) by the formula
(
)
p(w, z) = det wI − M(z) .
(1.10)
(1.11)
This function determines the stability properties (absolute stability) of GLMs
(1.5) with respect to the test equation (1.8). The GLM (1.5) is said to have
Runge-Kutta stability if the stability function p(w, z) has only one nonzero
root, i.e., if p(w, z) assumes the form
(
)
p(w, z) = wr−1 w − R(z) ,
(1.12)
where R(z) is an approximation of order p to the exponential function exp(z).
Here, p is the order of the method (1.5).
The GLMs were first introduced by Butcher [13] using somewhat different notation, and the modern theory of these methods was developed in
[14, 16, 17, 42, 43, 45, 65]. These methods include as special cases many
numerical methods for ODEs, for example, linear multistep methods [42–
44, 55, 56, 58, 59], predictor-corrector methods [55, 56, 58, 59], Runge-Kutta
(RK) methods [14, 16, 40, 42, 55, 56], diagonally implicit multistage integration methods (DIMSIMs) [15, 24–28, 30–35], Nordsieck methods [7, 20, 29],
two-step Runge-Kutta (TSRK) methods [6, 7, 46–50, 52, 53, 60–62], GLMs
with inherent Runge-Kutta stability (IRKS) [31, 35, 65, 66], and peer methods [9, 54, 57, 63, 64]. For additional examples we refer to [45] and the
references therein. Some special cases of GLMs are reviewed in Section 2.
The paper is organized as follows. Section 2 contains a short review of
some classes of GLMs. Section 3 deals with the derivation of local discretization errors of GLMs by Taylor series expansion. The main results on order
conditions are illustrated in Section 4. In Section 5 these theoretical results
are used to derive order conditions up to order six. Examples of application of the order conditions on various methods are given in Section 6. Last
section contains some concluding remarks.
4
2
Some special cases of GLMs
In this section we will review briefly some special cases of GLMs (1.5) which
will be used later to illustrate the general theory of order conditions.
2.1
Linear multistep methods
The class of linear multistep methods is defined by
yn+1 =
k
∑
αj yn+1−j + h
j=1
k
∑
βj f (yn+1−j ),
(2.1)
j=0
n = k − 1, k, . . . , N − 1, where y0 , y1 , . . . , yk−1 are given starting values which
are approximations to y(x0 ), y(x1 ), . . . , y(xk−1 ). Putting Y [n+1] = yn+1 and
[
]T
y [n] = yn yn−1 · · · yn−k+1 hf (yn ) hf (yn−1 ) · · · hf (yn−k+1 )
the method (2.1) can be represented as GLM with
c = c1 = 1, and the coefficient matrices given by

β0 α1 · · · αk−1 αk β1

 β0 α1 · · · αk−1 αk β1


 0 1 ···
0
0 0
 .
.
.
..
..
.

..
..
..
[
]  ..
.
.

A U
=
1
0 0
 0 0 ···
B V

 1 0 ···
0
0 0

 0 0 ···
0
0 1

 .
.
.
..
..
.
 ..
..
..
..
.
.

0 0 ···
0
0 0
r = 2k, s = 1, and with
···
βk−1 βk


βk−1 βk 


···
0
0 
..
.. 
..

.
.
. 

···
0
0 
.

···
0
0 

···
0
0 

..
.. 
..
.
.
. 

···
1
0
···
This method has order p with respect to the starting procedure
y [0] = q0 y(x0 ) + q1 hy ′ (x1 ) + . . . + qp hp y (p) (x0 ) + O(hp+1 ),
with q0 , q1 , . . . , qp given by
[
q0 = 1 1 · · ·
1 0 0 ···
5
]T
0
,
[
q1 =
[
0 −1 · · ·
−(k − 1) 1 1 · · ·
]T
1
,
]T
1
2l
(k − 1)l
,
−
··· −
0 1 2l−1 · · · (k − 1)l−1
0 −
l
l
l
l = 2, 3, . . . , p, compare [45]. This representation with r = 2k and s =
1 was first proposed by Burrage and Butcher [12] (compare also [18]). A
more compact representation of these methods with r = k and s = 1 was
discovered recently by Butcher and Hill [23], see also [45]. This more compact
representation may accommodate a simpler analysis of methods.
We refer to the monographs [42–44, 55, 56, 58, 59] for a thorough discussion of linear multistep methods.
(−1)l−1
ql =
(l − 1)!
2.2
RK methods
The RK methods with s stages are defined by

s
∑

[n+1]
[n+1]


Y
=
y
+
h
aij f (Yj
), i = 1, 2, . . . , s,
n

 i
j=1
s
(2.2)
∑

[n+1]


y
=
y
+
h
bj f (Yj
),
n+1
n


j=1
[n+1]
n = 0, 1, . . . , N − 1, where Yi
is an approximation to y(xn + ci h) and yn
is an approximation to y(xn ). These methods are usually represented by the
Butcher tableau
c1 a11 · · · a1s
.. . .
.
..
. ..
c A
.
.
=
.
bT
cs as1 · · · ass
b1
···
They can be represented as GLMs (1.5) with
c = [c1 , . . . , cs ]T , the coefficient matrices

a11
[
] [
]  .
 ..
A U
A e

=
=
T
 as1
B V
b 1

b1
6
bs
r = 1 by the abscissa vector
· · · a1s
.
..
. ..
···
ass
···
bs

1
.. 
. 

,
1 

1
and q0 = 1, q1 = 0. We refer to the monographs [14, 16, 40, 42, 55, 56] for
a thorough discussion of these methods.
2.3
DIMSIMs
DIMSIMs (with r = s) are the GLMs defined by

i−1
∑

[n+1]
[n+1]
[n+1]
[n]


=
h
aij f (Yj
) + hλf (Yi
) + yi , i = 1, 2, . . . , s,
Y

 i
j=1
s
s
∑
∑


[n+1]
[n+1]
[n]

y
=
h
b
f
(Y
)
+
vj y j ,

ij
j
 i
j=1
(2.3)
i = 1, 2, . . . , s,
j=1
n = 1, 2, . . . , N − 1. For these methods the coefficient matrix A is lower
triangular with the same element λ ≥ 0 on the diagonal. If λ = 0 these
methods are explicit and will be referred to as type 1 methods. If λ >
0 these methods are implicit and will be referred to as type 2 methods.
The coefficients matrix V is a rank one matrix of the form V = evT , with
vT e = 1. The methods (2.3) can be represented by the abscissa vector
c = [c1 , . . . , cs ]T , the coefficient matrices given by


1 0 ··· 0
λ


 a21 λ
0 1 ··· 0 


.. . .
.. .. . .
.. 
 ..
.
. . 
 .
.
. .

