ASYMPTOTIC BEHAVIOR FOR DISCRETIZATIONS
OF A SEMILINEAR PARABOLIC EQUATION WITH
A NONLINEAR BOUNDARY CONDITION
NABONGO DIABATE
THÉODORE K. BONI
Université d’Abobo-Adjamé
UFR-SFA, Dépt. de Math. et Informatiques
16 BP 372 Abidjan 16, (Côte d’Ivoire).
EMail: nabongo_diabate@yahoo.fr
Inst. Nat. Polytech. Houphouët-Boigny de Yamoussoukro
BP 1093 Yamoussoukro,
(Côte d’Ivoire).
Received:
16 October, 2007
Accepted:
17 March, 2008
Communicated by:
C. Bandle
2000 AMS Sub. Class.:
35B40, 35B50, 35K60, 65M06.
Key words:
Semidiscretizations, Semilinear parabolic equation, Asymptotic behavior, Convergence.
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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Abstract:
This paper concerns the study of the numerical approximation for the following initialboundary value problem:

p−1
u, 0 < x < 1, t > 0,
 ut = uxx − a|u|
ux (0, t) = 0 ux (1, t) + b|u(1, t)|q−1 u(1, t) = 0, t > 0,
(P)

u(x, 0) = u0 (x) > 0, 0 ≤ x ≤ 1,
where a > 0, b > 0 and q > p > 1. We show that the solution of a semidiscrete form
of (P ) goes to zero as t goes to infinity and give its asymptotic behavior. Using some
nonstandard schemes, we also prove some estimates of solutions for discrete forms of
(P ). Finally, we give some numerical experiments to illustrate our analysis.
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Contents
1
Introduction
3
2
Semidiscretizations Scheme
5
3
Asymptotic Behavior
8
4
Convergence
14
5
Full Discretizations
17
6
Numerical Results
24
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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1.
Introduction
Consider the following initial-boundary value problem:
(1.1)
(1.2)
ut = uxx − a|u|p−1 u,
0 < x < 1, t > 0,
q−1
ux (0, t) = 0 ux (1, t) + b|u(1, t)|
u(1, t) = 0,
t > 0,
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
(1.3)
u(x, 0) = u0 (x) > 0,
0 ≤ x ≤ 1,
where a > 0, b > 0, q > p > 1, u0 ∈ C 1 ([0, 1]), u00 (0) = 0 and u00 (1) +
b|u0 (1)|q−1 u0 (1) = 0.
The theoretical study of the asymptotic behavior of solutions for semilinear parabolic
equations has been the subject of investigation for many authors (see [2], [4] and the
references cited therein). In particular, in [4], when b = 0, the authors have shown
that the solution u of (1.1) – (1.3) goes to zero as t tends to infinity and satisfies the
following :
(1.4)
0 ≤ ku(x, t)k∞ ≤
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1
1
(ku0 (x)k∞ + a(p − 1)t) p−1
for
t ∈ [0, +∞),
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(1.5)
lim t
t→∞
1
p−1
ku(x, t)k∞ = C0 ,
1
p−1
1
where C0 = a(p−1)
. The same results have been obtained in [2] in the case
where b > 0 and q > p > 1.
In this paper we are interested in the numerical study of (1.1) – (1.3). At first, using a semidiscrete form of (1.1) – (1.3), we prove similar results for the semidiscrete
solution. We also construct two nonstandard schemes and show that these schemes
allow the discrete solutions to obey an estimation as in (1.4). Previously, authors
have used numerical methods to study the phenomenon of blow-up and the one of
extinction (see [1] and [3]). This paper is organized as follows. In the next section,
we prove some results about the discrete maximum principle. In the third section,
we take a semidiscrete form of (1.1) – (1.3), and show that the semidiscrete solution
goes to zero as t tends to infinity and give its asymptotic behavior. In the fourth
section, we show that the semidiscrete scheme of the third section converges. In
Section 5, we construct two nonstandard schemes and obtain some estimates as in
(1.4). Finally, in the last section, we give some numerical results.
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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2.
