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SIAM J. NUMER. ANAL.
Vol. 31, No. 2, pp. 552-571, April 1994
1994 Society for Industrial and Applied Mathematics
015
L-STABLE PARALLEL ONE-BLOCK METHODS FOR ORDINARY
DIFFERENTIAL EQUATIONS*
P. CHARTIERt
Abstract. In this contribution, the author considers the one-block methods designed by Sommeijer, Couzy, and van der Houwen for the purpose of solving ordinary differential equations (ODEs)
on a parallel computer. The author also derives a new set of order conditions, studies the stability,
and exhibits a new class of parallel methods which contains L-stable schemes up to order eleven.
Key words. ODEs, one-block methods, L-stability, high order, parallelism
AMS subject classifications. 65M10, 65M20
1. Introduction. The purpose of this paper is to numerically solve the initial-
value problem
y’(x)
(1)
where
f
.
f (x, y (x)),
(x0)
x
e [x0, X],
is assumed to be continuous and to satisfy a Lipschitz condition on region
[x0, X] x R
In the literature only a few parallel algorithms for solving (1) have been proposed. Generally speaking, speeding up the integration of (1) can be achieved by
partitioning the tasks either "across the system of equations" or "across the method,"
as exemplified by Franklin in [10]. In addition to these two types of parallelism, Bellen
and Zennaro have introduced a third one called "across the time" [1]. The equation
segmentation method is straightforward and widely used by engineers in the field of
dynamic system simulation. However, its scope is rather limited, since automated
partitioning seems feasible only for application-oriented codes in which the structure
of f is accessible. Parallelism "across the time" means that each processor evaluates f
for different values of x. These values are combined into a recurrence that carries the
information from the initial point. This kind of parallelism may yield large speedups
as far as a large number of processors is available. However, Vermiglio reports numerical simulations that suggest severe limitations [19]. Actually, this approach seems
feasible only for equations whose solution does not depend too strongly .on the initial
condition [5].
In this paper, we focus on parallelism "across the method." More specifically,
we consider parallel block methods. Block methods can be seen as a set of linear
multistep methods simultaneously applied to (1) and then combined to yield a "better" approximation. Numerous block methods have been proposed: Shampine and
Watts have constructed A-stable implicit one-block methods for very high orders [17],
whereas Chu and Hamilton have studied the predictor-corrector formulation of multiblock schemes [7].
More recently, Sommeijer, Couzy, and van der Houwen have extended some of the
strong stability properties of the backward differentiation formulae (BDF) (A-stability
Received by the editors May 11, 1992; accepted for publication (in revised form) February 19,
1993. This work was supported by SIMULOG, rue James Joule, 78182 St. Quentin Yvelines Cedex,
France.
Institute de Recherche en Informatique et Systmes Aleatoires
35042 Rennes Cedex, France (chart +/-er0mat. aukun+/-, ac. nz).
552
(IRISA), Campus de Beaulieu,
553
L-STABLE PARALLEL ONE-BLOCK METHODS
or A(c)-stability) to a new class of parallel one-block methods. Their procedure
consists in segmenting the total work per step into a few independent tasks, so that
each task actually requires the same amount of work as a sequential execution of a
BDF. By numerically scanning the space of free coefficients, they have obtained very
promising results with respect to A-stability. From a slightly different point of view,
the search for higher order A-stable multistep methods has been carried out in two
directions: the use of the second derivative of the solution [3], [4], [8], [9] and the
study of new "general linear methods" [15]. These methods have a better numerical
behavior (A-stability up to order six) but are fully implicit and consequently not
appropriate for parallel computers.
Our purpose is to carry on the work of [18] and more specifically to construct
high-order A-stable schemes that are easy to implement on a parallel computer. In
2, we recall the definition of parallel implicit one-block methods introduced in [18]
and we derive a new set of order conditions. Besides, we apply some fairly classic
results on stability to get a practical criterion for A-stability. This criterion and order
conditions are then built into a Maple program which enables us to solve the resulting
system of algebraic equations. This leads to the explicit construction of a third-order
L-stable block method.
Section 3 discusses a rather different approach. The complexity of order and
stability conditions have led us to restrict ourselves to a subset of the methods under
investigation. Rather than trying to solve this large number of algebraic conditions,
we designed a new class of methods by making strong simplifying assumptions. These
methods exhibited surprisingly good stability properties for very high orders. An
analytical proof of their L-stability up to order 11 has been given.
Finally, 4 reports results of accuracy tests and stability tests. These two sets of
tests are in perfect agreement with the theoretical results.
2. Parallel one-block methods. We present a class of methods, introduced by
Sommeijer, Couzy, and van der Houwen in [18], which are a direct generalization of
the implicit one-step method:
Yn+l
ayn
+ hbf(yn) + hdf(yn+l)
as well as a restriction of the multiblock methods of Chu and Hamilton
[7].
2.1. Definition. Let us introduce the following vector:
c--
(Cl,...,Ck) T
with cl
1 (the cis are assumed to be pairwise distinct), and let yn,i denote the
numerical approximation of the exact solution value y(Xn + (ci 1)h) at step n, and
yTn,k) T The final approximation of the solution y at
Yn denote the vector (ynT,1
the point Xn+l is given by Yn+I,1, i.e., the first component of the vector Yn+I. Finally,
let I denote the rn rn identity matrix.
DEFINITION 2.1. We call a k-dimensional parallel block method, a method defined
by the recursion
Yn+l
(A (R) I)Yn + h(B (R) I)F(Yn) + h(D (R) I)F(Yn+I)
where A, B, and D are k-by-k real matrices and D is assumed to be diagonal.
According to the usual convention, for any given vector v (vlT,..., v kT)T F(v)
denotes the vector (f(vl)T,..., f(vk)T) T. Let us emphasize that D is diagonal, so
554
P: CHARTIER
that it is possible to decouple the various components of the solution (see Remark 2.2
below). We can also notice that c is allowed to have k- 1 noninteger components and
that no restriction on their localization in R is assumed (provided they are distinct).