[
] 

 a
A U
 s1 as2 · · · λ 0 0 · · · 1 
=
,
 b11 b12 · · · b1s v1 v2 · · · vs 
B V




 b21 b22 · · · b2s v1 v2 · · · vs 
 .
.. . .
..
.. .. . .
.. 

 .
.
.
.
.
. .
. 
 .
bs1 bs2 · · · bss v1 v2 · · · vs
and the vectors q0 , q1 , . . . , qp defined in Section 1. These methods were
introduced by Butcher [15] and further investigated in [24–28, 30–35].
2.4
DIMSIMs in Nordsieck representation
The Nordsieck representation of DIMSIMs (2.3) was introduced in [20] to
simplify the implementation of these methods in variable stepsize variable
7
order environments. In this representation r = s+1 and the vector of external
approximations, denoted by z [n] , approximates directly the Nordsieck vector
z(x, h) defined by


y(x)

 hy ′ (x)

z(x, h) = 
..

.

s (s)
h y (x)



.


As demonstrated in [20, 21, 45] this leads to methods given by

i−1
s+1
∑
∑


[n+1]
[n+1]
[n+1]
[n]

Yi
=h
aij f (Yj
) + hλf (Yi
)+
uij zj ,




j=1
j=1





i = 1, 2, . . . , s,

s
s+1
∑
∑
[n+1]
[n+1]
[n]

z
=
h
b
f
(Y
)
+
vj zj ,

1j
1
j



j=1
j=1


s

∑

[n+1]
[n+1]


=h
bij f (Yj
), i = 2, 3, . . . , s + 1,
 zi

(2.4)
j=1
n = 1, 2, . . . , N − 1, with v1 = 1. These methods can be represented by the
abscissa vector c = [c1 , . . . , cs ]T and the coefficient matrices given by


u11 u12 · · · u1,s+1
λ


 a21
u
u
λ
21 u22 · · ·
2,s+1 


..
..
.. . .
.. 
 ..
...
.
 .
.
.
.
. 

[
] 
 a

A U
a
·
·
·
λ
u
u
·
·
·
u
s2
s1
s2
s,s+1 
 s1
=
.
 b11

B V
b
·
·
·
b
1
v
·
·
·
v
12
1s
2
s+1




b22 · · ·
b2s
0
0 ···
0 
 b21
 .
..
..
..
.. . .
.. 
...
 .

.
.
.
.
.
. 
 .
bs+1,1 bs+1,2 · · · bs+1,s 0
0 ···
0
Assuming that the method (2.4) has order p = s the vectors q0 , q1 , . . . , qp
are given by
q0 = e1 , q1 = e2 , . . . , qp = ep+1 ,
where e1 , e2 , . . . , ep+1 is the canonical basis in Rp+1 .
8
2.5
TSRK methods
TSRK methods depend on the stage values at two consecutive steps and
assume the form

s (
)
∑

[n+1]
[n+1]
[n]


Y
=
(1
−
u
)y
+
u
y
+
h
a
f
(Y
)
+
b
f
(Y
)
,
i n
i n−1
ij
ij

j
j
 i
j=1
(2.5)
s (
)
∑

[n+1]
[n]


vj f (Yj
) + wj f (Yj ) ,

 yn+1 = (1 − ϑ)yn + ϑyn−1 + h
j=1
[n+1]
i = 1, 2, . . . , s, n = 0, 1, . . . , N − 1. Here, the stage values Yi
are approximations to y(xn + ci h), i = 1, 2, . . . , s, and yn is an approximation to y(xn ).
These methods can be represented by the abscissa vector c = [c1 , . . . , cs ]T
and the tableaux of its coefficients
u A
B
ϑ vT wT
=
u1 a11 · · · a1s b11 · · · b1s
.. . .
.
.. . .
.
..
. ..
. ..
.
.
.
us as1 · · ·
ϑ
···
v1
ass bs1 · · ·
bss
w1 · · ·
ws
vs
.
The TSRK methods (2.5) can be represented as GLMs (1.5) with 2s internal
stages and r = s + 2 external stages. This representation take the form


 

0
0 I
Y [n]
hf (Y [n] )
0
0
 [n+1]  


 Y
  B A e − u u 0   hf (Y [n+1] ) 


 


  T


T
(2.6)
 yn+1  =  w v 1 − ϑ ϑ 0  
,
yn


 

 y

  0


0
1
0 0 
yn−1
n

 

[n+1]
[n]
Y
B A e−u u 0
Y
n = 0, 1, . . . , N − 1, and

1

q0 = 
 1
e
the vectors q0 and q1 are given by



0



 ∈ Rs+2 , q1 =  −1  ∈ Rs+2 ,



c−e
9
compare [6, 49]. A more compact representation of TSRK methods (2.5) as
GLMs (1.5) with s internal stages and r = s + 2 external stages is given by






Y [n+1]


A e−u u
  T
  v
=
 
yn
  0
I
hf (Y [n+1] )
yn+1
B

hf (Y [n+1] )


1 − ϑ ϑ wT 
yn


1
0 0 
yn−1
0
0 0
hf (Y [n] )



,


n = 0, 1, . . . , N − 1, and the vectors q0 and q1 are
 


1
0
 


 ∈ Rs+2 , q1 =  −1  ∈ Rs+2 ,
q0 = 
1
 


0
e
(2.7)
(2.8)
compare [45, 65]. TSRK methods (2.5) were introduced by Jackiewicz and
Tracogna [49] and further investigated in [6, 7, 46–48, 50, 52, 53, 60–62] and
the monograph [45].
2.6
GLMs with IRKS
Consider the GLMs with r = s = p + 1,

s
s
∑
∑

[n+1]
[n+1]
[n]


=h
aij f (Yj
)+
uij yj , i = 1, 2 . . . , s,

 Yi
j=1
s
j=1
s
∑
∑

[n+1]
[n+1]
[n]


y
=
h
b
f
(Y
)
+
vij yj ,
ij

i
j

j=1
(2.9)
i = 1, 2 . . . , s,
j=1
n = 0, 1, . . . , N − 1, where the vector of external approximations y [n] approximates the Nordsieck vector defined, similarly as in Section 2.4, by


y(x)


 hy ′ (x) 


z(x, h) = 
.
..