Semidiscretizations Scheme
In this section, we give some lemmas which will be used later. Let I be a positive integer, and define the grid xi = ih, 0 ≤ i ≤ I, where h = 1/I. We
approximate the solution u of the problem (1.1) – (1.3) by the solution Uh (t) =
(U0 (t), U1 (t), . . . , UI (t))T of the semidiscrete equations
(2.1)
d
Ui (t) = δ 2 Ui (t) − a|Ui (t)|p−1 Ui (t),
dt
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
0 ≤ i ≤ I − 1, t > 0,
Théodore K. Boni
(2.2)
d
2b
UI (t) = δ 2 UI (t) − a|UI (t)|p−1 UI (t) − |UI (t)|q−1 UI (t),
dt
h
vol. 9, iss. 2, art. 33, 2008
t > 0,
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Ui (0) =
(2.3)
Ui0
> 0,
0 ≤ i ≤ I,
Contents
where
Ui+1 (t) − 2Ui (t) + Ui−1 (t)
, 1 ≤ i ≤ I − 1,
h2
2U1 (t) − 2U0 (t)
2UI−1 (t) − 2UI (t)
δ 2 U0 (t) =
, δ 2 UI (t) =
.
2
h
h2
The following lemma is a semidiscrete form of the maximum principle.
δ 2 Ui (t) =
0
I+1
1
Lemma 2.1. Let ah (t) ∈ C ([0, T ], R ) and let Vh (t) ∈ C ([0, T ], R
that
d
(2.4)
Vi (t) − δ 2 Vi (t) + ai (t)Vi (t) ≥ 0, 0 ≤ i ≤ I, t ∈ (0, T ),
dt
(2.5)
Vi (0) ≥ 0,
0 ≤ i ≤ I.
Then we have Vi (t) ≥ 0 for 0 ≤ i ≤ I, t ∈ (0, T ).
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I+1
) such
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Proof. Let T0 < T and let m = min0≤i≤I,0≤t≤T0 Vi (t). Since for i ∈ {0, . . . , I},
Vi (t) is a continuous function, there exists t0 ∈ [0, T0 ] such that m = Vi0 (t0 ) for a
certain i0 ∈ {0, . . . , I}. It is not hard to see that
dVi0 (t0 )
Vi (t0 ) − Vi0 (t0 − k)
= lim 0
≤ 0,
k→0
dt
k
(2.6)
(2.7)
(2.8)
δ 2 Vi0 (t0 ) =
δ 2 Vi0 (t0 ) =
V1 (t0 ) − V0 (t0 )
≥ 0 if
h2
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
i0 = 0,
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
Vi0 +1 (t0 ) − 2Vi0 (t0 ) + Vi0 −1 (t0 )
≥ 0 if
h2
1 ≤ i0 ≤ I − 1,
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(2.9)
δ 2 Vi0 (t0 ) =
VI−1 (t0 ) − VI (t0 )
≥ 0 if
h2
i0 = I.
λt
Define the vector Zh (t) = e Vh (t) where λ is large enough such that ai0 (t0 )−λ > 0.
A straightforward computation reveals:
(2.10)
dZi0 (t0 )
− δ 2 Zi0 (t0 ) + (ai0 (t0 ) − λ)Zi0 (t0 ) ≥ 0.
dt
dZi0 (t0 )
dt
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2
We observe from (2.6) – (2.9) that
≤ 0 and δ Zi0 (t0 ) ≥ 0. Using (2.10),
we arrive at (ai0 (t) − λ)Zi0 (t0 ) ≥ 0, which implies that Zi0 (t0 ) ≥ 0. Therefore,
Vi0 (t0 ) = m ≥ 0 and we have the desired result.
Another form of the maximum principle is the following comparison lemma.
Lemma 2.2. Let Vh (t), Uh (t) ∈ C 1 ([0, ∞), RI+1 ) and f ∈ C 0 (R × R, R) such that
for t ∈ (0, ∞),
(2.11)
dVi (t) 2
dUi (t) 2
−δ Vi (t)+f (Vi (t), t) <
−δ Ui (t)+f (Ui (t), t),
dt
dt
Vi (0) < Ui (0),
(2.12)
0 ≤ i ≤ I,
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
0 ≤ i ≤ I.
Théodore K. Boni
Then we have Vi (t) < Ui (t), 0 ≤ i ≤ I, t ∈ (0, ∞).
vol. 9, iss. 2, art. 33, 2008
Proof. Define the vector Zh (t) = Uh (t) − Vh (t). Let t0 be the first t > 0 such that
Zi (t) > 0 for t ∈ [0, t0 ), i = 0, . . . , I, but Zi0 (t0 ) = 0 for a certain i0 ∈ {0, . . . , I}.
