COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.4(2004), No.3, pp.368–383
c 2004 Editorial Board of the Journal “Computational Methods in Applied Mathematics” Joint Co. Ltd.
°
AN ALMOST SIXTH-ORDER
FINITE-DIFFERENCE METHOD FOR
SEMILINEAR SINGULAR PERTURBATION
PROBLEMS1
RELJA VULANOVIĆ
Kent State University Stark Campus
6000 Frank Ave. NW, Canton, OH 44720-7599, USA
Abstract — The discretization meshes of the Shishkin type are more suitable for highorder finite-difference schemes than Bakhvalov-type meshes. This point is illustrated
by the construction of a hybrid scheme for a class of semilinear singularly perturbed
reaction-diffusion problems. A sixth-order five-point equidistant scheme is used at most
of the mesh points inside the boundary layers, whereas lower-order three-point schemes
are used elsewhere. It is proved under certain conditions that this combined scheme is
almost sixth-order accurate and that its error does not increase when the perturbation
parameter tends to zero.
2000 Mathematics Subject Classification: 65L10; 65L12; 65L50.
Keywords: finite differences, semilinear boundary value problem, Shishkin mesh, singular perturbation.
1. Introduction
We consider the singularly perturbed semilinear reaction-diffusion problem
−ε2 u00 + b(x, u) = 0 for x ∈ X = [0, 1], u(0) = u(1) = 0,
(1)
where ε is the perturbation parameter, 0 < ε ¿ 1, and b is a sufficiently smooth function
satisfying
bu (x, u) > b∗ > 0, x ∈ X, u ∈ IR.
(2)
Under condition (2), problem (1) has a unique solution which in general exhibits two boundary layers of exponential type near x = 0 and x = 1. Singularly perturbed boundary-value
problems arise in many applications, see [2, 3, 9], for instance. Problem (1) has been used
frequently as a model for testing different numerical methods for singular perturbation problems. The methods based on special discretization meshes, particularly the Shishkin (S)
mesh [14], have recently gained popularity because of their simplicity and applicability to
more complicated problems in several dimensions, cf. [4, 10, 13]. S meshes were preceded by
1
This work was presented in June, 2003, to Professor Herceg’s Numerical Analysis Group at the Institute
of Mathematics, University of Novi Sad. I would like to thank the audience for their interest and useful
discussion.
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the Bakhvalov (B) meshes [1], which are somewhat more complicated. Both types of meshes
have been subjected to various modifications: B meshes are generalized and simplified in [16],
some improvements of S meshes are considered in [19], and a combination of the two mesh
types is developed in [12] (see [7] as well).
S meshes are piecewise equidistant. For problems like (1), the S mesh typically consists
of three equidistant parts: two dense parts in the layers and one coarse part in between.
There are two transition points between the fine parts and the coarse part of the mesh. S
meshes are therefore simpler than B meshes which have fully nonequidistant dense parts.
A B mesh is generated by a suitable function which maps equidistant points into layercondensed mesh points. However, better theoretical error estimates can be proved on B
meshes, which also give numerically superior results, see [18]. Since B meshes are not
that much more complicated than S meshes, their use certainly pays off. Nevertheless, S
meshes have an advantage if more complicated higher-order schemes are to be applied. When
schemes with more than three points are to be used to discretize (1), their nonequidistant
generalizations are needed on the B mesh, and are therefore more difficult to construct and
analyze theoretically. At the same time, the simple equidistant stencil suffices on the S
mesh at all points except for a few, where some simpler (non)equidistant schemes can be
used. These exceptional points are the transition points, a couple of points adjacent to them,
and perhaps a few points in the vicinity of x = 0 and x = 1. This approach has already
been applied in [20, 21] to some nonlinear problems with two small parameters, which are
discretized by four-point third-order schemes.
One of the main purposes of the present paper is to point out this idea again, this
time as applied to problems of the type given in (1). The problem under consideration
is relatively simple (and there are other robust numerical methods, like the automatic hpadaptive schemes, that can solve it efficiently), but our approach can certainly be applied
to more complicated problems, including two-dimensional singular perturbation problems.
Moreover, we explore here how far we can go in the theoretical discussion of an elementary
method such as finite-differences. We do this by constructing and analyzing a five-point
sixth-order equidistant Hermitian approximation of (1). This, to best of our knowledge, new
scheme is a sixth-order counterpart of the better-known fourth-order three-point Hermitetype approximation, which is also known as the Numerov approximation. The sixth-order
scheme is applied at all the points of the fine mesh where possible and simpler three-point
schemes are used at the remaining points. It is proved under some additional conditions that
this hybrid scheme is stable uniformly with respect to ε and that its solution approximates
the solution of (1) with the following error:
Ã
µ
¶6 !
ln N
ε2
+
O
,
N4
N
where N is the number of mesh steps. This is a considerable improvement over the (almost)
fourth-order accuracy obtained in [6, 15, 17, 18] by different nonequidistant versions of the
fourth-order Hermite scheme on B (and S) meshes. The above error estimate gives the
accuracy of almost sixth order if ε 6 (ln N )3 /N , which is a mild practical constraint.