Remark 2.2. Parallelism comes from the possibility to decouple the nonlinear
system that has to be solved at each step (this represents the bulk of computations):
Yn+l h(D (R) I)F(Yn+I)
(A (R) I)Yn + h(B (R) I)F(Yn).
Its Jacobian matrix is indeed of the form
I hDl,1 Of
oy (X n + Cl h, Yn+ 1,1) 0
0
0
i.e., block diagonal. Each block is of dimension rn and corresponds to one subvector
of Y,+I. The corresponding subsystem can be solved independently.
2.2. Formulation as a "general linear method". In the sequel, we will frequently refer to results derived for "general linear methods." Hence, for ease of
presentation, we briefly give the main characteristics of one-block methods considered
as "general linear methods." Following the book of Hairer, Norsett, and Wanner (see
[12, p. 386]), we may define a forward step procedure and a correct value function as
follows:
Forward step procedure:
Vn
Un+I=(.,4(R)I) Un+h (13(R)I) F(Vn),
( h.F(Yn) )
V =(A(R)I)Un+h (l(R)I) F(V),
where
0
I
0
=D,R x
A=
A B
)
R x2
Correct value function:
z(x,h)
---(yT(x q-(Cl 1)h),...,yT(x + (Ck 1)h),
h.y’T(x + (cl 1)h),... ,h.y’T(x + (ck 1)h)) T
The correct value function
be approximated by U,.
Z(Xn, h) gives the interpretation of the method and
is to
2.3. Reduction to an autonomous system. Let e denote the vector (1,...,
Rk. If we assume that A, B, and D satisfy the relations
(3)
ne=e,
A(c- e) + Be + De
c,
then it is equivalent to apply a block method to problem
autonomous system
(4)
1) T
Y’= F(Y),
Y0 (x0, y)T,
(1)
or to the reformulated
555
L-STABLE PARALLEL ONE-BLOCK METHODS
(xiyT) T E Rm+l and F(Y)
dimensional block method to (4), we obtain
where Y
(5)
-
Yn+l
(A
Xne
hc--
When applying a k-
I)Yn 2t- h(B (R) I)F(Yn) + h(D (R) I)F(Yn+I),
xnAe + h[A(c- e)+ Be + De].
Owing to (3), the last line of
Yn+l
(1, f(x,y)T) T
(5)
is satisfied for any h, so that
(5) reduces to
(A (R) I)Yn + h(B (R) I)F(Yn) + h(D (R) I)F(Yn+l),
which is exactly the recursion we get when we apply a block method to (1).
It should be emphasized that conditions (3) are not too restrictive, since they are
satisfied for any method of order p >_ 2 (see Theorem 2.6). In the sequel, we therefore
restrict ourselves to the study of k-dimensional block methods on autonomous systems.
In particular, this will allow us to apply Theorem 8.14 from [12], which is only valid
for autonomous systems.
2.4. Pth-order consistency conditions. Further results will require the expansion of the components of the correct value function as Butcher series. This
motivates the following computations. Readers unfamiliar with Butcher’s series can
refer to [2], [12], and [13].
As a first observation, we can notice that all coefficients z(t) in the expansion of
ith component of the correct value function depend only on the order of the tree t.
In terms of the exact solution y(x) (which is assumed to be smooth enough), we have
y’ i,. + (c- 1).h) yJ(x) + (c 1)(yJ(x)) (1) +...
n
+ (Ci-n!1)nh (YJ(x))(n) +""
h.(yJ(x + (c 1).h)) (1) 0 + h.(yJ(x)) (1) +."
n+l
(yJ(x)) (n+l) +
+ (n + 1) (c-(n+1)nh
1)!
....
denote the only tree of order one and T n denote any tree of order n. According
to Theorem II.2.6. and Definition II.11.1 of [12] we consequently have
Let
zi((R))
1,
(6)
(Ci- 1) n
Zi(T n)
for all from 1 to k and
z,(e) =0,
1,
(7)
Zi(T n)
n.(c- l) n-l
for all from k + 1 to 2k. Expressing relations
(8)
0
z(-)
c-ee
(6) and (7)
in a vector form, we get
and Vn >_ 2, z(Tn):
n(c e) n-1
556
P. CHARTIER
where powers of vector are meant to be componentwise powers. The so-called preconsistency conditions (see [12, p. 397]), which ensure that the method makes sense,
have here a very simple form:
z(e) and z(e)
(,Az(e)
e)
Ae
e.
Now,
in order to present the order conditions, we need two additional definitions.
DEFINITION 2.3. A k-dimensional block method with matrix,4 is called zero-stable
if A is uniformly bounded for all n.
DEFINITION 2.4. Let us consider a zero-stable and preconsistent k-dimensional
block method, and let T be such that the Jordan canonical form of A is
4=T.diag
"..
"..
,J .T-,
"..
where i are the eigenvalues of Ji of modulus one and the Jordan block associated
with eigenvalues of modulus strictly less than one. Then, E is defined as the spectral
projector onto the invariant subspace associated with 1"
E- T.diag(I, 0,..., 0).T -1.
Remark 2.5. No eigenvalue of can lie outside the unit disc since the method
is assumed to be zero-stable. Moreover, preconsistency conditions imply that 1 is an
eigenvalue of .4. It is consequently relevant to decompose A as in Definition 2.4.
The matrix E of Definition 2.4 plays an important role in the definition of the
order of convergence according to Skeel (see Definition 8.10 of [12, p. 393]) that we
have retained here. Although it seems more complex, this definition is less restrictive
than the componentwise definition of the order given in [18] and leads to a global
error of the same order. Finally, we define, as in [18], the following vectors:
Co
Ae
e
and Cj
A(c e) j + j[B(c- e) j-1 + Dcj-l]
cj,
j
1, 2,...
that will be used for the sake of clarity. We are now prepared to give the following.