We observe that
dZi0 (t0 )
Zi (t0 ) − Zi0 (t0 − k)
= lim 0
k→0
dt
k
 Z (t )−2Z (t )+Z (t )
i0 +1 0
i0 0
i0 −1 0

≥ 0 if

h2



2Z1 (t0 )−2Z0 (t0 )
δ 2 Zi0 (t0 ) =
≥0
if
h2




 2ZI−1 (t0 )−2ZI (t0 )
≥0
if
h2
Contents
≤ 0.
1 ≤ i0 ≤ I − 1,
i0 = 0,
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i0 = I,
which implies:
dZi0 (t0 )
− δ 2 Zi0 (t0 ) + f (Ui0 (t0 ), t0 ) − f (Vi0 (t0 ), t0 ) ≤ 0.
dt
But this inequality contradicts (2.11).
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3.
Asymptotic Behavior
In this section, we show that the solution Uh of (2.1) – (2.3) goes to zero as t → +∞
and give its asymptotic behavior. Firstly, we prove that the solution tends to zero as
t → +∞ by the following:
Theorem 3.1. The solution Uh (t) of (2.1) – (2.3) goes to zero as t → ∞ and we
have the following estimate
0 ≤ kUh (t)k∞ ≤
1
(kUh (0)k1−p
∞ + a(p − 1)t)
1
p−1
for
t ∈ [0, +∞).
Proof. We introduce the function α(t) which is defined as
α(t) =
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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1
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1
(kUh (0)k1−p
∞ + a(p − 1)t) p−1
and let Wh be the vector such that Wi (t) = α(t). It is not hard to see that
dWi (t)
− δ 2 Wi (t) + a|Wi (t)|p−1 Wi (t) = 0,
dt
0 ≤ i ≤ I − 1, t ∈ (0, T ),
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2b
dWI (t)
− δ 2 WI (t) + a|WI (t)|p−1 WI (t) + |WI (t)|q−1 WI (t) ≥ 0,
dt
h
Wi (0) ≥ Ui (0), 0 ≤ i ≤ I,
t ∈ (0, T ),
where (0, T ) is the maximal time interval on which kUh (t)k∞ < ∞. Setting Zh (t) =
Wh (t) − Uh (t) and using the mean value theorem, we see that
dZi (t)
− δ 2 Zi (t) + ap|θi (t)|p−1 Zi (t) = 0,
dt
0 ≤ i ≤ I − 1, t ∈ (0, T )
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dZI (t)
2b
2
p−1
q−1
− δ ZI (t) + ap|θI (t)|
+ |θI (t)|
ZI (t) ≥ 0,
dt
h
Zi (0) ≥ 0,
t ∈ (0, T ),
0 ≤ i ≤ I,
where θi is an intermediate value between Ui (t) and Wi (t). From Lemma 2.1, we
have 0 ≤ Ui (t) ≤ Wi (t) for t ∈ (0, T ). If T < ∞, we have
kUh (T )k∞ ≤
1
1
(kUh (0)k1−p
∞ + a(p − 1)T ) p−1
< ∞,
which leads to a contradiction. Hence T = ∞ and we have the desired result.
Remark 1. The estimate of Theorem 3.1 is a semidiscrete version of the result established in (1.4) for the continuous problem.
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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Let us give the statement of the main theorem of this section.
Theorem 3.2. Let Uh be the solution of (2.1) – (2.2). Then we have
1
lim t p−1 kUh (t)k∞ = C0 ,
t→∞
where C0 =
1
a(p−1)
1
p−1
.
The proof of Theorem 3.2 is based on the following lemmas. We introduce the
function
µ(x) = −λ(C0 + x) + (C0 + x)p ,
1
p−1
1
where C0 = a(p−1)
.
Firstly, we establish an upper bound of the solution for the semidiscrete problem.
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Lemma 3.3. Let Uh be the solution of (2.1) – (2.3). For any ε > 0, there exist
positive times T and τ such that
Ui (t + τ ) ≤ (C0 + ε)(t + T )−λ + (t + T )−λ−1 ,
0 ≤ i ≤ I.
Proof. Define the vector Wh such that
Wi (t) = (C0 + ε)t−λ + t−λ−1 .