When we formally set ε equal to 0 in (1), we get the very simple reduced problem
b(x, u) = 0, x ∈ X. By considering the equation b(x, u) = 0 at all the values of x which are
outside the layers and which are of interest, we get a set of uncoupled nonlinear equations.
Each of them can be solved with arbitrary accuracy using some nonlinear solver. However,
the fact that −ε2 u00 is ignored, albeit outside the layers, still produces an O(ε2 )-error, which
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370 R. Vulanović
can compete with our method only if ε 6 (ln N/N )3 . Besides, we still have to use some
highly accurate method to solve the problem within the layers. This is why we are not
pursuing this approach here.
The remaining sections are organized as follows. Precise assumptions on (1) and properties of its solution are presented in the first subsection of Section 2, followed by the description
of two discretization meshes of the Shishkin type and of finite-difference schemes defined on
them. The proof of the almost sixth-order accuracy is given in Section 3. The method and
the technique of proof are discussed additionally in Section 4. Finally, Section 5 presents
results of some numerical experiments.
2. Preliminaries
2.1. Continuous problem
Problem (1) is considered under the assumptions described below.
Let b, bu ∈ C(X, IR) and let (2) hold true. Then there exist two constants, u∗ > 0 and
u∗ < 0, such that b(x, u∗ ) > 0 > b(x, u∗ ), x ∈ X. This means that u∗ and u∗ are respectively
the upper and lower solutions of (1). Therefore, problem (1) has a unique solution, uε ,
belonging to C 2 (X). This solution satisfies
uε (x) ∈ U ∗ := [u∗ , u∗ ], x ∈ X.
Let also U = [u, u] ⊃ U ∗ with u < u∗ and u > u∗ . Then there exist constants b̄ and b such
that
b̄ > bu (x, u) > b > b∗ , x ∈ X, u ∈ U.
Different additional smoothness assumptions are required in different lemmas and theorems below. The most that is needed is b ∈ C 8 (X × U ), so that uε ∈ C 10 (X). This smoothness is required to prove the following well-known derivative-estimates for uε (see [16] for
instance):
p
£
¤
−k −mx/ε
−k m(x−1)/ε
|u(k)
(x)|
6
M
1
+
ε
e
+
ε
e
,
x
∈
X,
m
=
b∗ ,
(3)
ε
where k = 0, 1, . . . , 8. Here and throughout the paper, M denotes any positive constant
independent of both ε and the number of mesh steps, N . Some specific constants of this
kind will be indexed.
2.2. Mesh
Let X N be a general discretization mesh with the points 0 = x0 < x1 < · · · < xN = 1. Let
also hi = xi − xi−1 , i = 1, 2, . . . , N , and ~i = (hi + hi+1 )/2, i = 1, 2, . . . , N − 1. By {wi } we
denote an arbitrary mesh function on X N \ {0, 1}, which we identify with the column vector
wN = [w1 , w2 , . . . , wN −1 ]T .
For any mesh function, we formally set w0 = wN = 0. Let also
T
`N = [1, 1, . . . , 1]T and uN
ε = [uε (x1 ), uε (x2 ), . . . , uε (xN −1 )] .
We use the maximum vector norm, kwN k = max16i6N −1 |wi |, and denote the corresponding
subordinate matrix norm in the same way.
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Two special Shishkin-type meshes are now defined. For simplicity, we assume that N is
even and that xN −i = 1 − xi , i = 1, 2, . . . , N . It therefore suffices to describe the mesh on
the interval [0, 1/2]. Let L∗ = L∗ (N ) be the solution of the equation exp(−L∗ ) = L∗ /N and
let L = L(N ) denote any quantity satisfying
L∗ 6 L 6 ln N so that e−L 6
L
.
N
Then define τ = (a/m)εL, where a is a positive constant to be determined and m is as in
(3). We assume that τ < 1/2, since N is unrealistically large otherwise. Let J = qN be a
positive integer such that q < 1/2 and 1/q 6 M . Then we form a Shishkin-type mesh S(L)
by dividing the interval [0, τ ] into J equidistant subintervals and the interval [τ, 1/2] into
N/2 − J equidistant subintervals. Thus,

τ

i = 1, 2, . . . , J,
 h := ,
J
hi =
1 − 2τ

 H :=
, i = J + 1, J + 2, . . . , N/2,
N − 2J
where h and H are respectively the fine and the coarse mesh step-sizes.
The standard Shishkin mesh is S(ln N ). The use of L < ln N enables meshes with
a greater density in the layers, which improves accuracy of numerical results. This is of
practical importance only since, theoretically, any L behaves like ln N as N → ∞, see [21].
It should also be noted that the constant a is another parameter for controlling the mesh
density in the layers; the smaller the value of a, the greater the density. However, accuracy
of the scheme dictates how small a can be.
We shall see in Section 3 that our technique of proof requires a Shishkin-type mesh that
changes smoothly in the transition from the fine part to the coarse part. Such a modified
S mesh, denoted here as S̃(L), was introduced in [21]. It can be defined by xi = κ(i/N ),
where κ is a mesh generating function in C 2 [0, 1/2],
τ

if t ∈ [0, q],
 t
q
κ(t) =
τ

 p(t − q)3 + (t − q) + τ if t ∈ [q, 1/2].
q
The above coefficient p is determined from κ(1/2) = 1/2, and it follows that 0 < p 6 M .