THEOREM 2.6. A zero-stable and preconsistent k-dimensional block method is of
order p >_ 1 if and only if
(9)
(10)
Vj,
O<_j<_p-1,
Cj=O,
T
(0, O) T.
E(Cp, O)
Proof. Suppose that the components of the local error (see [12, p. 391] for a
definition of the local error of a general linear method) possess an expansion as a
Butcher series, and let di(t) denote the coefficient of the ith component, and d(t)
(dl(t),..., dk(t)) T. Similarly, let v(t) (vl (t),..., vk(t)) T denote the coefficients of
the expansion of Vn. Theorem 2.6 is nothing else but Theorem 8.14 from [12, p. 398]
where d(t) is computed as follows. As mentioned above, the coefficients z(t) of the
correct value function depend only on the order #(t) of the tree t, so that relations
(8.27) of [12, p. 398] can be written as
j=,(t)
(11)
j=0
557
L-STABLE PARALLEL ONE-BLOCK METHODS
and
+
(12)
Inserting (8)into
(11)
(13)
(
d(t)=
(12),
and
we get
[A(c- e)(t) + #(t)B(c- e)(t)-l + Dv’(t)]
t(t)-I v’(t)
c(t)
and
(14)
A(c e) "(t) + #(t)B(c e) "(t)-I + Dv’(t).
v(t)
Now, let us consider a zero-stable pth-order k-dimensional block method. The
following conditions
Vt, #(t) <_ p-1, d(t)
of Theorem 8.14 of
[12,
p.
398], together with conditions (13), imply on the one hand
(15)
Vje [0,p-
i.e., conditions
(9),
1],
Cj -O,
and on the other hand
Vt,
(16)
O
#(t)c"(t)-l.
#(t) _< p- 1, v’(t)
If t is a tree of order less than p- 1, (14) and (16) lead to v(t) C(t) + c"(t), so that
from (15) we get v(t) c "(t). If t is now a tree of order p, according to Corollary 11.7
from [12, p. 246], we have v’(t) -pv(tl).. "v(tm), where t-It1,...,tm] denotes the
tree which leaves over the trees ti when its root and adjacent branches are chopped
off (see Definition 2.12 from [12, p. 152]). Since v(t) ct’(t) for any tree of order less
than p- 1, we have v’(t)
0 of Theorem 8.14 is
pcp-l, so that condition Ed(t)
equivalent to condition (10).
Conversely, let us consider a zero-stable method for which conditions (9) and (10)
are satisfied for an integer p _> 1. Then an induction argument based on (13) and (14)
implies that
Vt, #(t) <_ p-1, v(t)
v("(t))
c
and also consequently that
Vt, #(t) < p, d(t)
d(Tt(t))
( -Cry(t)
0
so that, according to Theorem 8.14, the method is of order p.
COROLLARY 2.7. A sufficient condition for a k-dimensional block method to be
order
p >_ 1 is given by
of
Vj e [O,pl,
(17)
Cj
O.
Remark 2.8. The result stated in Corollary 2.7 has been obtained by Sommeijer,
Couzy, and van der Houwen in [18] by using a componentwise definition of the order.
However, conditions (9) and (10) of Theorem 2.6 are less restrictive than conditions
(17). The
number of scalar equations involved in
(17)
with j
p is indeed k; this
558
P. CHARTIER
number drops to rank(E) for (10) (which is less or equal to k and in most cases strictly
less than k, since E is a projection matrix). The additional free parameters can be
used to improve stability for a given k and a given order. The method of Example
2.17 benefits from this approach and compares favorably with method (3.10) of [18].
Incidentally, method (3.11) of [18] implicitly uses the conditions of Theorem 2.6.
Another application of Theorem 2.6 is given in 3, where methods A//(k, r) reach
order k by using condition (10).
In order to completely define the order conditions, we need to compute matrix
E. However, its decomposition as T.diag(I, 0,..., 0).T -1 is not of practical interest,
since we aim at deriving purely formal algebraic conditions. The following theorem,
together with the next proposition, gives us an easy way to compute E from the
characteristic polynomial of A.
THEOREM 2.9. Let us assume that the method is stable and preconsistent. Let
be the multiplicity of the eigenvalue 1 of A, and let q be the polynomial defined by
( 1)tq()
(18)
det(A I),
k identity matrix. We may then write E as
where I is here the k
0
q- (q(A)
q(A)B
1
E=
(19)
0
I
Proof. The spectral projector E onto the invariant subspace associated with the
eigenvalue 1 of A is characterized by
{
(20)
Vx e n
E =E,
and
(Ex
(Ax x)
E(A-2-)=O,
where 2" is the 2k 2k identity matrix. From
plicity of 1 we have
x)
(18) and the assumption on the multi-
(A-I)q(A)=O.
Therefore,
Aq(A)
q(A) and q(A) 2
q(1)q(A).
Using these properties, it is easy to check that the matrix given by
(19) then satisfies
(0).
2.5. Stability.
PROPOSITION 2.10. A k-dimensional block method is zero-stable
there exists a constant K such that
Vn E N,
Proof.
An
K.
Since
VnEN
An+
( An+
0
An B
0
I
if and
only
if
559
L-STABLE PARALLEL ONE-BLOCK METHODS
[:]
it is easily seen that this property is equivalent to Definition 2.3.
The linear stability of block methods can be investigated by applying the method
to the test equation yl Ay. This leads to a recursion of the form
Yn+ M(Z)Yn,
(21)
where M(z) (I-zD)-(A+zB) is called the amplification matrix and where z Ah
is defined as usual. Following the familiar notions of stability, we will successively
define the following.
DEFINITION 2.11.. The stability region of a block method is the region of the
complex plane, where the sequence of vectors (Yn)neN remains bounded for any initial
vector Yo.
DEFINITION 2.12. A k-dimensional block method is called A-stable if the stability
region contains the left half plane C- {z C; Re(z) <_ 0}.
DEFINITION 2.13. A k-dimensional block method is called L-stable if it is A-stable
and if it has an amplification matrix with vanishing eigenvalues at infinity.