A straightforward computation reveals that
dWi
− δ 2 Wi + a|Wi |p−1 Wi
dt
= −λ(C0 + ε)t−λ−1 − (λ + 1)t−λ−2 + a((C0 + ε)t−λ + t−λ−1 )p
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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= t−λ−1 (−λ(C0 + ε) − (λ + 1)t−1 + a(C0 + ε + t−1 )p ),
because λp = λ + 1. Using the mean value theorem, we get
(C0 + ε + t−1 )p = (C0 + ε)p + ξi t−1 ,
where ξi (t) is a bounded function. We deduce that
dWi
− δ 2 Wi + a|Wi |p−1 Wi = t−λ−1 (µ(ε) − (λ + 1)t−1 + ξi t−1 ),
dt
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dWI
2b
− δ 2 WI + a|WI |p−1 WI + |WI |q−1 WI
dt
h
2b −qλ+λ+1
−λ−1
−1
−1
−1 q
=t
µ(ε) − (λ + 1)t + ξi t + t
(C0 + ε + t ) .
h
p−q
Obviously −qλ + λ + 1 = p−1
< 0. We also observe that µ(0) = 0 and µ0 (0) = 1,
which implies that µ(ε) > 0. Therefore there exists a positive time T such that
dWi
− δ 2 Wi + a|Wi |p−1 Wi > 0,
dt
0 ≤ i ≤ I − 1, t ∈ [T, +∞),
dWI
2b
− δ 2 WI + a|WI |p−1 WI + |WI (t)|q−1 WI (t) > 0, t ∈ [T, +∞),
dt
h
T −λ C0
.
Wi (T ) >
2
Since from Theorem 3.1 limt→∞ Ui (t) = 0, there exists τ > T such that Ui (τ ) <
T −λ C0
< Wi (T ). We introduce the vector Zh (t) such that Zi (t) = Ui (t + τ − T ),
2
0 ≤ i ≤ I. We obtain
dZi
− δ 2 Zi + a|Zi |p−1 Zi > 0,
dt
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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0 ≤ i ≤ I − 1, t ≥ T,
dZI
2b
− δ 2 ZI + a|ZI |p−1 ZI + |ZI (t)|q−1 ZI (t) > 0,
dt
h
Zi (T ) = Ui (τ ) < Wi (T ).
t ≥ T,
We deduce from Lemma 2.2 that Zi (t) ≤ Wi (t), that is to say
(3.1)
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Ui (t + τ − T ) ≤ Wi (t) for
t ≥ T,
which leads us to the result.
The lemma below gives a lower bound of the solution for the semidiscrete problem.
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Lemma 3.4. Let Uh be the solution of (2.1) – (2.3). For any ε > 0, there exists a
positive time τ such that
Ui (t + 1) ≥ (C0 − ε)(t + τ )−λ + (t + τ )−λ−1 ,
0 ≤ i ≤ I.
Proof. Introduce the vector Vh such that
Vi (t) = (C0 − ε)t−λ + t−λ−1 .
A direct calculation yields
dVi
− δ 2 Vi + a|Vi |p−1 Vi = −λ(C0 − ε)t−λ−1 − (λ + 1)t−λ−2 + a((C0 − ε)t−λ + t−λ−1 )p
dt
= t−λ−1 (−λ(C0 − ε) − (λ + 1)t−1 + a(C0 − ε + t−1 )p )
because λp = λ + 1. From the mean value theorem, we have
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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(C0 − ε + t−1 )p = (C0 − ε)p + χi (t)t−1 ,
where χi (t) is a bounded function. We deduce that
dVi
− δ 2 Vi + a|Vi |p−1 Vi = t−λ−1 (µ(−ε) − (λ + 1)t−1 + χi t−1 ),
dt
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2b −qλ+λ+1
−1 q
−1
−1
µ(ε) − (λ + 1)t + χi t + t
(C0 − ε + t ) .
h
Obviously −qλ + λ + 1 < 0. Also, since µ(0) = 0 and µ0 (0) = 1, it is easy to see
that µ(−ε) < 0. Hence there exists T > 0 such that
dVi
− δ 2 Vi + a|Vi |p−1 Vi < 0,
dt
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dVI
2b
− δ 2 VI + a|VI |p−1 VI + |VI |q−1 VI
dt
h
= t−λ−1
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0 ≤ i ≤ I − 1, t ∈ [T, +∞),
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dVI
2b
− δ 2 VI + a|VI |p−1 VI + |VI |q−1 VI < 0, t ∈ [T, +∞).
dt
h
Since Vi (t) goes to zero as t → +∞, there exists τ > max(T, 1) such that Vi (τ ) <
Ui (1). Setting Xi (t) = Vi (t + τ − 1), we observe that
dXi
− δ 2 Xi + a|Xi |p−1 Xi < 0,
dt
0 ≤ i ≤ I − 1, t ≥ 1,
2b
dXI
− δ 2 XI + a|XI |p−1 XI + |XI |q−1 XI < 0,
dt
h
Xi (1) = Vi (τ ) < Ui (1).
t ≥ 1,
We deduce from Lemma 2.2 that
(3.2)
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
Title Page
Ui (t) ≥ Vi (t + τ − 1) for t ≥ 1,
which leads us to the result.