The fine parts of S(L) and S̃(L) are identical, but the coarse part of S̃(L) is a smooth
continuation of the fine mesh and is no longer equidistant. Numerical results of Section
5 are the same on the two meshes, which indicates that S̃(L) is indeed needed for technical/theoretical reasons only.
2.3. Discretization
We are now going to describe the schemes used to form the discretization of (1). The simple
second-order central scheme will be used outside the layers in some cases,
T2 wi := −ε2 D2 wi + bi ,
where
1
D2 wi =
~i
µ
wi−1 − wi wi+1 − wi
+
hi
hi+1
¶
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372 R. Vulanović
and bi = b(xi , wi ). The accuracy of D2 can be improved by combining it with an appropriate linear combination of bi−1 , bi , and bi+1 . This is the fourth-order Hermite scheme
(see [6, 15, 17, 18]). The version from [18] is used here to discretize (1),
T4 wi := −ε2 D2 wi +
2hi − hi+1
5
2hi+1 − hi
bi−1 + bi +
bi+1 .
12~i
6
12~i
An analogous sixth-order scheme can be constructed, but this is done below only on equidistant mesh. Start from the following equidistant fourth-order five-point scheme (see [8],
for instance) approximating u00 (xi ):
D4 wi =
1
(−wi−2 + 16wi−1 − 30wi + 16wi+1 − wi+2 ).
12h2
Then the sixth-order discretization of (1) is
T6 wi := −ε2 D4 wi +
1
(−bi−2 + 4bi−1 + 84bi + 4bi+1 − bi+2 ).
90
T4 is a three-point scheme and therefore easier to derive on a nonequidistant mesh.
However, not all nonequidistant versions of this scheme are equally amenable to theoretical
analysis (see [18]). It therefore seems pointless to try to construct a nonequidistant generalization of the more complicated T6 . However, there is no problem to use it at almost all
points of the fine part of S(L) or S̃(L).
Two discretizations of problem (1) on S(L) or S̃(L) are discussed in the rest of the paper.
Both are of the form
T wN = 0,
(4)
where


T4 wi




 T6 wi
T wi :=
for
for
i = 1,
2 6 i 6 J − 2,
T̃ wi for J − 1 6 i 6 N/2,





 symmetrical scheme w.r.t. x
N/2
=
1
2
for N/2 + 1 6 i 6 N − 1.
The first discretization, denoted by T6,2 uses T̃ = T2 , and the other one, T6,4 , is with T̃ = T4 .
3. Error estimates
Discretization (4) uses the fourth-order scheme T4 at x1 and xN −1 . In order to prove accuracy
of almost sixth order, these two equations are multiplied by M0 δ 2 , where δ = L/N and M0 is
some suitably chosen constant independent of ε and N . Let the corresponding modification
of the discrete operator T be denoted by T̄ . Then the two resulting operators, T̄6,2 and T̄6,4 ,
are discussed using the standard principle that consistency and stability give convergence.
The consistency error estimate is of the following form in this paper:
|T̄ uε (xi )| 6 M δ s , i = 1, 2, . . . , N − 1,
with some fixed s > 0. Then we say that the order of accuracy/consistency of the scheme T̄
is almost s. Stability can be expressed as
kwN − v N k 6 M kT̄ wN − T̄ v N k,
(5)
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373
where wN and v N are two arbitrary mesh functions. The order of convergence is the same
as that of consistency. In this discussion, N may be assumed sufficiently large independently of ε.
Various (N − 1)–dimensional open balls centered at uN
ε are used in the forthcoming
analysis. Let for a ρ > 0,
β(ρ) = {wN | kwN − uN
ε k < ρ}.
3.1. Scheme T6,2
The operator T̄6,2 is easier to analyze and we do this first. Since T6 is complemented with
the central scheme, convergence of almost sixth order can be proved if ε is very small in
comparison to 1/N . More precisely, the following is required:
ε6M
L3
.
N2
(6)
Theorem 3.1. Let (6) hold true and consider T = T6,2 on the S̃(L) mesh with a > 10.
Then,
6
kT̄6,2 uN
ε k 6 M∗ δ ,
where M∗ is some constant independent of both ε and N .
Proof. The proof follows a fairly standard technique. Because of the symmetry of the
mesh, the scheme, and the estimates in (3), it suffices to prove
|T̄6,2 uε (xi )| 6 M δ 6
(7)
for i = 1, 2, . . . , N/2. At most of these points, Taylor’s expansion of T̄6,2 uε (xi ) is used
together with a simplified form of (3),
−k
−mx/ε
|u(k)
, x ∈ [0, 1/2] .
ε (x)| 6 M [1 + ε y(x)], y(x) := e
If i = 1,
|T̄6,2 uε (x1 )| = M0 δ 2 |T4 uε (x1 )| 6 M δ 2 ε2 h4 (1 + ε−6 ) 6 M δ 6
and (7) holds true in this case.