.


p (p)
h y (x)
10
This implies that the vectors q0 , . . . , qs−1 are given by
q0 = e 1 ,
q1 = e 2 ,
...,
qs−1 = es ,
where e1 , . . . , es is the canonical basis in Rs . The GLM (2.9) is said to possess
IRKS property if its coefficients satisfy the algebraic constraints
BA ≡ XB,
BU ≡ XV − VX,
for some matrix X ∈ Rs×s , and the condition on the characteristic polynomial
of the matrix V
det(wI − V) = wr−1 (w − 1).
Here, the notation M1 ≡ M2 means that these matrices are identical except
possibly their first rows. The significance of IRKS conditions follows from the
result, which was proved in [35, 65], that for the methods (2.9) which satisfy
these conditions (and the preconsistency condition Ve1 = e1 discussed in
Section 3) the stability function p(w, z) have the form (1.12), i.e., the GLMs
have RK stability.
GLMs with IRKS were investigated in [31, 35, 65, 66] and the monograph
[45] with additional assumption that the stage order q is equal to the order
p and that the coefficient matrix V has the form
[
]
1 vT
V=
,
0 V
where the matrix V ∈ Rp×p has spectral radius equal to zero. In [31] this
matrix V was assumed to be upper triangular with the zero on the diagonal.
3
Local discretization errors of GLMs
To simplify the notation we assume that m = 1, i.e., that (1.1) is a scalar
differential equation. The general case corresponding to m > 1 follows easily
by replacing A, U, B, V, and qi by A ⊗ I, U ⊗ I, B ⊗ I, V ⊗ I, and
qi ⊗ I, where I is the identity matrix of dimension m - the dimension of the
differential system (1.1).
For GLMs (1.5), local discretization errors of stage values Y [n+] and exˆ n+1 ), respecternal approximations y [n+1] , denoted by hd(xn + ch) and hd(x
tively, are defined as the residua obtained by replacing in (1.5), Y [n+1] by
11
y(xn + ch), and y [n+1] , y [n] by z(xn+1 ), z(xn ). Taking into account that
f (y(xn + ch)) = y ′ (xn + ch) this leads to

 y(xn + ch) = hAy ′ (xn + ch) + Uz(xn ) + hd(xn + ch),
(3.1)
 z(x ) = hBy ′ (x + ch) + Vz(x ) + hd(x
ˆ n+1 ),
n+1
n
n
n = 0, 1, . . . , N − 1. Put e = [1, . . . , 1] ∈ Rs . Expanding y(xn + ch) and
y ′ (xn + ch) into Taylor series around the point xn we obtain
p
∑
∑
ck
ck−1
(xn )
= hA
hk y (k) (xn )
k!
(k − 1)!
k=1
p
∑
+ U
hk y (k) (xn )qk + hd(xn + ch) + O(hp+1 ),
p
k (k)
h y
k=0
k=0
and
p
∑
k (k)
h y
(xn+1 )qk = hB
p
∑
hk y (k) (xn )
k=1
k=0
+ V
p
∑
ck−1
(k − 1)!
ˆ n+1 ) + O(hp+1 ).
hk y (k) (xn )qk + hd(x
k=0
This leads to
hd(xn + ch) = (e − Uq0 )y(xn )
)
p ( k
∑
c
Ack−1
+
−
− Uqk hk y (k) (xn ) + O(hp+1 ),
k! (k − 1)!
k=1
(3.2)
ˆ n+1 ) = (q0 − Vq0 )y(xn )
hd(x
)
p ( k
∑
∑ ql
Bck−1
+
−
− Vqk hk y (k) (xn ) + O(hp+1 ).
(k − l)! (k − 1)!
k=1
l=0
(3.3)
and
Introducing the notation