Remark 2.14. It is easily seen by considering the form of M(z) that
-
-D B;
M(z)
lim
therefore, the amplification matrix of an L-stable matrix is such that p(D -B) O.
Here we observe that a point z belongs to the stability region if and only if the
eigenvalues of the amplification matrix M(z) have a modulus less or equal to one
and the eigenvalues of modulus one are nondefective. However, it seems difficult to
investigate the stability region from this single characterization. A more appropriate
result establishes sufficient conditions for A-stability. This will enable us to "verify"
that a method is A-stable.
PROPOSITION 2.15. Let a k-dimensional zero-stable block method be defined by
the matrices A, B and D R k, with D > O. Let P(, z) det(M(z)- I) denote
the characteristic polynomial of M(z), and let ay(y),j
0,... ,2k be the coefficient
j in
ofT
W(w’Y)=(w-1)2tcp(w+l
w-i’iY ) P (
w+l
iy
w-l’
)
Finally, let the matrix Tl be
O2k-1
O2k
?-/(Y)
where
o fo j < o,
0
0
O2k-3
OZ2k-2
O2ka2a
0
0
nd
02k-2
O-2k+l
O-2k-t-2
O-2k+3
a-2k+4
0
ao
O2k-5
O2k-4
(2k-3
() dt((())_<,_<_) t (2 )th pnpt
of four conditions is sufficient for A-stability.
minor. Then the following set
()
By*10,+cx[, VweC, W(w,y*)=O=Re(w)<O,
e ]0, +[,
o() # 0,
e ]0, +[,
() # 0,
e ]0, +[,
fl() # 0.
v
v
v
560
P. CHARTIER
Proof. Let us assume that conditions (22) are satisfied. Similarly to Corollary 1 of
[11], we can prove that for any y E ]0, +cx[, the 2 k roots wj(y) of W(w, y) lie in the
left half plane Re(w) < 0. Now, since the transformation
(w + 1)/(w- 1) maps
the unit disc (of the C-plane) onto the left half plane, the roots of P(, iy)P(,-iy)
lie in the unit disk for any y E ]0, +c[. The same obviously holds for the roots of
P(, iy). This implies that
el0,
p(M(iy)) < 1.
In particular, the method is stable on the imaginary axis (by assumption, 0 belongs to
the stability region). The conclusion now follows from the maximum principle applied
E!
to p(M(z)).
COROLLARY 2.16. If p(D-1B) O, then a sufficient condition for L-stability is
that all coefficients of co(y), c2k(y) and (y) are strictly positive.
For small values of k (k _< 3), the criteria of Proposition 2.15 give the opportunity to derive formal stability conditions in terms of free parameters. The set of all
conditions (stability and order conditions) lead to a large but solvable system, so that
methods can be constructed explicitly (see Example 2.17). However, it does not seem
possible to compute and solve the corresponding system for large values of k, owing
to the expensive computations involved by conditions (22) when all parameters are
kept free. In the next section, Proposition 2.15 and its corollary will be used to verify
stability of a restricted class of methods (methods AJ(k, r)).
Example 2.17. Solving the order and stability conditions (see [6]) for k 2 leads
to the following third-order method:
A
B
D
3
147-- 1880b2t
2s009/ 5
+2000b
-2760b2 -49
120 2
(7+20b2)
s00bb’l_27a0b2,-49
b21
120
b2 (-11+20b2)
2S00b-TaOb2-a9
0
--8
400b 1- 360b2 t-49
2800b 1-2760b2-49
-4/5
3 49+280b2 +400bl
2800b221-2760b21-49
-7/40- 1/2b2
0
)
)
)
where c 3/2 and b2 RootOf(4361 36480Z + 158800Z 2 + 48000Z3), i.e., b2
-3.530865570574838. The condition of Proposition 2.15 is fulfilled, so that A-stability
is achieved. Since the eigenvalues of the amplification matrix vanish at infinity, the
method is L-stable.
3. Generalized backward differentiation formulas. It soon becomes iinpossible to solve the system of order conditions and stability constraints. Even powerful
symbolic manipulation software such as Maple cannot tackle such a huge task for the
time of computation increases exponentially. A way to overcome these obstacles is to
perform a numerical search in the space of free coefficients, once the conditions for
zero-stability and order p have been derived. This technique has been investigated in
[18]. The authors have reached the fifth order with almost A-stability. However, no
formal proof of these results was given. In addition to this, it seems hard to go further,
owing to the considerable amount of work needed to find optimal coefficients. This is
why we turned to the more simple case where B is null, even if it dramatically reduces
the number of free coefficients: for a given block-size k and a given vector c R k,
561
L-STABLE PARALLEL ONE’BLOCK METHODS
there are (k + 1)k unknowns coefficients in A and D instead of (2k + 1)k. Since every
order condition Cj 0 corresponds to k linear equations, we do not expect more than
kth-order methods. However, by sacrificing many coefficients, we got the number of
free parameters to a manageable level.
3.1. Construction. We now attack the explicit construction of a restricted class
of one-block methods. An interesting question is how to choose the matrices B and
D of Definition 2.1 and vector c so as to obtain good stability properties in addition
to high order. Our search for a good choice of parameters is based onthe following
ideas:
1. Set B 0 to ensure stability at infinity (see Remark 2.14);
2. Choose A such that A.U V- D.W, where
U
g
W
[e, (- e), (c- e)2,..., (c- )k-1],
[,c,c2,...,ck-1],
[0, e, 2c,..., (k 1)ck-2],
to ensure a reasonably high order with respect to the "Daniel-Moore" conjecture (see
Theorem 4.17 of [13]). The relation A.U V- D.W is nothing else but conditions
j
(17) of Corollary 2.7 expressed in a matrix form. The assumption ci cj for
ensures that U is nonsingular so that A is determined once D and c are given.
3. Set D (1/r)(I+diag(c)) where I is the k k identity matrix and diag(c) the
k k diagonal matrix with diagonal elements ci, to ensure zero-stability (see Remark
3.1 below).
