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ASYMPTOTIC BEHAVIOR FOR DISCRETIZATIONS OF A SEMILINEAR PARABOLIC EQUATION WITH A NONLINEAR BOUNDARY CONDITION Discretizations of a Semilinear Parabolic Equation Nabongo Diabate and Théodore K. Boni NABONGO DIABATE THÉODORE K. BONI vol. 9, iss. 2, art. 33, 2008 Université d’Abobo-Adjamé Inst. Nat. Polytech. Houphouët-Boigny de Yamoussoukro UFR-SFA, Dépt. de Math. et Informatiques BP 1093 Yamoussoukro, 16 BP 372 Abidjan 16, (Côte d’Ivoire). (Côte d’Ivoire). Title Page EMail: nabongo_diabate@yahoo.fr Contents Received: 16 October, 2007 Accepted: 17 March, 2008 Communicated by: C. Bandle 2000 AMS Sub. Class.: 35B40, 35B50, 35K60, 65M06. Page 1 of 26 Key words: Semidiscretizations, Semilinear parabolic equation, Asymptotic behavior, Conver- gence. Go Back Abstract: This paper concerns the study of the numerical approximation for the following initial- boundary value problem: Full Screen p−1 ut = uxx − a|u| u, 0 < x < 1, t > 0, Close (P) ux (0, t) = 0 ux (1, t) + b|u(1, t)|q−1 u(1, t) = 0, t > 0, 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. Contents 1 Introduction 3 2 Semidiscretizations Scheme 5 3 Asymptotic Behavior 8 Discretizations of a Semilinear Parabolic Equation Nabongo Diabate and 4 Convergence 14 Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 5 Full Discretizations 17 6 Numerical Results 24 Title Page Contents Page 2 of 26 Go Back Full Screen Close 1. Introduction Consider the following initial-boundary value problem: (1.1) ut = uxx − a|u|p−1 u, 0 < x < 1, t > 0, Discretizations of a q−1 Semilinear Parabolic Equation (1.2) ux (0, t) = 0 ux (1, t) + b|u(1, t)| u(1, t) = 0, t > 0, 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]), u0 (0) = 0 and u0 (1) + Title Page b|u0 (1)|q−1 u0 (1) = 0. Contents 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 : Page 3 of 26 1 Go Back (1.4) 0 ≤ u(x, t) ∞ ≤ 1 for t ∈ [0, +∞), ( u0 (x) ∞ + a(p − 1)t) p−1 Full Screen Close 1 (1.5) lim t p−1 u(x, t) ∞ = C0 , t→∞ 1 1 p−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, us- ing 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, Discretizations of a Semilinear Parabolic Equation we take a semidiscrete form of (1.1) – (1.3), and show that the semidiscrete solution Nabongo Diabate and goes to zero as t tends to inﬁnity and give its asymptotic behavior. In the fourth Théodore K. Boni section, we show that the semidiscrete scheme of the third section converges. In vol. 9, iss. 2, art. 33, 2008 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. Title Page Contents Page 4 of 26 Go Back Full Screen Close 2. Semidiscretizations Scheme In this section, we give some lemmas which will be used later. Let I be a pos- itive 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 Discretizations of a d (2.1) Ui (t) = δ 2 Ui (t) − a|Ui (t)|p−1 Ui (t), 0 ≤ i ≤ I − 1, t > 0, Semilinear Parabolic Equation dt Nabongo Diabate and Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 d 2b (2.2) UI (t) = δ 2 UI (t) − a|UI (t)|p−1 UI (t) − |UI (t)|q−1 UI (t), t > 0, dt h Title Page (2.3) Ui (0) = Ui0 > 0, 0 ≤ i ≤ I, Contents where Ui+1 (t) − 2Ui (t) + Ui−1 (t) δ 2 Ui (t) = , 1 ≤ i ≤ I − 1, h2 2U1 (t) − 2U0 (t) 2UI−1 (t) − 2UI (t) δ 2 U0 (t) = 2 , δ 2 UI (t) = . Page 5 of 26 h h2 The following lemma is a semidiscrete form of the maximum principle. Go Back 0 I+1 1 I+1 Lemma 2.1. Let ah (t) ∈ C ([0, T ], R ) and let Vh (t) ∈ C ([0, T ], R ) such Full Screen that Close 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 ). 