 γ0 = e − Uq0 ,
ck
Ack−1

 γk =
−
− Uqk ,
k! (k − 1)!
12
(3.4)
k = 1, 2, . . . , p,
and


γ̂ = q0 − Vq0 ,

 0


 γ̂k =
k
∑
l=0
ql
Bck−1
−
− Vqk ,
(k − l)! (k − 1)!
(3.5)
k = 1, 2, . . . , p,
the relations (3.2) and (3.3) for the local discretization errors take the simple
form
p
∑
hd(xn + ch) =
γl hl y (l) (xn ) + O(hp+1 ),
(3.6)
l=0
and
ˆ n+1 ) =
hd(x
p
∑
γ̂l hl y (l) (xn ) + O(hp+1 ).
(3.7)
l=0
If the GLM (1.5) is to have order p then by considering the solution of
y ′ (x) = f (x) by (1.2), it follows that a necessary (but far from sufficient)
condition is that the local discretization error of the vector of external approximations satisfies
ˆ n+1 ) = O(hp+1 ).
hd(x
This leads to
γ̂i = 0,
i = 0, 1, . . . , p.
(3.8)
The condition γ̂0 = 0 or
Vq0 = q0
(3.9)
is called the preconsistency condition, and the condition γ̂1 = 0 or
Be + Vq1 = q0 + q1
(3.10)
is called the consistency condition, compare [45]. We will also always assume
the stage preconsistency condition γ0 = 0, or
Uq0 = e,
(3.11)
and stage consistency condition γ1 = 0, or
Ae + Uq1 = c,
13
(3.12)
compare again [45]. Assuming (3.9), (3.10), (3.11), and (3.12) the local
discretization errors now take the form
hd(xn + ch) =
p
∑
γl hl y (l) (xn ) + O(hp+1 ),
(3.13)
l=2
and
ˆ n+1 ) =
hd(x
p
∑
γ̂l hl y (l) (xn ) + O(hp+1 ).
(3.14)
l=2
Assuming in addition to (3.8), (3.11), and (3.12), that
γl = 0,
l = 2, 3, . . . , p − 1 or l = 2, 3, . . . , p
leads to the GLMs of order p with stage order q = p − 1 or q = p, respectively. Such high order and stage order methods with some desirable
stability properties (large regions of absolute stability for explicit methods,
A-, L-, and algebraic stability for implicit methods) were investigated in a
series of papers [8, 10, 11, 15, 24–28, 30–39] and the monograph [45]. In
the next section we will derive the general order conditions for GLMs (1.5)
without restrictions on stage order (except stage preconsistency and stage
consistency conditions (3.11) and (3.12)) using the approach proposed by
Albrecht [1–5] in the context of RK, composite and linear cyclic methods,
and generalized by Jackiewicz and Tracogna [49, 50, 60], to TSRK methods
for ODEs, by Jackiewicz and Vermiglio [51] to general linear methods with
external stages of different orders, and by Garrappa [41] to some classes of
Runge-Kutta methods for Volterra integral equations with weakly singular
kernels. A very readable summary of this approach for RK methods for
ODEs was presented by Lambert [56].
4
General order conditions for GLMs
In [51] an extension of Albrecht [2–5] was used to derive general order conditions for GLMs with external stages of different orders. This derivation is
technically very complicated and in this paper we restrict our attention to
GLMs, where all components of the vector of external approximations have
the same order p. Although order conditions for such methods can be obtained using the theory presented in [51], for the sake of completeness we
present the derivation of order conditions for this simpler class of GLMs.
14
In this section we will derive general order conditions for GLMs (1.5) for
which the coefficient matrix V is such that there exists a limit
Ṽ = lim Vn .
n→∞
This is a case, for example, for RK methods (2.2) for which V = [1], DIMSIMs (2.3) for which V = evT , DIMSIMs in Nordsieck representation (2.4),
TSRK methods (2.5), GLMs with IRKS (2.9), and for GLMs (1.5) which are
strictly zero-stable.
As observed in Section 3 for the GLM (1.5) to have order p, it is necessary
that
ˆ n+1 ) = O(hp+1 ),
hd(x
which leads to the algebraic conditions (3.8) expressed in terms of the coefficients of the methods. In order to obtain necessary and sufficient conditions
for the method (1.5) to have order p we consider the global truncation errors
q [n+1] and q̂ [n+1] of the vectors of internal and external approximations Y [n+1]
and y [n+1] , which are defined by
q [n+1] = y(xn + ch) − Y [n+1] ,
q̂ [n+1] = z(xn+1 ) − y [n+1] ,
n = 0, 1, . . . , N − 1. We also define the vector t[n+1] by
(
)
t[n+1] = f y(xn + ch) − f (Y [n+1] ),
n = 0, 1, . . . , N − 1. Subtracting (1.5) (with m = 1) from (3.1) we obtain
q [n+1] = hAt[n+1] + Uq̂ [n] + hd(xn + ch),
(4.1)
ˆ n+1 ),
q̂ [n+1] = hBt[n+1] + Vq̂ [n] + hd(x
(4.2)
and
n = 0, 1, . . . , N − 1. By an easy induction argument (4.2) leads to
q̂
[n]
n [0]
= V q̂
+h
n−1
∑
V
n−k
Bt
[k]
+ hBt
k=1
[n]
+h
n
∑
ˆ k ),
Vn−k d(x
(4.3)
k=1
n = 1, 2, . . . , N , where q̂ [0] = z(x0 )−y [0] is the error of the starting procedure.
The above expression leads to the following general order criterion.
15
Theorem 4.1. Assume that there exists a limit Ṽ = limn→∞ Vn . Moreover,
assume that q̂ [0] = O(hp ), and that
ˆ k ) = O(hp+1 ),
hd(x
k = 1, 2, . . . , N.
Then the GLM (1.5) is convergent with order p, i.e.,
q̂ [n] = O(hp ),
n = 1, 2, . . . , N,
if and only if the following conditions hold
Bt[k] = O(hp−1 ),
k = 1, 2, . . . , N,
(4.4)
ṼBt[k] = O(hp ),
k = 1, 2, . . . , N.
(4.5)
Moreover, the global errors q [n+1] of the internal stages are given by
q [n+1] = hAt[n+1] + hd(xn + ch) + O(hp ),
n = 0, 1, . . . , N − 1.
(4.6)
Proof. It follows from the assumption of the theorem that the relation (4.3)
corresponding to n = N takes the form
q̂
[N ]
= hBt(xN ) + h
N
−1
∑
VN −k Bt(xk ) + O(hp ),
n = 1, 2, . . . , N,
k=1
where we have written t(xk ) instead of t[k] . Since k = (xk − x0 )/h we have
q̂
[N ]
= hBt(xn ) + h
N
−1
∑
V(X−xk )/h Bt(xk ) + O(hp ),
k=1
and approximating the resulting sum by the corresponding Riemann integral
we obtain
∫ X
[N ]
q̂ = hBt(xN ) +
V(X−s)/h Bt(s)ds + O(hp ),
x0
where t(s) is the continuous analogue of t(xk ). Hence, for h → 0, we obtain
∫ X
[N ]
ṼBt(s)ds + O(hp ).
q̂ = hBt(xN ) +
x0
This relation implies (4.4) and (4.5). The relation (4.6) follows easily from
(4.1) and the relation q̂ [n] = O(hp ).
16
The relations (4.4) and (4.5) are the general order conditions for GLMs
(1.5). In what follows we will try to reformulate them in terms of the coefficients of the methods.
Similarly as in the theory of RK methods [2, 4] we will need the following
lemma for recursive computation of order conditions.
Lemma 4.1. Assume that the function f is sufficiently smooth. Then
t[n+1] =
∞
∑
(
)l
Gl (xn ) q [n+1]
(4.7)
l=1
with
∞
(
) ∑
1 k k (k)
Gl (xn ) := diag gl (xn + c1 h), . . . , gl (xn + cs h) =
h Γc gl (xn ),
k!
k=0
where
gl (x) =
)
(−1)l+1 ∂ l f (
y(x) ,
l
l!
∂y
Γc = diag(c1 , . . . , cs ).
Proof. We have
[n+1]
ti
(
)
( [n+1] )
= f y(xn + ci h) − f Yi
(
)
(
)
[n+1]
= f y(xn + ci h) − f y(xn + ci h) + Yi
− y(xn + ci h)
∞
)l
∑
)( [n+1]
1 ∂lf (
= −
( y(xn + ci h) Yi
− y(xn + ci h) ,
l! ∂y l
l=1
i = 1, 2, . . . , s, and using the definition of q [n+1] and gl (x) we obtain
[n+1]
ti
=
∞
∑
(−1)l+1 ∂ l f (
l=1
l!
∂y l
∞
)( [n+1] )l ∑
( [n+1] )l
gl (xn + ci h) qi
,
y(xn + ci h) qi
=
l=1
i = 1, 2, . . . , s. This is equivalent to (4.7).
Introduce the notation

 u := q [n+1] − hAt[n+1] − hd(xn + ch) + O(hp ),
)2
(

v := t[n+1] − G1 (xn )q [n+1] − G2 (xn ) q [n+1] − · · · ,
17
(4.8)
where (q [n+1] )2 stands for componentwise exponentiation. Then




∂u
∂u
−hA
I
 ∂t[n+1] ∂q [n+1] 
 = det 
,
det 
 ∂v
∂v 
[n+1]
I
−G1 (xn ) − 2G2 (xn )q
− ···
∂t[n+1] ∂q [n+1]
and it follows that

∂u
∂u
 ∂t[n+1] ∂q [n+1]
det 
 ∂v
∂v
[n+1]
∂t
∂q [n+1]

 
 (

t[n+1] =0,q [n+1] =0,h=0

) = det
0
I
I −G1 (xn )