At all other points we consider here, T̄6,2 = T6,2 . Similarly to the case i = 1, for
2 6 i 6 J − 2, we have
|T6,2 uε (xi )| = |T6 uε (xi )| 6 M ε2 h6 [1 + ε−8 ] 6 M δ 6 .
The central scheme T2 is used at the remaining points. The scheme is equidistant at
xJ−1 but this case will be treated together with the nonequidistant part of the mesh. For
J − 1 6 i 6 N/2, it holds that
|T2 uε (xi )| 6 M ε2 (Yi + Zi ),
where
−3
Yi = (hi+1 − hi )|u(3)
ε (xi )| 6 M (hi+1 − hi )[1 + ε y(xi−1 )]
and
Zi = h2i+1
−4
2
max |u(4)
ε (x)| 6 M hi+1 [1 + ε y(xi−1 )].
[xi−1 ,xi+1 ]
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374 R. Vulanović
Note that for i = N/2, the term ε−4 exp[−m(1 − xi+1 )] resulting from (3) is not neglected
when estimating Zi , but included in the above estimate since 1 − xN/2+1 = xN/2−1 . Then, Yi
and Zi are estimated further using
y(xi−1 ) 6 y(τ − 2h) = e−aL · e2mh/ε 6 M δ a ,
hi+1 6 N −1
max
κ0 (t) 6 M N −1 ,
[(i−1)/N,(i+1)/N ]
and
hi+1 − hi 6 N −2
max
[(i−1)/N,(i+1)/N ]
κ00 (t) 6 M N −2 .
After applying (6) and a > 10, it follows that
|T2 uε (xi )| 6 M (δ 6 + ε−2 N −2 δ 10 ),
which implies (7) if N −1 δ 2 6 ε.
The remaining case of the proof is therefore ε 6 N −1 δ 2 when J − 1 6 i 6 N/2. |T2 uε (xi )|
is estimated differently in this case,
|T2 uε (xi )| 6 2ε2
max |u00ε (x)| 6 M [ε2 + y(xi−1 )] 6 M (N −2 δ 4 + δ 10 ) 6 M δ 6 .
[xi−1 ,xi+1 ]
In order to prove (5), we first need the following lemma.
Lemma 3.1. On S(L) and S̃(L) with a > 1, it holds that
|uε (xi ) − uε (xi−1 )| 6 M δ, i = 1, 2, . . . , N.
Proof. See [15].
Theorem 3.2. Consider T = T6,2 on either S(L) or S̃(L). Then for all sufficiently
large N , independent of ε, T̄ = T̄6,2 satisfies the stability inequality (5) for any wN , v N ∈
β(ρ0 ), where ρ0 is sufficiently small but independent of both ε and N .
Proof. For the technique used here, see [15, 18, 21]. Let wN ∈ β(ρ0 ), where ρ0 is chosen
small enough to give β(ρ0 ) ⊂ U N −1 . Then
b̄ > bu (xi , wi ) > b > b∗ , i = 1, 2, . . . , N − 1.
Next, let F = T̄ 0 (wN ) denote the Fréchet derivative of T̄ = T̄6,2 at wN . We first prove that
F = [fij ] is an inverse-monotone matrix. It is easy to see that
fii > 0 and fi,i±1 6 0
(8)
when T2 is used at xi . At all other points, the sign of fij is determined by the sign of the
corresponding coefficient resulting from the discretization of −ε2 u00 (x). For instance, if T6 is
used at xi , then
2
4ε2
fi,i−1 = − 2 + bu,i−1 6 M (b̄ − N 2 ) 6 0
3h
45
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since N is sufficiently large. Thus, (8) is satisfied for all i = 1, 2, . . . , N − 1 (we formally set
f10 = fN,N +1 = 0). However, fi,i±2 > 0 when T6 is applied at xi and because of this, F is
not an L-matrix.
F is nevertheless an inverse-monotone matrix because of the Lorenz standard decomposition (see [8]). D4 is one of the schemes considered in [8] and shown to yield an inversemonotone matrix. F meets all the conditions for the standard decomposition, the following
inequalities being crucial:
4fi,i−2 fi−1,i−1 6 fi,i−1 fi−1,i−2 ,
4fi,i+2 fi+1,i+1 6 fi,i+1 fi+1,i+2 , i = 2, 3, . . . , J − 2,
(with f2,0 := 0). These inequalities hold true provided N is sufficiently large independently
of ε. This is so because the coefficients of D4 satisfy the corresponding inequalities strictly
and ε2 /h2 behaves like M N 2 /L2 . Finally, the following inequality, which should be understood componentwise, holds true:
F `N > b∗ `N .
(9)
It is easy to see that (F `N )i > b for all i such that the central scheme T2 is used at xi .
Furthermore, if i = 1, we have
·³ ´
¸
³ mq ´2 11
ε 2 5
1
N
2
(F ` )1 = M0 δ
+ bu1 + bu2 > M0
+ bδ 2 > b∗
h
6
12
a
12
if M0 = a2 /q 2 . When T6 is used at xi , it follows that
(F `N )i >
1
(−bu,i−2 + 4bu,i−1 + 84bu,i + 4bu,i+1 − bu,i+2 ) .