4. Set c (1,..., k) T in order to make the implementation easier. The ith component is indeed a good initial guess for the determination of the (i- 1)th component
of Yn+l by Newton’s method (the determination of the kth component of Yn+I will
require special attention).
5. Determine the remaining coefficient r so as to maximize the order of accuracy.
Within the class of values of r for which L-stability is achieved, choose the value that
leads to the smallest error constant (the procedure will be described in details in 3.5).
Remark 3.1. This particular and somewhat surprising choice of D as a linear
combination of the two matrices I and diag(c) is motivated by the following observation. A close look at the order conditions of Corollary 2.7 for order (k- 1) suggests
that A can be made upper-triangular in the basis {e, c, c2,..., C k-l} if Dcj-1 belongs
to Span{e, c,..., cJ} for j
1,..., k. This goal is easily reached by taking D as a
aI +/3diag(c)
first degree polynomial of matrix diag(c). Generally speaking, D
leads to a zero-stable method for 0 </3 <_ 2/(k- 1) (see Theorem 3.3 for the details).
0, we get
Furthermore, setting
=
va > 0,
Vz e c-/{0},
p(M(z)) p((I
=
czl)-lA)
=11 -z[-lp(A)
<-
0. Taking this
so that stability is then ensured in the left half plane except at z
of
term
the
a
as
I, which brings
perturbation
result into account, /3diag(c) appears
have
simulations
numerical
domains
of
The
determination
by
stability
zero-stability.
shown that among the best set of parameters was a =/3 1/r.
Combining these ideas we finally give the definition of methods AJ(k,r).
562
P. CHARTIER
DEFINITION 3.2. By 3d(k,r) we mean a special k-dimensional parallel block
of (1), where
method
c-(1,2,...,k) TE
B=0;
D (1/r)(I + diag(c)), r e ]0, +(x[;
A V.U -1 D.W.U
-.
3.2. Zero-stability and stability at infinity. The following theorem sets minimal properties for the methods Jtd(k, r).
TH OaEM 3.3. Let us assume that r >_ (k- 1)/2. Then the method M(k,r)
satisfies
M (k, r) is of order at least k 1;
M(k,r) is zero-stable;
Ad(k, r) is stable in the left half plane x <_ r/(k + 1)Proof. By definition of A, the method is of order p k- 1. Since (Ci)l<i<k
are
pairwise distinct e, c,..., c k-1 is a basis of R k where matrix A has a reduced form.
According to the order conditions of Corollary 2.7 we indeed have
Ae=e and Vje [1,...,p], Ac2- 1- j
c
r
j-jc j-I-E(})(-1)j-Ac
r
l=0
.
Hence,
an easy induction argument shows that A is upper-triangular. From the
expression of A we can see that its eigenvalues are 1 j/r for j 0,..., p. They are
pairwise distinct and of modulus less or equal to one, so that Ad(k, r) is zero-stable.
Finally, by considering the Laplace transform g of g(t) e -tD-1D-1A we get
e-tD-ID-iAe’tdt
.g(-z)
fo
e-t(D--ZI)D-1Adt
(I- z.D)-IA
M(z).
Provided
II.II is subordinated to a vector norm, it follows that
IlM(z)ll <
<
where c
max<i<k ci
k and z
IID-1All
e-(X+-r)tdt
IID-1All
x+r/(c+l)’
-x + iy.
3.3. Error constant. In [18] the normalized error vectors have been introduced
in order to compare their components with the error constants corresponding to conventional linear multistep methods:
gJ
j!De’
where .the division of vectors is meant componentwise..The motivation for such a
choice was the similarity with the definition of error constant for multistep methods. Incidentally, BDF methods written in block format have an error vector of
563
L-STABLE PARALLEL ONE-BLOCK METHODS
(0,.,., 0,-1/(k-- 1)) T, whose last component is nothing but the usual error constant
of BDFs. However, it seems more relevant to define an error constant according to
the asymptotic expansion of the global error. Since our block methods fits into the
class of general linear methods, we may study the asymptotic expansion for a strictly
zero-stable method, i.e., zero-stable and having 1 as the only eigenvalue of A with
modulus one. This previous assumption greatly simplifies the discussion and is fulfilled for all methods A//(k, r), provided r > (k- 1)/2. Theorem 9.1 of [12, p. 406] now
ensures the existence of an asymptotic expansion of the global error with principal
term ep(X) (see [12, p. 409]) for a method of order p satisfying
.ep(X)= EG(x)ep(X)- Edp+l(x)-(I- A + E)-Idp(X),
ep(Xo)- E’),p- (I- d + E)-ldp(xo)
where di (Xn) is the coefficient of h in the Taylor expansion of the local error at x Xn,
p the coefficient of hp in the expansion of do z0- (I)(h) ((I) is the starting procedure)
and G is a function depending on both the problem and the method. If we assume
that j/l(k, r) is of order p k- 1, then dp(x) 0 and dp+(x) -(1/k!)y(k)(x)Ck.
An appropriate definition of the error constant C is consequently given by
1
_
C.e
-.EC.
However, this definition cannot be used when p k, i.e., ECk O, since,dp(x) is
then no longer null and dp+l(X) no longer depends on only derivatives of y. From
a different point of view, we may define an error constant from the defect of the
exponential when inserted into the numerical scheme j4(k, r). For the test equation
y’ Ay and stepsize h, Yo (1,eZ,..., e(k-1)z) T (Z Ah), and we have
0 for all i, 0
Since Ci
A Yo + z D Y
Y1
_< k- 1,
we have
(A ezI + zeZD)Yo
(23)
Let R(, z)
-.Ckzk + O(zk+ 1). P(,
I + zCD) det(I zD)P(, z), where
det(d
teristic polynomial of
M(z).