 ̸= 0.
Moreover, u and v are analytical in a neighborhood of (0, 0, 0). Therefore,
it follows from the implicit function theorem that the system (u, v) = (0, 0),
where u and v are defined by (4.8), has a unique solution
( [n+1] [n+1] ) ( [n+1]
)
t
,q
= t
(h), q [n+1] (h)
in a neighborhood of h = 0, and the following Taylor series expansions exist
q [n+1] = r2 (xn )h2 + r3 (xn )h3 + · · · + rp−1 (xn )hp−1 + O(hp ),
(4.9)
t[n+1] = w2 (xn )h2 + w3 (xn )h3 + · · · + wp−1 (xn )hp−1 + O(hp ),
(4.10)
with h0 and h1 terms equal to zeros. To see why r1 (xn ) = w1 (xn ) = 0, observe
first that q [n+1] = O(h) (compare (4.6)), which implies that t[n+1] = O(h)
(compare (4.7)). The formula (4.1) with (3.12) now implies that q [n+1] =
O(h2 ), and then using (4.7) again we obtain t[n+1] = O(h2 ). These observations lead to expansions of the form (4.9) and (4.10).
The order conditions (4.4) and (4.5) reduce now to
Bwi (xk ) = 0,
i = 2, 3, . . . , p − 2,
k = 0, 1, . . . , N,
(4.11)
and
ṼBwp−2 (xk ) = 0,
k = 0, 1, . . . , N.
(4.12)
The vectors ri (xk ) and wi (xk ), i = 2, 3, . . . , p − 1, can be computed in
the same way as was done in the theory of order condition for RK methods
[2, 4, 56]. We have the following result.
18
Theorem 4.2. The vectors ri (xk ) and wi (xk ), i = 2, 3, . . . , p − 1, satisfy the
following recursions
ri (xk ) = γi y (i) (xk ) + Awi−1 (xk ),
w1 (xk ) := 0,
(4.13)
i−2
∑
1 l (l)
wi (xk ) =
Γ g (xk )ri−l (xk )
l! c 1
l=0
+
+
i−4
∑
∑
l=0
m,n≥2
m+n+l=1
i−6
∑
l=0
(
)
1 l (l)
Γc g2 (xk ) rm (xk ) · rn (xk )
l!
∑
m,n,s≥2
m+n+s+l=i
(4.14)
(
)
1 l (l)
Γc g3 (xk ) rm (xk ) · rn (xk ) · rs (xk )
l!
+ ···
Proof. Substituting (4.9) and (4.10) into (4.6) and (4.7) and taking into
account (3.13) we obtain
p−1
∑
rl (xk )h
l
=
l=2
=
p−1
∑
(l)
l
γl y (xk )h + A
p−2
∑
l=2
l=2
p−1
∑
p−1
∑
γl y (l) (xk )hl + A
wl (xk )hl+1 + O(hp )
wl−1 (xk )hl + O(hp ),
l=2
l=2
where we have used the convention that w1 (xk ) := 0. Comparing the hi
terms leads to the recurrence (4.13). We have also
(
)
p−1
∑
h2 2 ′′
l
′
wl (xk )h =
g1 (xk )I + hΓc g1 (xk ) + Γc g1 (xk ) + · · ·
2!
l=2
(
)
× r2 (xk )h2 + r3 (xk )h3 + · · ·
(
)
h2 2 ′′
′
+
g2 (xk )I + hΓc g2 (xk ) + Γc g2 (xk ) + · · ·
2!
(
)2
× r2 (xk )h2 + r3 (xk )h3 + · · ·
(
)
h2 2 ′′
′
+
g3 (xk )I + hΓc g3 (xk ) + Γc g3 (xk ) + · · ·
2!
(
)3
× r2 (xk )h2 + r3 (xk )h3 + · · ·
+ ··· .
19
Comparing the hi terms we obtain the recurrence relation (4.14). This completes the proof.
5
Derivation of order conditions up to order
six
The vectors ri (xk ) and wi (xk ) can be computed from the recursions (4.13)
and (4.14) in exactly the same way as was done for RK methods [2, 4] and
for TSRK methods [45, 49]. Once the vectors wi (xk ), i = 2, 3, . . . , p − 1, are
computed, the order conditions for GLMs (1.5) can be obtained by substituting these vectors into (4.11). In what follows we will illustrate this process
by computing order conditions for GLMs (1.5), up to the order p = 6
For p = 6 the expanded form of recursions (4.13) and (4.14) takes the
form
r2 h2 + r3 h3 + r4 h4 + r5 h5 = Aw2 h3 + Aw3 h4 + Aw4 h5
+ γ2 y ′′ h2 + γ3 y ′′′ h3 + γ4 y (4) h4 + γ5 y (5) h5 + O(h6 ),
and
w 2 h2 + w 3 h3 + w 4 h4 + w 5 h5
(
)
)
1 2 2 ′′ 1 3 3 ′′′ ( 2
′
= g1 I + hΓc g1 + h Γc g1 + h Γc g1 r2 h + r3 h3 + r4 h4 + r5 h5
2!
6
(
)(
)
2
+ g2 I + hΓc g2′ r2 h2 + r3 h3 + O(h6 ),
where ri = ri (xk ) and wi = wi (xk ). Comparing the successive powers of h
leads to the following relations
r2 = γ2 y ′′ ,
w2 = g1 r2 = γ2 g1 y ′′ ,
r3 = γ3 y ′′′ + Aw2 = γ3 y ′′′ + Aγ2 g1 y ′′ ,
w3 = g1 r3 + Γc g1′ r2 = γ3 g1 y ′′′ + Aγ2 g12 y ′′ + Γc γ2 g1′ y ′′ ,
r4 = γ4 y (4) + Aw3 = γ4 y (4) + Aγ3 g1 y ′′′ + A2 γ2 g12 y ′′ + AΓc γ2 g1′ y ′′ ,
20
1
w4 = g1 r4 + Γc g1′ r3 + Γ2c g1′′ r2 + g2 r22
2
= γ4 g1 y (4) + Aγ3 g12 y ′′′ + A2 γ2 g13 y ′′ + AΓc γ2 g1 g1′ y ′′
1
+ Γc γ3 g1′ y ′′′ + Γc Aγ2 g1′ g1 y ′′ + Γ2c γ2 g1′′ y ′′ + γ22 g2 (y ′′ )2 ,
2
r5 = γ5 y (5) + Aw4
= γ5 y (5) + Aγ4 g1 y (4) + A2 γ3 g12 y ′′′ + A3 γ2 g13 y ′′ + A2 Γc γ2 g1 g1′ y ′′
1
+ AΓc γ3 g1′ y ′′′ + AΓc Aγ2 g1′ g1 y ′′ + AΓ2c γ2 g1′′ y ′′ + Aγ22 g2 (y ′′ )2 ,