90
For j = i ± 1, i ± 2, we now use
bu,j = bu,i + gj , where |gj | 6 M (h + |wj − wi |).
Applying Lemma 3.1 and the fact that wN ∈ β(ρ0 ), we get
|wj − wi | 6 |wj − uε (xj )| + |uε (xj ) − uε (xi )| + |uε (xi ) − wi | 6 2ρ0 + M δ.
It therefore follows that
(F `N )i > bu,i + O(ρ0 + δ) > b∗ ,
provided N is large enough, independently of ε, and provided ρ0 is chosen sufficiently small
but independent of both ε and N . Thus, (9) is proved. This completes the proof of inverse
monotonicity of F .
Since F is inverse monotone, it immediately follows from (9) that
kF −1 k 6
1
.
b∗
(10)
This, on the other hand, implies (5) with M = 1/b∗ , since
Z 1
¡
¡
¢¢
N
N
T̄ 0 v N + s wN − v N ds · (wN − v N ).
T̄ w − T̄ v =
0
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376 R. Vulanović
It is now possible to prove the convergence result.
Theorem 3.3. Consider the discretization (4) with T = T6,2 on S̃(L) with a > 10.
Let N be sufficiently large, independently of ε, and let (6) hold true. Then (4) has a solution
in β(ρ∗ ), where ρ∗ = γδ 6 and γ < M∗ /b∗ is independent of both ε and N . This solution is
unique in β(ρ0 ) introduced in Theorem 3.3.2.
Proof. For a sufficiently large N , β(ρ∗ ) ⊂ β(ρ0 ) and (10) holds true. Then,
1
M∗ 6
kT̄6,2 uN
δ < ρ∗ ,
ε k 6
b∗
b∗
which means that the local version of the Hadamard theorem, [11; p. 138], guarantees that
the discrete problem (4) with T = T6,2 has a solution in β(ρ∗ ). This solution is unique in
β(ρ0 ) by virtue of (5).
3.2. Scheme T6,4
The constraint (6) is too restrictive and it is important to relax it. This can be done when
the scheme T6,4 is used on S̃(L) to discretize (1). The relationship between ε and N still has
to be restricted by assuming
L3
ε6
.
(11)
N
This condition is discussed in Remark 4.1 of the following section, where it is also shown
that the present method remains highly accurate even when (11) is removed.
In this subsection, we also assume that (6) is not satisfied, since the scheme T6,2 suffices
otherwise. We can therefore use
δ 2 6 ε.
(12)
Theorem 3.4. Let (11) and (12) hold true and consider T6,4 on the S̃(L) mesh
with a > 10. Then,
6
kT̄6,4 uN
ε k 6 M∗∗ δ ,
where M∗∗ is some constant independent of both ε and N .
Proof. The proof is similar to that of Theorem 3.3.1. The only difference is that T4 is
used instead of T2 as the middle part of the discretization. The consistency error at xi ,
i = J − 1, J, . . . , N/2, is now estimated in one step,
|T4 uε (xi )| 6 M ε2 (Pi + Qi + Ri ),
where (see [18])
−4
(1 + ε−4 δ 10 ),
Pi = (hi+1 − hi )2 |u(4)
ε (xi )| 6 M N
−4
(1 + ε−5 δ 10 ),
Qi = (hi+1 − hi )(h2i + h2i+1 )|u(5)
ε (xi )| 6 M N
and
−4
(1 + ε−6 δ 10 ).
Ri = (h4i + h4i+1 ) max |u(6)
ε (x)| 6 M N
[xi−1 ,xi+1 ]
Thus,
|T4 uε (xi )| 6 M ε2 N −4 (1 + ε−6 δ 10 ) 6 M δ 6 ,
where we have used (11) and (12).
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The proof of stability requires the following lemma.
Lemma 3.2. If (12) holds true, then 2hi > hi+1 for i = 1, 2, . . . , N/2 on S̃(L).
Proof. The assertion is trivially true on the fine part of the mesh and for i = N/2. Let
J 6 i 6 N/2 − 1. Then hi > hi+1 − hi follows if
µ
¶
µ
¶
i−1
i+1
0
00
Nκ
>κ
,
N
N
that is, if
τN2
ω(z) := 3pz − 6pz +
− 12p > 0,
q
where z = i − 1 − J > −1. It is easy to verify that the discriminant of the quadratic function
ω is nonpositive if τ N 2 > 15pq. Since (12) definitely implies the last inequality, it follows
that ω(z) > 0 for all z.
2
An additional constraint on the function b is also required,
θ := 5b − b̄ > 0.
(13)
Theorem 3.5. Consider T = T6,4 on S̃(L). Let N be sufficiently large, independently
of ε, and let (12) and (13) hold true. Then T̄ = T̄6,4 satisfies the stability inequality (5) for
any wN , v N ∈ β(ρ1 ), where ρ1 is sufficiently small but independent of both ε and N .
0
Proof. Let G = T̄6,4
(wN ) be the Fréchet derivative of T̄6,4 at an arbitrary wN ∈ β(ρ1 ).
Similarly to ρ0 in the proof of Theorem 3.3.2, ρ1 is chosen so that β(ρ1 ) ⊂ U N −1 .