Now, we are in a position to give the proof of the main result of this section.
Proof of Theorem 3.2. From Lemma 3.3 and Lemma 3.4, we deduce
Ui (t)
Ui (t)
(C0 − ε) ≤ lim inf
≤ lim sup
≤ (C0 + ε),
t→∞
t→∞
tλ
tλ
and we have the desired result.
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4.
Convergence
In this section, we will show that for each fixed time interval [0, T ], where u is
defined, the solution Uh (t) of (2.1) – (2.3) approximates u, when the mesh parameter
h goes to zero.
Theorem 4.1. Assume that (1.1) – (1.3) has a solution u ∈ C 4,1 ([0, 1] × [0, T ]) and
the initial condition at (2.3) satisfies
kUh0 − uh (0)k∞ = o(1)
(4.1)
as
h → 0,
Théodore K. Boni
where uh (t) = (u(x0 , t), . . . , u(xI , t))T . Then, for h sufficiently small, the problem
(2.1) – (2.3) has a unique solution Uh ∈ C 1 ([0, T ], RI+1 ) such that
(4.2)
max kUh (t) − uh (t)k∞ = O(kUh0 − uh (0)k∞ + h2 ) as
0≤t≤T
h → 0.
vol. 9, iss. 2, art. 33, 2008
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Proof. Let K > 0 and L be such that
2kuxxx k∞
K
≤ ,
3
2
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
kuxxxx k∞
K
≤ ,
12
2
kuk∞ ≤ K,
ap(K + 1)p−1 ≤ L,
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(4.3)
2q(K + 1)q−1 ≤ L.
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The problem (2.1) – (2.3) has for each h, a unique solution Uh ∈ C 1 ([0, Tqh ), RI+1 ).
Let t(h) the greatest value of t > 0 such that
(4.4)
kUh (t) − uh (t)k∞ < 1f ort ∈ (0, t(h)).
The relation (4.1) implies that t(h) > 0 for h sufficiently small. Let t∗ (h) =
min{t(h), T }. By the triangular inequality, we obtain
kUh (t)k∞ ≤ ku(x, t)k∞ + kUh (t) − uh (t)k∞
f or t ∈ (0, t∗ (h)),
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which implies that
(4.5)
kUh (t)k∞ ≤ 1 + K,
f or t ∈ (0, t∗ (h)).
Let eh (t) = Uh (t)−uh (x, t) be the error of discretization. Using Taylor’s expansion,
we have for t ∈ (0, t∗ (h)),
h2
d
ei (t) − δ 2 ei (t) = uxxxx (e
xi , t) − apξip−1 ei (t),
dt
12
d
2
2h2
h2
eI (t) − δ 2 eI (t) = qθIq−1 eI +
uxxx (e
xI , t) + uxxxx (e
xI , t) − apξIp−1 eI (t),
dt
h
3
12
where θI ∈ (UI (t), u(xI , t) and ξi ∈ (Ui (t), u(xi , t). Using (4.3) and (4.5), we arrive
at
(4.6)
(4.7)
d
ei (t) − δ 2 ei (t) ≤ L|ei (t)| + Kh2 , 0 ≤ i ≤ I − 1,
dt
L|eI (t)|
deI (t) (2eI−1 (t) − 2eI (t))
−
≤
+ L|eI (t)| + Kh2 .
2
dt
h
h
Consider the function
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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2
z(x, t) = e((M +1)t+Cx ) (kUh0 − uh (0)k∞ + Qh2 )
where M , C, Q are constants which will be determined later. We get
zt (x, t) − zxx (x, t) = (M + 1 − 2C − 4C 2 x2 )z(x, t),
zx (0, t) = 0, zx (1, t) = 2Cz(1, t),
2 z(x, 0) = eCx (Uh0 − uh (0)∞ + Qh).
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By a semidiscretization of the above problem, we may choose M, C, Q large enough
that
(4.8)
d
z(xi , t) > δ 2 z(xi , t) + L|z(xi , t)| + Kh2 , 0 ≤ i ≤ I − 1,
dt
(4.9)
L
d
z(xI , t) > δ 2 z(xI , t) + |z(xI , t)| + L|z(xI , t)| + Kh2 ,
dt
h
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
z(xi , 0) > ei (0), 0 ≤ i ≤ I.