2
1
1
w5 = g1 r5 + Γc g1′ r4 + Γ2c g1′′ r3 + Γ3c g1′′′ r2 + Γc g2′ r22 + g2 r2 r3 + g2 r3 r2 .
2
6
We have
r2 r3 = γ2 γ3 y ′′ y ′′′ + γ2 Aγ2 y ′′ g1 y ′′ ,
r3 r2 = γ3 γ2 y ′′′ y ′′ + Aγ22 g1 (y ′′ )2 ,
and substituting the relations for r5 , r4 , r3 , r2 and the above relations for
r2 r3 and r3 r2 into the formula for w5 we obtain
w5 = γ5 g1 y (5) + Aγ4 g12 y (4) + A2 γ3 g13 y ′′′ + A3 γ2 g14 y ′′
1
+ A2 Γc γ2 g12 g1′ y ′′ + AΓc γ3 g1 g1′ y ′′′ + AΓc Aγ2 g1 g1′ g1 y ′′ + AΓ2c γ2 g1 g1′′ y ′′
2
+ Aγ22 g1 g2 (y ′′ )2 + Γc γ4 g1′ y (4) + Γc Aγ3 g1′ g1 y ′′′ + Γc A2 γ2 g1′ g12 y ′′
1
1
1
+ Γc AΓc γ2 (g1′ )2 y ′′ + Γ2c γ3 g1′′ y ′′′ + Γ2c Aγ2 g1′′ g1 y ′′ + Γ3c γ2 g1′′′ y ′′
2
2
6
+ Γc γ22 g2′ (y ′′ )2 + 2γ2 γ3 g2 y ′′ y ′′′ + γ2 Aγ2 g2 y ′′ g1 y ′′ + Aγ22 g2 g1 (y ′′ )2 .
Observe that the coefficients of g1 g2 (y ′′ )2 and g2 g1 (y ′′ )2 are the same and
equal to Aγ22 .
(j)
The above expressions for ri and wi contain various combinations of gi
and y (j) . These combinations are called recursive differentials [2, 4]. They
play a similar role to elementary differentials in the Butcher theory of order
conditions for RK methods which is based on rooted trees and elementary
weights.
21
Order Recursive differentials
Corresponding order conditions
p=1
y
′
γ̂1 = 0
p=2
y ′′
γ̂2 = 0
p=3
y ′′′
γ̂3 = 0
g1 y ′′
ṼBγ2 = 0 or Bγ2 = 0
y (4)
γ̂4 = 0
g1 y ′′′
ṼBγ3 = 0 or Bγ3 = 0
g12 y ′′
ṼBAγ2 = 0 or BAγ2 = 0
g1′ y ′′
ṼBΓc γ2 = 0 or BΓc γ2 = 0
y (5)
γ̂5 = 0
g1 y (4)
ṼBγ4 = 0 or Bγ4 = 0
g12 y ′′′
ṼBAγ3 = 0 or BAγ3 = 0
g13 y ′′
ṼBA2 γ2 = 0 or BA2 γ2 = 0
g1 g1′ y ′′
ṼBAΓc γ2 = 0 or BAΓc γ2 = 0
g1′ y ′′′
ṼBΓc γ3 = 0 or BΓc γ3 = 0
g1′ g1 y ′′
ṼBΓc Aγ2 = 0 or BΓc Aγ2 = 0
g1′′ y ′′
ṼBΓ2c γ2 = 0 or BΓ2c γ2 = 0
g2 (y ′′ )2
ṼBγ22 = 0 or Bγ22 = 0
p=4
p=5
Table 1: Recursive differentials and order conditions for p ≤ 5
22
Order Recursive differentials Corresponding order conditions
p=6
y (6)
γ̂6 = 0
g1 y (5)
ṼBγ5 = 0
g12 y (4)
ṼBAγ4 = 0
g13 y ′′′
ṼBA2 γ3 = 0
g14 y ′′
ṼBA3 γ2 = 0
g12 g1′ y ′′
ṼBA2 Γc γ2 = 0
g1 g1′ y ′′′
ṼBAΓc γ3 = 0
g1 g1′ g1 y ′′
ṼBAΓc Aγ2 = 0
g1 g1′′ y ′′
ṼBAΓ2c γ2 = 0
g1 g2 (y ′′ )2 , g2 g1 (y ′′ )2
ṼBAγ22 = 0
g1′ y (4)
ṼBΓc γ4 = 0
g1′ g1 y ′′′
ṼBΓc Aγ3 = 0
g1′ g12 y ′′
ṼBΓc A2 γ2 = 0
(g1′ )2 y ′′
ṼBΓc AΓc γ2 = 0
g1′′ y ′′′
ṼBΓ2c γ3 = 0
g1′′ g1 y ′′
ṼBΓ2c Aγ2 = 0
g1′′′ y ′′
ṼBΓ3c γ2 = 0
g2′ (y ′′ )2
ṼBΓc γ22 = 0
g2 y ′′ y ′′′
ṼBγ2 γ3 = 0
g2 y ′′ g1 y ′′
ṼBγ2 Aγ2 = 0
Table 2: Recursive differentials and order conditions for p = 6
23
Order conditions for GLMs (1.5) for which there exists a limit Ṽ,
Ṽ = lim Vn ,
n→∞
can now be obtained by imposing the condition
ˆ k ) = O(hp+1 ),
hd(x
k = 1, 2, . . . , N,
compare Theorem 4.1, and equating to zero the coefficients of recursive differentials in the expressions
Bwi = 0,
i = 2, 3, . . . , p − 2,
Ṽ Bwp−1 = 0,
compare (4.11) and (4.12). The recursive differentials and the resulting order
conditions up to the order p = 6 are listed in Tables 1 and 2. We always
assume the preconsistency condition γˆ0 = 0, or
Vq0 = q0 ,
compare (3.9), and the stage preconsistency condition γ0 = 0, or
Uq0 = e,
compare (3.11). Moreover, we will always assume stage consistency condition
γ1 = 0, or
Ae + Uq1 = c,
compare (3.12). In the last column of Table 1 and Table 2, when there is
a couple of conditions separated by ’or’, the first condition refer to order p
methods, while the second condition refers to methods with order greater
than p.
6
Application of the order conditions to special cases of GLM
In this section we will illustrate the application of the theory of order conditions derived in Section 4 and 5 to verify the order p and stage order q of
some classes of GLMs discussed in Section 2, and some other GLMs from the
literature on the subject.
24
6.1
Linear multistep methods
In this subsection we will illustrate the derivation of order conditions for
linear multistep methods with k = 3 and order p = 4. These methods take
the form
yn+1 =
3
∑
αj yn+1−j + h
j=1
3
∑
with given starting values y0 , y1 , and y2 .
follows from Section 2.1 that