The standard decomposition argument does not apply to G. Rather, it is combined with
the technique from [21]. G is decomposed as G = A + K, where K = [kij ] and
ki,i−1 =
2hi − hi+1
2hi+1 − hi
bu,i−1 , ki,i+1
bu,i+1 , i = J, J + 1, . . . , N − J,
12~i
12~i
all other elements of K being equal to 0. Because of Lemma 3.2 and the symmetry of the
mesh, ki,i±1 > 0 and it follows that
µ
¶
2hi − hi+1 2hi+1 − hi
b̄
kKk 6 b̄ max
+
= .
J6i6N −J
12~i
12~i
6
On the other hand, A is inverse monotone, which can be shown in the same way as the
inverse monotonicity of F in the proof of Theorem 3.3.2. Moreover, it holds that
kA−1 k 6
6
,
5b
cf. (10). Then,
b̄
6 b̄
· =
< 1,
5b 6
5b
where the last inequality follows from (13). This implies that the matrix I + A−1 K is
nonsingular, and so is G = A(I + A−1 K). Furthermore,
kA−1 Kk <
kG−1 k = k(I + A−1 K)−1 A−1 k 6
1
kA−1 k
= ,
−1
1 − kA Kk
θ
where θ is from (13). Therefore, (5) holds true with M = 1/θ.
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The convergence result for T6,4 is finally proved analogously to Theorem 3.3.
Theorem 3.6. Consider the discretization (4) with T = T6,4 on S̃(L) with a > 10. Let
N be sufficiently large, independently of ε, and let (11), (12), and (13) hold true. Then (4)
has a solution in β(ρ∗∗ ), where ρ∗∗ = ηδ 6 and η < M∗∗ /θ is independent of both ε and N .
This solution is unique in β(ρ1 ) introduced in Theorem 3.5.
4. Remarks
Remark 4.1. For problem (1), we have developed a numerical method, which is of
almost sixth order when condition (11) holds true. The method can be described also as
discretization (4) in which the operator T uses
(
2T2 wi
if ε < δ 2 ,
T̃ wi =
(14)
T4 w i
if ε > δ 2 .
By Theorems 3.2 and 3.5, this discretization is stable uniformly in ε, provided N is sufficiently
large, but independent of ε, and provided (13) holds true. Theorem 3.6 guarantees that the
error of the numerical solution does not increase when ε → 0. However, Theorem 3.6 does
not give convergence uniform in ε because of the condition (11). The following result, which
can be proved without assuming (11), shows that the discretization is still highly accurate.
Theorem 4.1. Consider discretization (4) with T̃ like in (14) on S̃(L) with a > 10. Let
N be sufficiently large, independently of ε, and let (13) hold true. Then (4) has a solution
in β(ρ∗ ), where
ρ∗ 6 M (ε2 N −4 + δ 6 ).
This solution is unique in β(min{ρ0 , ρ1 }), where ρ0 and ρ1 are introduced in Theorems 3.2
and 3.5 respectively.
This theorem indicates that the present method converges uniformly in ε with order which
is at least 4, and therefore, it cannot be asymptotically worse than any of the discretizations
in [6,15,17,18]. In fact, because of the ε2 -factor multiplying N −4 in the above error estimate,
our method is expected to be superior to the above-mentioned fourth-order ones.
On the other hand, (11) is not a serious practical constraint. For instance, the inequality
(11) is still valid for N = 3.5 · 106 , when 2−10 6 ε 6 2−30 , which is the range used in the
numerical experiments presented in the next section. Moreover, ε and N are usually such
that
ε| ln ε| 6 M N −1 ,
(15)
which means that no mesh point lies inside the layers when the entire mesh is equidistant.
It is obvious that (15) implies (11).
Remark 4.2. The condition (13), which is needed in Theorems 5, 6, and 7, seems to be
purely theoretical. We have tested T6,4 also on problems violating (13) without detecting any
instability. A condition of the same kind, but even more restrictive than (13), is also assumed
in [6] when discussing the stability of a nonequidistant fourth-order Hermite scheme. We
show next that it is possible to construct a discretization operator T with a different T̃ ,
which does not require (13) in the proof of Theorem 3.5. At the same time, the simpler S(L)
mesh suffices.
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Let D4 (h) indicate the previously introduced scheme D4 on an equidistant mesh with
step-size h. The following, then, is a fourth-order discretization of (1) on the coarse part of
the mesh:
T4∗ wi := −ε2 D4 (H)wi + bi .
Two other fourth-order schemes are also needed on the coarse mesh. Both are of the form
T4∗ (α−1 , α5 )wi := −ε2 D∗ (α−1 , α5 )wi + bi ,
5
1 X
D (α−1 , α5 )wi = 2
αj wi+j .
H j=−1
∗
One of these schemes uses α−1 = 0 and the other α5 = 0, whereas the remaining coefficients αi
are determined by the O(H 4 ) accuracy of the schemes. Then the T̃ part of the discretization
(4) on S(L) can be defined as

3T4 wi



 T ∗ (0, α )w
5
i
4
T̃ ∗ wi :=
∗

 T4 (α−1 , 0)wi

 ∗
T4 wi
for
for
for
for
i = J − 1,
i = J,
i = J + 1,
J + 2 6 i 6 N/2.