(4.10)
It follows from Lemma 3.4 that
z(xi , t) > ei (t) f or t ∈ (0, t∗ (h)),
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0 ≤ i ≤ I.
By the same way, we also prove that
z(xi , t) > −ei (t) f or t ∈ (0, t∗ (h)),
0 ≤ i ≤ I,
which implies that
kUh (t) − uh (t)k∞ ≤ e(M t+C) (Uh0 − uh (0)∞ + Qh2 ), t ∈ (0, t∗ (h)).
Let us show that t∗ (h) = T . Suppose that T > t(h). From (4.4), we obtain
(4.11)
1 = kUh (t(h)) − uh (t(h))k∞ ≤ e(M T +C) (Uh0 − uh (0)∞ + Qh2 ).
Since the term in the right hand side of the inequality goes to zero as h goes to zero,
we deduce from (4.11) that 1 ≤ 0, which is impossible. Consequently t∗ (h) = T ,
and we obtain the desired result.
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5.
Full Discretizations
In this section, we study the asymptotic behavior, using full discrete schemes (explicit and implicit) of (1.1) – (1.3). Firstly, we approximate the solution u(x, t) of
(n)
(1.1) – (1.3) by the solution Uh = (U0n , U1n , . . . , UIn )T of the following explicit
scheme
(n+1)
(n)
Ui
− Ui
(n) p−1 (n+1)
(n)
(5.1)
= δ 2 Ui − a Ui Ui
, 0 ≤ i ≤ I − 1,
∆t
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
(n+1)
(5.2)
UI
(n)
− UI
∆t
=δ
2
(n)
UI
(n) p−1 (n+1) 2b (n) q−1 (n+1)
− a UI UI
− UI UI
,
h
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(0)
Ui
(5.3)
= φi > 0,
0 ≤ i ≤ I,
2
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where n ≥ 0, ∆t ≤ h2 . We need the following lemma which is a discrete form of
the maximum principle for ordinary differential equations.
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Lemma 5.1. Let f ∈ C 1 (R) and let an and bn be two bounded sequences such that
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(5.4)
(5.5)
an+1 − an
bn+1 − bn
+ f (an ) ≥
+ f (bn ),
∆t
∆t
a0 ≥ b 0 .
Then we have an ≥ bn , n ≥ 0 for h small enough.
n ≥ 0,
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Proof. Let Zn = an − bn . We get
Zn+1 − Zn
+ f 0 (ξn )Zn ≥ 0,
∆t
(5.6)
where ξn is an intermediate value between an and bn . Obviously
Zn+1 ≥ Zn (1 − ∆tf 0 (ξn )).
(5.7)
Since an and bn are bounded and f ∈ C 1 (R), there exists a positive M such that
|f 0 (ξn )| ≤ M . Let j be the first integer such that Zj < 0. From (5.5), j ≥ 0. We
have Zj ≥ Zj−1 (1 − ∆tM ). Since ∆tM goes to zero as h → 0 and Zj−1 ≥ 0, we
deduce that Zj ≥ 0 as h → 0 which is a contradiction. Therefore, Zn ≥ 0 for any n
and we have proved the lemma.
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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Now, we may state the following.
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(n)
Theorem 5.2. Let Uh be the solution of (5.1) – (5.3). We have Uh ≥ 0 and
1
(n) Uh ≤ 1 ,
p−1
∞
(0) 1−p
+ A(p − 1)n∆t
Uh ∞
where A =
a p−1 .
(0) 1+a∆tUh II
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∞
Proof. A straightforward calculation yields
(n)
(n)
∆t (n)
2∆t
U
+
1
−
Ui + Ui−1
2
2
(n+1)
i+1
h
h
,
(5.8)
Ui
=
(n) p−1
1 + a∆t Ui JJ
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1 ≤ i ≤ I − 1,
(n+1)
U0
(5.9)
(n+1)
UI
(5.10)
=
2∆t (n)
U1
h2
(n)
+ 1 − 2∆t
U0
h2
,
p−1
(n) 1 + a∆t U0 2∆t (n)
UI−1
h2
(n)
+ 1 − 2∆t
UI
2
h
=
p−1
.