Relation
z)
is the charac-
(23) implies that
z
R,
+
Since 1 is a simple eigenvalue of A, we can consider the analytic continuation
1 in a neigborhood of the origin (called the principal root). This gives
e
with
C
(z)
Cz
1 (Z) of
+ O(z
(OR/O(1, 0))-(. Now, considering the Laurent series of (A- i)-1
neigborhood of
in a
1, one can easily show that
0)
so that the two definitions of the error constant are consistent when the order is k- 1.
When we get an extra order with ECk O, the second definition carries on since 1 is
still a simple eigenvalue of A, so that we will retain the definition of C given by the
principal term of R(e z). It is clear thatthis definition is fully acceptable only when
the order of .M(k, r) is k- 1. Like Runge-Kutta methods the principal error term is
indeed composed of several "elementary.differentials" when the order is k.
,
564
P. CHARTIER
3.4. Algebraic stability. One can ask whether jJ(k,r) can be made algebraically stable or not (see, for instance, Definition 9.3 of [13, p. 386] for a definition
of algebraic stability). The following proposition shows that the free parameter r can
not be used to achieve algebraic stability.
PROPOSITION 3.4. For k _> 3, A//(k,r) is not algebraically stable.
To prove Proposition 3.4 we need the following lemma.
LEMMA 3.5. For k >_ 3 and for all x in the set {1,2,...,k}, the following
inequality holds:
x k-1
(4)
(k- 1). x_
Proof. Let y
._
x k-2
(x- )-1
> + 1.
x- 1. Straightforward calculations lead to
l--k-2
( + 1)
(4)
-1
l--k-1
(_)
+ (- 1)
(_).
>_
l=O
l=O
Now, for 0 _< _< k- 3, it is clear that (k- 1)(_2) >_ 2(_1). Consequently, (24) is
ensured provided y >_ 1. As for y 0, we may see directly that (24) holds.
Proof of Proposition 3.4. A necessary condition for a preconsistent general linear
method to be algebraically stable is given by Lemma 9.5 of [13]. A diagonal positive
matrix A and a symmetric positive definite matrix G should exist such that
Ae-BTGo,
(- )
_
(2)
0.
Here we have ,4= A, B- D and @-
e. Let us now denotev- Ge. Vector v is
a nonzero vector whose components are all positive (due to (25)). Now, (25) implies
that v ATv. Using the order conditions for j 1 and for j k- 1, we get
{
(26)
-
[(- ) + ;( + ) c] 0,
vT[(c )k-1 k:r. _ .l (ck-1 t_ ck-2) ck-1]
For (26) to hold, the components of (c- e)+ 7 (C
e)
same sign, so that r should belong to the interval ]2; k +
(c- e) k-1 + ((k 1)/r)(ck-1 + ck-2) 5 k-1 implies that
(k- 1)
l<i<k
min
ck + ck-1
k-1
Ci
(5
1) k-1
<r<
(k- 1)
l<i<k
C
O.
should not have all the
similar argument for
1[. A
ck + ck-1
max
c
(ci
1) k-l"
Now, due to Lemma 3.5, this would lead to a contradiction.
3.5. Survey of methods. The estimation of Theorem 3.3 shows that when
Re(z) tends to infinity, [IM(z)ll tends to zero. In fact, this result is also valid when
Iz tends to infinity, since p(D-1B) 0 (see Remark 2.14). It follows that L-stability
is implied by A-stability for methods A/I(k,r). Nevertheless, A-stability cannot be
easily obtained, and we actually did not find any direct proof. All we can show is that
methods j4(k, r) satisfy the conditions of Proposition 2.15 for certain values of r with
respect to k. This can be done by using an efficient symbolic calculus system. We
present in [6] a program written in Maple whose purpose is to build methods AJ (k, r)
for any value of k and r. This program then assembles matrix 7-/and computes the
L-STABLE PARALLEL ONE-BLOCK METHODS
565
(2k- 1)th principal minor, which is known to be a polynomial of the real variable y.
This program is used in the sequel to investigate the L-stability of methods M(k, r)
for particular values of r.
According to 3.1, it still remains to choose parameter r. One special choice seems
interesting: since 1 is a simple eigenvalue of A, the projection matrix E is of rank
one. As already mentioned in Remark 2.8, conditions (10) of Theorem 2.6 for order
k consequently sum up to the following scalar equation:
-
0,
where E(r) and Ck(r) stand for E and Ck as functions of r and the index indicates
that we consider the first component only. For r 0, the left part of (27) becomes a
kth-degree polynomial whoose roots can be easily computed: for any value r of these
roots, JM(k, r) is of order k. We now choose the solution of (27) according to point 5
of 3.1, and we denote it by rk"
2 < k < 6. The values r 4 and r 5 are the solution of (27), respectively,
for k 2 and k 4, which lead to the smallest error constant. Since they are rational
and since B 0, one can easily test the condition of Corollary 2.16. For other values
of k, the smallest error constant is obtained for an irrational solution of (27), so that
f, c0, and c2 have to be computed in terms of the formal parameter r. In that case,
we have to test the positivity of the coefficients of gt, c0, and c2k by applying the
following technique" for a given k, coefficients of gt, c0, and c2k are computed in terms
of r. Then, using an idea of Roy and Szpirglas (see [16]) and the corresponding Maple
program from F. Cucker 1, it is possible to determine the exact sign of each coefficient
for every solution of (27). It turned out that the solutions of (27) minimizing the
error constant lead to L-stable methods.
7 <_ k < 10. There is no rational solution of (27) and the size of the polynomials involved in conditions (22) prevent us from using the technique previously
explained on the machines at our disposal. It is, however, possible to choose r among
the solutions of (27) for which AJ(k, r) is stable in a neighborhood of the origin (see
Lemma 3 of [6]) and then to pick up the value rk minimizing the error constant. For
those methods, L-stability is consequently not proven but only conjectured (owing
to numerical tests). Besides, we have approximated rk with rational values up to 17
digits and checked L-stability for the resulting method. With such an approximation,
the method obtained is numerically equivalent to the method Ad(k, r), since its error
constant (corresponding to order k- 1) is less than the machine precision (10-16).