 
0
0
1



 
 1
 −1 
 1 

 2

 



 
 2
 −2 
 1 





,
q
=
q0 =   , q1 = 
2

 0

 1 
 0 



 
 −1
 1 
 0 



 
0
n = 2, . . . , N − 1,
βj f (yn+1−j ),
(6.1)
j=0
For this method c = c1 = 1 and it





0
0






 −1 
 1 

 6 
 24 

 4 
 2 

 −3 


 , q3 = 
 , q4 =  3  .

 0 
 0 






 1 
 1 



 − 

 2 
 6 
−2
2
− 43
1
By imposing the order conditions:
γ̂i = 0, i = 0, 1, 2, 3, 4,
and stage preconsistency and consistency conditions γ0 = 0, γ1 = 0, we
obtain a two-parameters family of methods, with coefficients:
α3 = 1 − α1 − α2 ,
β2 =
β0 =
27 − 32α1 − 19α2
,
24
9 − α2
,
24
β3 =
β1 =
27 − 8α1 + 5α2
,
24
9 − 8α1 − 9α2
.
24
For these methods, by formula (3.4) find out that
γ2 = γ3 = γ4 = 0.
Hence, the method has also stage order q = 4, and the remaining conditions
of order 4, i.e.
ṼBγ3 = 0,
ṼBAγ2 = 0,
25
ṼBΓc γ2 = 0,
are automatically satisfied.
Putting α1 = 1 and α2 = 0 leads to
yn+1 = yn + h
3
∑
βj f (yn+1−j ),
n = 2, . . . , N − 1,
j=0
with
3
19
5
1
β0 = , β1 = , β2 = − , β3 = ,
8
24
24
24
and we obtain the Adams-Moulton method with k = 3 and p = 4 [56].
6.2
Almost Runge–Kutta method
We consider the explicit Almost Runge–Kutta method with s = 4 internal
and r = 3 external stages (see method (505a) of [16]):


1
1
0 0
0 0 1
 1

1 
7

0
0
0
1
16
16
8



] 
[
0 0 1 − 43 − 12 
 − 14 2


A U
2
1
1
.
=
0
1
0
0


3
6
6
B V


1
1
 0 23

0
1
0
6
6


 0 0

0
1
0
0
0


1
1
1
−6 0 −3 1 0 −2
0
By easy computations we find that matrix Ṽ = limn→∞ Vn is given by


1 16 0


.
Ṽ = 
0
0
0


0 0 0
This method satisfies the order conditions
γ̂i = 0,
i = 0, 1, 2, 3, 4, γi = 0,
i = 0, 1, 2,
ṼBγ3 = 0,
for c1 = c3 = c4 , c2 = c4 − 1/2. Hence, this method has order p = 4 and stage
order q = 2 which is in agreement with [16]. It can be also verified that the
26
vectors q0 , q1 , q2 , q3 , and q4 , are given by
 


1
c4 − 1
 


 , q1 =  1  ,
q0 = 
0
 


0
0


q3 = 

(c4 −1)3
6
(c4 −1)2
2
2c4 −3
4





,
q2 = 
c
−
1
4


1
2


,

(c4 −1)2
2

q4 = 


q41
(c4 −1)3
6
c24 −9c4 +7
12

,

where c4 and q41 are free parameters. Observe that ṼBγ3 = 0 and Bγ3 ̸= 0.
6.3
G-symplectic method
We analyze the order and stage order of the G-symplectic method with r =
s = 2 (see [17, 19]) defined by the abscissa vector c given by
[
c=
1
2
√
+
3
6
1
2
−
√
3
6
]T
and the coefficient matrices

[
A U
B V
]


=


√
3+ 3
√6
− 33
1
2
1
2
√
3
0
√
3+ 3
6
1
2
− 12
1 − 3+23
1
√
3+2 3
3
1
0
0
−1



.


The coefficient matrix V of this method is nonsingular, and we have
V2k = I,
V2k+1 = V,
k = 0, 1, . . . .
Hence, the limit of Vn as n → ∞ does not exist. However, by an argument
similar to that given in the proof of Theorem 4.1 we can show that in the
order conditions the matrix Ṽ can be replaced by the identity matrix I. In
particular, it is possible to verify that this method satisfies the conditions
γ̂i = 0,
i = 0, 1, 2, 3,
27
γi = 0,
i = 0, 1, 2,
and that
[
q0 =
1
]
0
[
,
q1 =
0
]
[
,
0
q2 =
0
]
1
√
4 3
[
,
q3 =
0
0
]
.
Moreover it results γ̂4 = 0 and
[
Bγ3 =
0
√
3
− 9+5
108
]T
̸= 0.
This means that this method has exactly order p = 3 and stage order q = 2
with respect to the starting procedure
y [0] = Sh y0 = q0 y(x0 ) + q1 hy ′ (x0 ) + q2 h2 y ′′ (x0 ) + q3 h3 y ′′′ (x0 ) + O(h4 ).
However, it was proved by Butcher [19] (see also [22]) that this method
has order p = 4 and stage order q = 2 with respect to a different starting
procedure which satisfies
[
]
y(x
)
0
√
√
y [0] = Sh y0 = √3 2 ′′
+ O(h5 ).
3 4 (4)
9+5 3 4 ∂f (4)
h y (x0 ) − 108 h y (x0 ) + 216 h ∂y y (x0 )
12
It was also demonstrated in [19, 22] that this starting procedure can be
generated by
]
[
y0
[0]
,
y = Sh y0 = 1
(Rh y0 + R−h y0 − y0 )
2
where Rh is the Runge-Kutta method which, when written as GLM, takes
the form [17, 19]









0
0
0
0
1
2
5
11
√
9− 3
72
0
0
0
0
0
0
6
11√
3
− 15+2
54
√
10 3
27
√
33+11 3
216
√
11 3
− 108
28
0
1
1


1 

1 
.

1 

1
6.4
DIMSIMs
Consider the class of DIMSIMs with s
and coefficient matrices

λ
] 
[
 a21
A U
=

B V
 b11
b21
= r = 2, abscissa vector c = [0, 1]T ,
1
λ
0
b12 v1
b22 v1

0

1 
,

v2 
v2
(6.2)
where v1 + v2 = 1. Since Ṽ = V, this method has order p = 4 and stage
order q = 2 if and only if
γ̃i = 0,
i = 0, 1, 2, 3, 4,
γi = 0,
i = 0, 1, 2,
VBγ3 = 0.
Solving these order and stage order conditions leads to a one-parameter family of methods with the following coefficients
[
]
[
]
5−12λ
1
λ
6(1−2λ)
6(1−2λ)
A = −144λ3 +204λ2 −102λ+17
, V=
,
5−12λ
1
λ
5−12λ
6(1−2λ)
6(1−2λ)
[
]
B
59−156λ
12(5−12λ
144λ2 +24λ−29
12(12λ−5)
5−12λ
12
−1728λ3 +2160λ2 −828λ+89
12(12λ−5)
,
λ ̸= 1/2, λ ̸= 5/12. Moreover, the vectors qi , i = 0, 1, 2, 3, 4, which define
the starting procedure
y [0] = Sh y0 = q0 y(x0 )+q1 hy ′ (x0 )+q2 h2 y ′′ (x0 )+q3 h3 y ′′′ (x0 )+q4 h4 y (4) (x0 )+O(h5 )
are given by
[
q0 =
1
1
]
[
,
q1 =
[
q3 =
]
−λ
−144λ3 +192λ2 −85λ+12
12λ−5
1
24
12λ−5
24
]
[
,
q4 =
[
,
q2 =
q41
1+12q41 −2λ
12
]
0
1−2λ
2
]
,
,
where q41 is a free parameter. It can be verified using the Schur
√ criterion
∗
that these methods are A-stable if and only if λ < λ < (3 + 3)/6, where
λ∗ ≈ 0.714633 is a root of
φ(λ) = 17 − 204λ + 840λ2 − 1440λ3 + 864λ4 .
Moreover, these methods are not L-stable for any value of λ.
29
6.5
DIMSIMs in Nordsieck form
Now we consider the Nordsieck form of DIMSIM method illustrated in the
previous subsection. This method can be derived from representation (6.2),
using the approach described in [21] (see also [45]). In this way we obtain
a DIMSIM with s = 2, r = 3, abscissa vector c = [0, 1]T , and coefficient
matrices