(n) q−1
(n) 1 + a∆t UI + 2 hb ∆t UI Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
Since 1 −
(n)
Uh
2 ∆t
h2
is nonnegative, using a recursive
argument, it is easy to see that
(n) (n)
≥ 0. Let i0 be such that Ui0 = Uh . From (5.8), we get
∞
(n+1) Uh
∞
≤
∆t (n)
U
h2 i0 +1
(n) (n)
+ 1−
Uh + Ui0 −1
∞
(n) p−1
1 + a∆t Uh 2∆t
h2
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if
1 ≤ i0 ≤ I − 1.
∞
Applying the triangle inequality and the fact that 1 − 2∆t
is nonnegative, we arrive
h2
at
(n) Uh (n+1) (5.11)
Uh
≤
∞ p−1 .
(n) ∞
1 + a∆t Uh ∞
We obtain
if i0 = 0 or i0 = I. Theinequality
(5.11)
that
the same
estimation
implies
(n+1) (n) (n) (0) Uh
≤ Uh and by iterating, we obtain Uh ≤ Uh . From
∞
∞
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∞
∞
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(5.11), we also observe that
(n+1) Uh
− kU (n) k∞
∞
∆t
(n) p
a Uh ≤−
∞p−1 .
(n) 1 + a∆t Uh ∞
(n) (0) Using the fact that Uh ≤ Uh , we have
∞
∞
(n+1) (n) Uh
− Uh (n) p
∞
∞
≤ −A Uh .
∆t
∞
We introduce the function α(t) which is defined as follows
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
Title Page
1
α(t) = 1 .
p−1
(0) 1−p
+ A(p − 1)t
Uh ∞
We remark that α(t) obeys the following differential equation
(0) α0 (t) = −Aαp (t), α(0) = Uh .
∞
Using a Taylor’s expansion, we have
(∆t)2 00 ˜
α (tn ),
α(tn+1 ) = α(tn ) + ∆tα0 (tn ) +
2
where t˜n is an intermediate value between tn and tn+1 . It is not hard to see that α(t)
is a convex function. Therefore, we obtain
α(tn+1 ) − α(tn )
≥ −Aαp (tn ).
∆t
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
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(n) From Lemma 5.1, we get Uh ∞
(n) Uh ∞
≤ α(tn ), which ensures that
1
≤
1 ,
p−1
1−p
(0) + A(p − 1)n∆t
Uh ∞
and we have the desired result.
Remark 2. The estimate of Theorem 5.2 is the discrete form of the one given in (1.4)
for the continuous problem.
Now, we approximate the solution u(x, t) of problem (1.1) – (1.3) by the solution
(n)
Uh of the following implicit scheme
(n+1)
(n+1)
(5.13)
UI
(n)
− Ui
∆t
Ui
(5.12)
(n+1)
= δ 2 Ui
(n)
− UI
∆t
=
(n+1)
δ 2 UI
(n) p−1 (n+1)
− Ui Ui
,
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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Contents
0 ≤ i ≤ I − 1,
(n) p−1 (n+1) 2b (n) p−1 (n+1)
− a UI UI
− UI UI
,
h
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(5.14)
(0)
Ui
= φi > 0,
0 ≤ i ≤ I,
where n ≥ 0. Let us note that in the above construction, we do not need a restriction
on the step time.
The above equations may be rewritten in the following form:
2∆t (n+1)
∆t
(n) p−1
(n)
(n+1)
U0 = − 2 U1
+ 1 + 2 2 + a∆t U0 U0
,
h
h
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∆t
∆t (n+1)
(n) p−1
(n+1) ∆t (n+1)
= − 2 Ui−1 + 1 + 2 2 + a∆t Ui Ui
− 2 Ui+1 , 1 ≤ i ≤ I−1,
h
h
h
2∆t (n+1)
∆t
(n) p−1 2b (n) q−1
(n)
(n+1)
UI = − 2 UI−1 + 1 + 2 2 + a∆t UI + ∆t UI UI
,
h
h
h
(n)
Ui
which gives the following linear system
(n+1)
A(n) Uh
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
(n)
= Uh
where A(n) is the tridiagonal matrix defined as follows

d0 −2∆t
0
0
···
0
h2
−∆t
−∆t
 h2
d1
0
···
0
h2

−∆t
−∆t
 0
d
0
·
··
2
h2
h2
 .
.
..
(n)
..
..
..
..
A =
.
.
.
.
 ..
 0
−∆t
−∆t
0
···
dI−2 h2

h2
 0
0
0
· · · −∆t
dI−1
h2
−2∆t
0
0
0
···
0
h2
with
di = 1 + 2
∆t
(n)
+ a∆t|Ui |p−1
h2
for
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
0
0
0
..