We now conclude with a survey of the results obtained up to k 10 (cf. Table 1)
and we give the matrix A as well as (27) for the different methods in the appendix from
A to G (from k 2 to 8). For k <_ 6, the methods presented have been completely
constructed according to the strategy described in 3.1. As mentioned above, this
strategy had to be partly modified for higher values of k. For k > 8 the coefficients of
A are large and may cause rounding errors, so that methods for k > 8 have not been
included.
Remark 3.6. For higher values of k we have investigated the L-stability of methods
Ad(k, r) for the particular set of values r k 2. It appeared that conditions (22)
are satisfied for k
11 and k 12. In particular, we have obtained a method of order
11.
This program is to be found in the "share" directory of Maple.
566
P. CHARTIER
TABLE
Error constants for methods JM(k, rk) and BDFs of the same order.
Error Constants (Ad(k, rk), BDFk)
k
rk
2
3
4
5
6
7
4
3.18
5
4.37.
3.92
5.54
(-5/6,-1/3)
(1.15,-1/4)
(93/60,-1/5)
(- 1.55, 1/6)
(1.67, 1/7)
(1.83, ()
8
7.25
(2.53, Q)
9
6.72
(-2.06, (R))
10
8.40
..
L-stability
Proved.
Proved.
Proved.
Proved.
Proved.
Conjectured (proved for
r
5.5438664370799515).
Conjectured (proved for
r
7.2474044915485680).
Conjectured (proved for
r
6.7217218560398991).
Conjectured (proved for
r
8.4009592670313347).
TABLE 2
(A) for Kaps’s problem
-s.
Number
of correct
Method
h=1/4
h=1/8
h=1/16
h=1/32
h=1/64
h=1/128
h=1/256
[(.,r)l
0.57
0.73
1.06
l’3S
1.71
2.03
2.29
2.66
2.88
3.24
3.66
4.66
1.11
.72
1.97
[
I.2.aa
2.58
2.’75
3.64
4’20
5.04
5.80
6.52
7.37
7.97
a.94
9.45
10.S3
2.94
3.18
4.53
5"11
6.24
7.01
8.01
3.54
3.79
5.43
6.02
7.44
8.22
9.51
10.40
.56
12.59
3.65
15.63
4.15
4.39
6.33
6.92
8.65
9.43
11.02
11.91
13.36
4.39
15.75
18.04
BDF2
ra)
BDF3
A/l(4, r4)
BDF4
3/1(5, r5)
BDF5
[3,t(a,
1(6,)
BDF
1(7,)1
I(S,)J
.I
digits
1.35
1.88
2"35
3’28
3.85
4.57
2.70
3.32
3.59
4.28
4.47
5.3
5.3
6.26
5.04
5.84
6.20
7.10
7.33
.47
8,89
9.76
X0"76
X.5
3.3
with e
10
--
3.6. Accuracy tests. In order to numerically verify the order of methods Ad (k, rk)
we integrate the test problem proposed by Kaps [14], and used in [18]:
y
Yl (0)
(0)
--(2 -1)yl -1y,
( + ),
,
1,
OxX,
with the following exact solution for any value of e:
The methods Ad(k, rk) used in this test are those of Table 1 and are exactly defined
at the end of the paper. In Table 2 we have listed the values of A, where A denotes
the number of correct digits of the numerical solution at the end of the interval
(i.e., A -logl0(llyi(Z -Yn,llloc/llYn,lllc)). All experiments have been done with
10 -s and X 4, in order to avoid the influence of the starting procedure (in this
e
567
L-STABLE PARALLEL ONE-BLOCK METHODS
case, the k first approximations are actually exact) 2. These values have been plotted
on a graph (see Fig. 1), and a good appropriateness between theoretical and real order
can be observed.
18
16
14
12
10
8
6
4
2
0
1.5
2
2.5
3.5
3
4
4.5
5
2
8
5,5
6
Stepsize in logarithmic scale
FIG. 1. Error distributions for the methods A/(k, rk), k
of Table
1.
3.7. Stability tests. The stability of the methods is tested by integrating a
problem whose Jacobian matrix J has purely imaginary eigenvalues:
y
y
yl(0)
(2s)
(0)
-ay2
ayl-
,
+ (1 + a)cos(x),
(1 + a)sin(x),
0,
O<_x<_X,
with exact solution
yl
sin(x),
Y2
cos (x)
for any value of the parameter a. Therefore, J has the following expression:
a
0
so that the eigenvalues of J are i.a and -i.a. In Table 3, we give the values of A,
which has been defined above, as a function of the stepsize. We finally illustrate the
better behavior of methods AJ(k, rk) over BDFs (Backward Differentiation Formulas)
for orders 3, 4, 5, and 6. The second-order BDF is indeed A-stable, whereas for orders
higher than 6, the BDF’s are no longer zero-stable. Overflow is indicated by c and
values of A corresponding to stepsizes that are theoretically unstable are underlined.
It should be noted that all methods behave correctly, even those for which stability
is only conjectured.
2
Experiments were performed with a Fortran written code in quadruple precision
(32 digits).
568
P. CHARTIER
TABLE 3
Number
of correct digits (A) for problem (28)
Method
(2,)
BDF2
J(3, r3)
BDF3
J(4, r4)
BDF4
(5,)
BDF5
(,)
BDF6
/I(7, rT)
(8, rs)
with X
100 and c
10.
h-4/5
h--2/5
h--1/5
h-1/10
h--1/20
h--1/40
1.49
2.16
2.82
3.47
4.09
4.72
1.91
2.83
3.58
oc
a
5.3__3
-0.4___1
x)
o
8.15
10.00
2.90
3.43
4.74
5.67
6.96
8.23
9.64
9.77
11.24
11.92
10.83
12.57
8.13
9.29
4. Concluding remarks. In this paper, we have tried to improve the results
obtained by Sommeijer, Couzy, and van der Houwen in [18]. They introduced a class of
seemingly very promising block methods suitable for integrating ordinary differential
equations on parallel computers. On a first step, we analyzed these methods by
considering them as "general linear methods." We applied the different results of
Butcher and Skeel theory, and we obtained a refinement of the order conditions. On
a second step, we got a practical condition for A-stability by applying well-known
results of complex analysis.