∗
Let T6,4
denote the operator T on S(L) with T̃ = T̃ ∗ .
This discretization is mentioned here only for theoretical reasons, since it is obviously
clumsy and less practical than the previously considered ones. It also requires that some other
theoretical assumptions be made somewhat stronger in order to prove a result analogous to
Theorem 3.6. The mesh S(L) has to be less dense, since a > 14 is needed, and (11) has to
be replaced with a slightly stronger condition,
ε 6 M1 N −1 ,
(16)
where M1 is a sufficiently small constant independent of ε and N .
The following theorem can be proved.
∗
Theorem 4.2. Consider discretization (4) with T = T6,4
on S(L) with a > 14. Let
N be sufficiently large, independently of ε, and let (16) hold true. Then (4) has a unique
solution in β(ρ) with ρ 6 M δ 6 .
Proof. The consistency-error estimate is proved analogously to the proofs of Theorems 1
and 4, but since neither (6) nor (12) can be used, a > 14 is needed. The proof of stability
requires (16) and can be done like in the proof of Theorem 3.5. The main difference is in the
decomposition of the Fréchet derivative. All the entries resulting from the second-derivative
approximations at xi , i = J, J + 1, . . . , N − J are stored in a matrix corresponding to K.
Remark 4.3. The only reason for using the mesh S̃(L) and not S(L) in Theorems 1
and 4 is the need for the estimate hJ+1 − hJ 6 M N −2 , which is not true on S(L), where
hJ+1 − hJ = H − h 6 M N −1 . Because of this, the technique used to prove Theorems 3
and 6 can give only almost fifth-order convergence results on S(L). For this accuracy, the
condition on a can be relaxed to a > 13/2.
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380 R. Vulanović
It is also possible to prove the result of Theorem 3.3 on S(L) by replacing D2 at xJ with
a four-point second-order approximation of u00 (xJ ), cf. [5],
u00 (xJ ) ≈
6H
h − 3H
2(2H − h)
h−H
uJ−1 +
uJ +
uJ+1 +
uJ+2 ,
2
2
h(h + H)(h + 2H)
hH
(h + H)H
(h + 2H)H 2
provided the following condition is assumed instead of (6):
ε 6 M2
L
,
N2
where M2 is sufficiently small but independent of ε and N . The stability result is then proved
like in Theorem 3.5 by decomposing the Fréchet derivative. The matrix corresponding to K
should contain only the elements resulting from the second-derivative scheme at xJ .
Remark 4.4. In some cases, like when the reduced problem b(x, u) = 0, x ∈ X, has a
constant solution, estimates sharper than (3) are available,
£ −mx/ε
¤
−k
|u(k)
e
+ e−m(1−x)/ε , x ∈ X, k = 0, 1, . . . , 8.
(17)
ε (x)| 6 M ε
If this holds true, Theorem 3.3 can be proved on S(L) with a > 6 without assuming (6).
This result means ε-uniform convergence of almost sixth order.
5. Numerical results
Numerical results are presented in this section for the same problem as in [18],
−ε2 u00 +
u−1
+ f (x) = 0,
2−u
u(0) = u(1) = 0,
where f (x) is chosen so that the exact solution is
uε (x) = 1 −
e−x/ε + e(x−1)/ε
.
1 + e−1/ε
The homogeneous version of this problem is used as a model of the Michaelis-Menten process
in biology [2], and its solution behaves like the above uε .
The tables below present the errors
N
EN = kuN
ε − u k,
where uN is the numerical solution on a mesh with N mesh steps. The following two
numerical orders of convergence are also calculated:
ON =
ln EN − ln E2N
ln 2
and ÕN =
ln EN − ln E2N
.
ln (2 ln N ) − ln(ln(2N ))
ON and ÕN estimate the convergence rate as a power of N and as a power of δ respectively.
The S meshes in our numerical experiments were used with m = 0.49, different values of
a and q, and either L = ln N or L = L∗ . According to Remark 4.4, a > 6 is theoretically
safe for the above test problem. For this problem, we can take b = 1/4 and b̄ = 1, thus
(13) is satisfied. Numerical experiments were run with ε = 2−j , j = 10, 15, 20, 25, 30, and
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An almost sixth-order method
381
were the same for all these values of ε. Also, EN was identical on S(L) and S̃(L), as well
as for T6,2 and T6,4 . The latter is to be expected for this test problem because of Remark
4.4. However, T6,2 and T6,4 produced the same errors even when they were applied to some
other problems, whose solutions satisfied (3) rather than (17), and when (6) could not be
assumed. This happened because in both cases the greatest pointwise error was attained
inside the layers where T6 was used.
Table 1 shows the errors produced by T6,4 (or T6,2 ) for different values of mesh parameters.