.





,

0 

−∆t 
h2
dI
0≤i≤I −1
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and
∆t
(n) p−1 2b (n) q−1
+
a∆t
U
∆t
U
+
I I .
h2
h
Let us remark that the tridiagonal matrix A(n) satisfies the following properties
dI = 1 + 2
(n)
(n)
Aii > 0 and Aij < 0 i 6= j,
Close
(n) X (n) Aij .
Aii >
i6=j
(n)
Uhn
These properties imply that
exists for any n and Uh ≥ 0 (see for instance [2]).
As we know that the solution of the discrete implicit scheme exists, we may state the
following.
(n)
(n)
Theorem 5.3. Let Uh be the solution of (5.12) – (5.14). We have Uh ≥ 0 and
1
(n) Uh ≤ 1 ,
p−1
1−p
∞
(0) + A(p − 1)n∆t
Uh Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
∞
where A =
a p−1 .
(0) 1+a∆tUh Title Page
∞
(n)
≥ 0 as we have seen above. Now, let us obtain
the above
(n) (n)
estimate to complete the proof. Let i0 be such that Ui0 = Uh . Using the
∞
equality (5.12), we have
∆t
∆t (n)
∆t (n)
(n) (n+1) (n) 1 + 2 2 + a∆t Uh Uh
≤ Uh + 2 Ui0 −1 + 2 Ui0 +1
h
h
h
∞
∞
∞
if 1 ≤ i0 ≤ I − 1.
Proof. We know that Uh
Applying the triangle inequality, we derive the following estimate
(n) Uh (n+1) Uh
≤
∞ p−1 .
(n) ∞
1 + a∆t Uh ∞
We obtain the same estimation if we take i0 = 0 or i0 = I. Reasoning as in the proof
of Theorem 5.3, we obtain the desired result.
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6.
Numerical Results
In this section, we consider the explicit scheme in (5.1) – (5.3) and the implicit
scheme in (5.12) – (5.14). We suppose that p = 2, q = 3, a = 1, b = 1, Ui0 =
2
0.8 + 0.8 ∗ cos(πhi) and ∆t = h2 . In the following tables, in the rows, we give the
first n when
(n)
n∆tUh − 1 < ε,
∞
the corresponding time T n = n∆t, the CPU time and the order(s) of method computed from
log((T4h − T2h )/(T2h − Th ))
s=
.
log(2)
Table 1: (ε = 10−2 ) Numerical times, numbers of iterations, CPU times (seconds),
and orders of the approximations obtained with the implicit Euler method
I
16
32
64
128
256
Tn
674.0820
674.2632
674.3085
674.3278
674.4807
n
345129
1.380890.
5.523.934
22095735
87383041
CPU time
103
660
6020
58290
574823
s
2.01
1.24
2.99
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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Table 2: (ε = 10−2 ) Numerical times, numbers of iterations, CPU times (seconds)
and orders of the approximations obtained with the explicit Euler method
I
16
32
64
128
256
Tn
674.3281
674.3452
674.3290
674.3187
674.3098
n
345.255
1.381.058
5.524.102
22845950
88237375
CPU time
90
720
10820
323528
19457811
s
0.08
0.65
0.21
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
vol. 9, iss. 2, art. 33, 2008
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References
[1] L. ABIA, J.C. LÓPEZ-MARCOS AND J. MARTINEZ, On the blow-up time
convergence of semidiscretizations of reaction-diffusion equations, Appl. Numer.
Math., 26 (1998), 399–414.
[2] T.K. BONI, On the asymptotic behavior of solutions for some semilinear
parabolic and elliptic equation of second order with nonlinear boundary conditions, Nonl. Anal. TMA, 45 (2001), 895–908.
Discretizations of a
Semilinear Parabolic Equation
Nabongo Diabate and
Théodore K. Boni
[3] T.K. BONI, Extinction for discretizations of some semilinear parabolic equations, C.R.A.S, Serie I, 333 (2001), 795–800.
vol. 9, iss. 2, art. 33, 2008
[4] V.A. KONDRATIEV AND L. VÉRON, Asymptotic behaviour of solutions of
some nonlinear parabolic or elliptic equation, Asymptotic Analysis, 14 (1997),
117–156.
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[5] R.E. MICKENS, Relation between the time and space step-sizes in nonstandard finite difference schemes for the fisher equation, Num. Methods. Part. Diff.
Equat., 13 (1997), 51–55.
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