Based on this analysis, we have constructed a class of diagonally implicit block
methods denoted by J4(k, rk). The structure of these schemes is such that k processors can be exploited to run them with the same effective computational cost as a BDF
of order k. The. implicit system to be solved at each step can indeed be decoupled
into k independent subsystems of dimension equal to the number of ODEs.
With respect to a real implementation, it is of interest to consider the loadbalancing. In order to avoid any degradation of the parallelism, we need to compute
an initial guess for the Newton process of the subsystem associated with the last
component of our vector of approximations. This may be done by using an explicit
scheme. The extra work required for this represents only a small part of the amount
of computations needed for the Newton process of a large system of ODEs. Another
possible improvement could consist in requesting different convergence tolerances for
the Newton process of different components. All components of the vector of approximations but the first one are indeed to be recomputed at next step.
The main advantage of methods A/I(k, r) is definitely their robustness. Using our
formula manipulation approach, we have proved that they are L-stable at least up
to order 6. Owing to their numerical behavior, L-stability has been conjectured for
methods of order 7 to 10. In addition to this, for orders 7 and 8, we have proved the
existence of L-stable methods which are practically equivalent to methods AA(7, rT)
and j4(8, rs), respectively. With respect to accuracy, the comparison with BDFs of
the same order shows that methods A/I(k, rk) loose about one digit. In a forthcoming
paper, we intend to check L-stability for orders 7 and 8 and to confirm this comparable
accuracy with extensive numerical testing.
569
L-STABLE PARALLEL ONE-BLOCK METHODS
(a) .M(2, 4).
Appendix.
(b) A/I(3, r3).
5/4
63r32 + 185ra
3
r3 is the root of 6r
value is 3.18.
r -1
2
2r-[2
3
2r
144
--r -1
1
6_
3
A=
_
1/2
( -1/4
A=
2r--9
16
r2-r20
6
r
0 whose approximated
)
(c) Ad(4, 5).
2/15 6/5 -2/5 1/15
-1/10 3/5 7/10 -1/5
4/15 -6/5 12/5-7/15
-a
-1/a
(d) A//(5, r5). r5 is the root of 120r55 3000r54 + 26650r53
86400 -0 whose approximated value is 4.37.
--10-6r
6r
2r
_2
4r
-
r -1
15
20--6 r
6r
20
1
r
3r
5
4r
24 r--300
24r
3
r
2
6
r
r
20
3r
30 r--366
6r
r
40 r--468
4r
60 r-642
6r
102850r52 + 166904r5
6r
1
4r
--1
24 r--250
24r
120 r--924
24r
(e) A//(6, r6). r6 is the root of 360r66-12420r +.161250r-992475r63 +2987647r4037154r6 + 1814400 0 whose approximated value is 3.92.
2
5-
52+24r
4
24 r
3
r
3
12---------
5
3r
15
2r
144 r-2268
24 r
5
12r
10
r
r
2
(f) J4(7, rT).
3
r
12r--16
2"---r --1
--2---0-
4r
6
5r
120r-1918
120
-12-12r
2
4
r
20
r
180 r-2772
12 r
r7 is the root of
30
240 r-3556
12 r
2
10r
3
4r
2
10r
5r
-1
r
130-24r
24r
30
r
360 r-4914
24 r
120 r- 1644
120r
720 r-7308
120 r
-87442264r73-254393394r7 + 18305175r
+ 218041817r72-2044770r + 101606400 + l14660rT-2520r77
0 whose approximated
value is 5.54.
3r
10r
lr
12r
5r
7
6r
720r--14112
720
--308-120r
120r
6
5r
5
10
r
3r
84+48 r
4
48r
r
a
_a
5r
2
3r
3
2r
42
5r
r
--16304840 r
120 r
5
2r
5
r
105
4r
--19296+1008 r
48 r
1
20
3r
10
r
140
3r
--23616+1260 r
36 r
570
P. CHARTIER
5
-57
2-7
3
2r
3
2
5r
3
5r
r
--14048r
20r
15r
__2
48r
15
__1
6r
r
924-- 120 r
120r
42
r
105
2r
-30368- 1680 r
48 r
A/l(8, r8).
15 r
--r --1
12348720 r
720 r
-42192T2520 r
120 r
3
r8 is the root of 2289061754rs
-4907763545r82 + 83919360r85 -6850200r86
-64224-5040 r
720 r
+ 5066650482rs-580819883r
5040r88
1828915200 + 292320r
0 whose
approximated value is 7.25.
2__
7r
14 r
4
105
2---2
35r
2088-720r
2r
4
r
180--144r
144 r
10
r
35
2r
280
3r
5
r
r
--564--240 r
240 r
12
5r
3
2r
2
r
2
5r
3-7
2r
5
r
144+144r
144 r
5
5
r
7
6r
8
7r
5040r--117612
5040 r
10
3r
_6
720 r
35
21
4r
168
5r
5r
28
3r
133488+5760 r
720 r
3r
70
r
154296-6720 r
182736-{-8064 r
240 r
3
2
5r
2-7
144 r
"21r
__I
_r-i
4r
4
15r
6
5-7
35r
35:
3
r
1128+240r
240
21
r
84
2r
21
2_
__1
r
7r
7308-720 r
720 r
56
104544+5040
r
r
-223884+10080 r
-288432+13440 r
-402408+20160 r
144 r
240 r
720 r
5040 r
-623376+40320 r
5040
Acknowledgments. I would like to thank Bernard Philippe of IRISA and Michel
Crouzeix of Universit6 de Rennes I, who provided the author with numerous ideas and
therefore brought an important contribution to this paper.
REFERENCES
AND VI. ZENNARO, Parallel algorithms for initial value problems, J. Comput. Appl.
Math., 25 (1989), pp. 341-350.
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