In general, the results are better in this example if the density of the mesh in the layers is
greater, that is, if L is less (witness the practical significance of using L∗ instead of ln N ), if
q is greater, and if a is less. This is why the third set of numerical results in Table 1 displays
better results than the previous two, while still using theoretically safe parameter values. The
even better errors in the fourth set of results are beyond the scope of the presented theory
since a = 2.5. The last set of results shows that the accuracy decreases if a is diminished
further. In all cases, but the last one, the values of ÕN are small in the beginning, but as N
increases, they get closer to the theoretical order of convergence, reaching between 5.50 and
5.67 for N = 1024. Greater (but impractical) values of N would be needed to improve this,
but a decrease in ÕN should be expected when N becomes extremely large so that (11) no
longer holds true.
Table 1. A comparison of mesh parameters for T6,4 and ε = 2−j , j = 10, 15, 20, 25, 30
Parameters
a = 6,
q = 1/4,
L = ln N
a = 6,
q = 1/4,
L = L∗
a = 6,
q = 3/8,
L = L∗
a = 2.5,
q = 3/8,
L = L∗
a = 2,
q = 3/8,
L = L∗
N
EN
ÕN
ON
EN
ÕN
ON
EN
ÕN
ON
EN
ÕN
ON
EN
ÕN
ON
64
1.92–2
2.63
2.05
8.65–3
3.30
2.56
2.20–3
4.04
3.14
4.22–5
4.97
3.87
2.52–5
3.67
2.86
128
4.64–3
3.78
3.05
1.46–3
4.23
3.41
2.50–4
4.71
3.80
2.89–6
5.28
4.26
3.48–6
3.78
3.05
256
5.61–4
4.64
3.85
1.37–4
4.86
4.03
1.79–5
5.14
4.27
1.51–7
5.49
4.55
4.20–7
3.84
3.19
512
3.89–5
5.18
4.39
8.39–6
5.25
4.45
9.30–7
5.42
4.60
6.41–9
5.58
4.73
4.60–8
3.89
3.30
1024
1.85–6
5.50
4.75
3.83–7
5.50
4.74
3.85–8
5.58
4.81
2.41–10
5.67
4.89
4.68–9
3.92
3.38
2048
6.91–8
—
—
1.43–8
—
—
1.37–9
—
—
8.11–12
—
—
4.48–10
—
—
In Table 2, we see some comparisons of the present method to the previously available
ones. The value of q is increased to 15/32 for this purpose. The very good second set of
numerical results, again for a = 2.5, is unsupported by the theory. Already for a = 6, the
errors of T6,4 are smaller than the T4 errors on the same S(L) mesh.
As for T4 on B meshes, this method gives O(N −4 ) ε-uniform accuracy and the results
are much better than on S meshes (see [18]). The present method is not available on B
meshes, so it is important to see whether T6,4 used on S(L) can surpass T4 on B meshes.
The same type of B meshes is used here as in [18]: the mesh points in [0, 1/2] are defined as
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382 R. Vulanović
xi = λ(i/N ), where
(
λ(t) =
ϕ(t) := εt/(q∗ − t)
if 0 6 t 6 α,
0
ϕ (α)(t − α) + ϕ(α) if α 6 t 6 1/2.
In this mesh generating function, q∗ is a parameter in (0, 1/2), which controls mesh density,
and α is the first coordinate of the point where the tangent line from (1/2, 1/2) touches ϕ.
Let us denote this kind of B mesh as B(q∗ ).
Table 2 shows that T4 on B(.4) gives better results than T6,4 on S(L∗ ) when theoretically
justified mesh parameters are used. However, it should be pointed out that this B mesh has
much greater density in the layers than S(L∗ ), thus the two meshes are not quite comparable.
Although the density of B(.2) is still greater than that of S(L∗ ), the scheme T6,4 for a = 6
surpasses T4 on B(.2). Moreover, T6,4 with the theoretically unsafe a = 2.5 is overall the
best scheme in Table 2.
Table 2. A comparison of methods for ε = 2−j , j = 10, 15, 20, 25, 30
Method
T6,4 ,
a = 6, L = L∗ ,
q = 15/32
a = 2.5,
q = 15/32,
L = L∗
T4
a = 6, L = L∗ ,
q = 15/32
T4 , on
B(.4)
T4 , on
B(.2)
N
EN
ÕN
ON
EN
ÕN
ON
EN
ÕN
ON
EN
ON
EN
ON
64
8.97–4
4.35
3.39
1.34–5
5.11
3.97
1.05–3
3.76
2.93
9.53–5
3.96
1.47–3
3.95
128
8.58–5
4.90
3.96
8.50–7
5.36
4.33
1.38–4
3.75
3.03
6.13–6
3.99
9.53–5
3.96
256
5.52–6
5.26
4.37
4.22–8
5.53
4.59
1.68–5
3.82
3.17
3.85–7
4.00
6.13–6
3.99
512
2.68–7
5.49
4.65
1.76–9
5.60
4.75
1.87–6
3.84
3.26
2.40–8
4.00
3.85–7
4.00
1024
1.06–8 3.74–10
5.60
—
4.83
—
6.54–11 2.17–12
5.69
—
4.91
—
1.96–7 1.95–8
3.86
—
3.33
—
1.50–9 9.38–11
4.00
—
2.40–8 1.50–9
4.00
—
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Received 8 Sep. 2003
Revised 27 Apr. 2004
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