λ
u11 u12 u13


[
]  a21 λ u21 u22 u23 


A U


=  b11 b12 1 v2 v3  ,


B V
 b
0
0 
 21 b22 0

b31 b32 0
0
0
with matrix A the same of method (6.2), and
[
U=
]
1
−λ
0
1
−144λ3 +192λ2 −85λ+12
1−2λ
2
12λ−5
v2 = −
,


B=

156λ−59
12(12λ−5)
0
−1
5
12


1 
,
1
2(3λ − 1)
1
, v3 = ,
12λ − 5
12
with λ ̸= 5/12.
Since Ṽ = V, this method has order p = 4 and stage order q = 2 if and
only if
γ̂i = 0,
i = 0, 1, 2, 3, 4,
γi = 0,
i = 0, 1, 2,
VBγ3 = 0.
It can be easily verified that these conditions hold, if the vectors qi , i =
0, 1, 2, 3, 4, which define the starting procedure
y [0] = Sh y0 = q0 y(x0 )+q1 hy ′ (x0 )+q2 h2 y ′′ (x0 )+q3 h3 y ′′′ (x0 )+q4 h4 y (4) (x0 )+O(h5 )
are given

1

q0 = 
 0
0
by


,



0
 

q1 = 
 1 ,
0


0
 

q2 = 
 0 ,
1
where q41 is a free parameter.
30

1
24



,
q3 = 
0


1
−2


q41


,
q4 = 
0


1
6
6.6
TSRK methods
The class of TSRK with s = 1 is defined by the abscissa c and by the tableau
of coefficients
u λ b
.
ϑ v w
It can be represented as a GLM with one internal stage and three external
stages (see (2.7)) with the following coefficients


λ 1−u u b
[
] 

 v 1−ϑ ϑ w 
A U
.
=


B V
1
0
0
0


1
0
0 0
This method has order p = q = 2 if and only if
γi = 0, i = 0, 1, 2,
γ̂i = 0, i = 0, 1, 2.
(6.3)
Solving these order and stage order conditions we derive a family of methods
depending on (λ, c, ϑ) with the tableau
−c2 +2c−2λ
2c−1
λ
c2 −2λc+c−λ
2c−1
ϑ
−2θc−2c+θ+3
2
2θc+2c+θ−1
2
,
for c ̸= 1/2. This is in agreement with the conclusions of [45], Sec. 5.4.2.
Then, if we are looking for a TSRK with p = 3, q = 2, i.e. if we impose
conditions (6.3) and γ̂3 = 0, we find a two-parameters family of methods
with tableau
−c2 +2c−2λ
c2 −2λc+c−λ
λ
2c−1
2c−1
,
2 −6c+2
2
3c
2
3c2 −1)
2
(
)
(
6c −12c+5
1−6c2
√
1−6c2
6c2 −1
for c ̸= 1/2 and c ̸= ± 6/6. This is the same result as that reported in [45],
Sec. 5.4.2. The vectors qi , i = 0, 1, 2, 3, which define the starting procedure
are given by (2.8) and




0
0




1



q2 =  2  , q3 =  − 16 
.
2
(c−1)
c−1
2
31
A discussion is due on order conditions for TSRK derived following Theorem 4.1. In the representation (2.7) of a TSRK as GLM, the external stage
is given by


yn


.
y [n] = 
y
n−1


[n]
hf (Y )
Therefore, if we impose that the method has order p with respect to the
external stage y [n] , we are requiring that the method has order p with respect
to the approximate solution yn and order p − 1 with respect to Y [n] . Thus,
coming back to the standard representation (2.5), we are imposing that the
method has order p and stage order q ≥ p − 1. Expressed in other words,
applying order conditions of theorems 4.1 and 4.2 to TSRK methods, we are
able to find only high stage order methods. Similar conclusions can be drawn
considering the representation (2.6).
6.7
GLMs with IRKS
The GLM with IRKS with s = r = 3, illustrated in [45] has abscissae vector
c = [0, 1/2, 1] and coefficient matrices


1
1
0
0
1
−
0
4

 4123
631
1
 −
1
0
0 
770
4
1540


[
]  1496 1391
1
756
690 
1 1163 − 3911 
 − 1987 1631
A U
4
,
=
 1376 315
68
1297
140 
B V
 − 1157 269 1277
1 1344 − 1009 


 − 1572 938
483
560 
0
0
−
 1093 531
719
1009 
55
103
151
− 24
0 0
0
48
48
It can be easily verified that the method has order p = q = 2, as stated in
[45] since it results
γ̂i = 0,
i = 0, 1, 2,
γi = 0,
i = 0, 1, 2,
Moreover it has also p = q = 3, with respect to the starting procedure defined
by the vectors qi , i = 0, 1, 2, 3 with q0 , q1 , q2 equal to the canonical basis of
R3 and
]T
[
194
676
47
,
q3 = − 11907
− 12287
2351
32
since it results γ3 = γ̂3 = 0.
The stability polynomial of this method takes the form
p(z, w) = w2 (w − R(z))
with R(z) = P (z)/(4 − z)4 , where P (z) is a polynomial of degree 4.
For the GLMs with IRKS we are not able to give examples of methods
with stage order different from the order of the method, since these methods
have p = q by design.
7
Concluding remarks
We derived general order conditions on GLMs with the aim to have separate conditions for the external and for the internal stages. Such conditions
leave more freedom in the choice of method parameters, since one can use
a low stage order method and use the remaining free parameters to obtain
some desirable properties, such as strong stability, G-symplecticity or the
conservation of some properties of the analytical solution. From the main
theoretical results, we derived order conditions up to order six and applied
these conditions to various classes of GLMs.
Acknowledgements. The results reported in this paper were obtained
during the visit of the first author to the Arizona State University from
January to May of 2013. This author wishes to express her gratitude to the
School of Mathematical and Statistical Sciences for hospitality during this
visit.
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