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) (2.6) = lim 0 ≤ 0, dt k→0 k Discretizations of a Semilinear Parabolic Equation V1 (t0 ) − V0 (t0 ) Nabongo Diabate and (2.7) δ 2 Vi0 (t0 ) = ≥ 0 if i0 = 0, Théodore K. Boni h2 vol. 9, iss. 2, art. 33, 2008 Vi0 +1 (t0 ) − 2Vi0 (t0 ) + Vi0 −1 (t0 ) (2.8) δ 2 Vi0 (t0 ) = ≥ 0 if 1 ≤ i0 ≤ I − 1, Title Page h2 Contents VI−1 (t0 ) − VI (t0 ) (2.9) δ 2 Vi0 (t0 ) = ≥ 0 if i0 = I. h2 Deﬁne the vector Zh (t) = eλt Vh (t) where λ is large enough such that ai0 (t0 )−λ > 0. Page 6 of 26 A straightforward computation reveals: Go Back dZi0 (t0 ) (2.10) − δ 2 Zi0 (t0 ) + (ai0 (t0 ) − λ)Zi0 (t0 ) ≥ 0. Full Screen dt Close dZi0 (t0 ) 2 We observe from (2.6) – (2.9) that ≤ 0 and δ Zi0 (t0 ) ≥ 0. Using (2.10), dt 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, ∞), dVi (t) 2 dUi (t) 2 (2.11) −δ Vi (t)+f (Vi (t), t) < −δ Ui (t)+f (Ui (t), t), 0 ≤ i ≤ I, dt dt Discretizations of a (2.12) Vi (0) < Ui (0), 0 ≤ i ≤ I. Semilinear Parabolic Equation Nabongo Diabate and 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}. Title Page We observe that dZi0 (t0 ) Zi (t0 ) − Zi0 (t0 − k) Contents = lim 0 ≤ 0. dt k→0 k Z (t )−2Z (t )+Z (t ) i0 −1 0 i0 +1 0 i0 0 h2 ≥ 0 if 1 ≤ i0 ≤ I − 1, Page 7 of 26 δ 2 Zi0 (t0 ) = 2Z1 (t0 )−2Z0 (t0 ) h2 ≥0 if i0 = 0, Go Back 2Z (t0 )−2Z (t0 ) ≥0 I−1 I h2 if i0 = I, Full Screen which implies: Close dZi0 (t0 ) − δ 2 Zi0 (t0 ) + f (Ui0 (t0 ), t0 ) − f (Vi0 (t0 ), t0 ) ≤ 0. dt But this inequality contradicts (2.11). 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 Discretizations of a have the following estimate Semilinear Parabolic Equation Nabongo Diabate and 1 Théodore K. Boni 0 ≤ Uh (t) ∞ ≤ 1−p 1 for t ∈ [0, +∞). vol. 9, iss. 2, art. 33, 2008 ( Uh (0) ∞ + a(p − 1)t) p−1 Proof. We introduce the function α(t) which is deﬁned as Title Page 1 α(t) = 1−p 1 Contents ( Uh (0) ∞ + 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, 0 ≤ i ≤ I − 1, t ∈ (0, T ), Page 8 of 26 dt Go Back dWI (t) 2b − δ 2 WI (t) + a|WI (t)|p−1 WI (t) + |WI (t)|q−1 WI (t) ≥ 0, t ∈ (0, T ), Full Screen dt h Wi (0) ≥ Ui (0), 0 ≤ i ≤ I, Close where (0, T ) is the maximal time interval on which Uh (t) ∞ < ∞. 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, 0 ≤ i ≤ I − 1, t ∈ (0, T ) dt dZI (t) 2b − δ 2 ZI (t) + ap|θI (t)|p−1 + |θI (t)|q−1 ZI (t) ≥ 0, t ∈ (0, T ), dt h Zi (0) ≥ 0, 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 Discretizations of a 1 Semilinear Parabolic Equation Uh (T ) ∞ ≤ 1−p 1 < ∞, Nabongo Diabate and ( Uh (0) ∞ + a(p − 1)T ) p−1 Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 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 estab- Title Page lished in (1.4) for the continuous problem. Let us give the statement of the main theorem of this section. Contents Theorem 3.2. Let Uh be the solution of (2.1) – (2.2). Then we have 1 lim t p−1 Uh (t) ∞ = C0 , t→∞ Page 9 of 26 1 1 p−1 Go Back where C0 = a(p−1) . Full Screen The proof of Theorem 3.2 is based on the following lemmas. We introduce the function Close µ(x) = −λ(C0 + x) + (C0 + x)p , 1 1 p−1 where C0 = a(p−1) . Firstly, we establish an upper bound of the solution for the semidiscrete problem. 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 . Discretizations of a Semilinear Parabolic Equation Nabongo Diabate and A straightforward computation reveals that Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 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 Title Page = t−λ−1 (−λ(C0 + ε) − (λ + 1)t−1 + a(C0 + ε + t−1 )p ), Contents because λp = λ + 1. Using the mean value theorem, we get (C0 + ε + t−1 )p = (C0 + ε)p + ξi t−1 , Page 10 of 26 where ξi (t) is a bounded function. We deduce that Go Back dWi − δ 2 Wi + a|Wi |p−1 Wi = t−λ−1 (µ(ε) − (λ + 1)t−1 + ξi t−1 ), Full Screen dt Close dWI 2b − δ 2 WI + a|WI |p−1 WI + |WI |q−1 WI dt h 2b −qλ+λ+1 = t−λ−1 µ(ε) − (λ + 1)t−1 + ξi t−1 + t (C0 + ε + t−1 )q . h p−q Obviously −qλ + λ + 1 = p−1 < 0. We also observe that µ(0) = 0 and µ (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, 0 ≤ i ≤ I − 1, t ∈ [T, +∞), dt dWI 2b − δ 2 WI + a|WI |p−1 WI + |WI (t)|q−1 WI (t) > 0, t ∈ [T, +∞), Discretizations of a dt h Semilinear Parabolic Equation Nabongo Diabate and T −λ C0 Théodore K. Boni Wi (T ) > . 2 vol. 9, iss. 2, art. 33, 2008 Since from Theorem 3.1 limt→∞ Ui (t) = 0, there exists τ > T such that Ui (τ ) < T −λ C0 2 < Wi (T ). We introduce the vector Zh (t) such that Zi (t) = Ui (t + τ − T ), Title Page 0 ≤ i ≤ I. We obtain Contents dZi − δ 2 Zi + a|Zi |p−1 Zi > 0, 0 ≤ i ≤ I − 1, t ≥ T, dt dZI 2b − δ 2 ZI + a|ZI |p−1 ZI + |ZI (t)|q−1 ZI (t) > 0, t ≥ T, dt h Page 11 of 26 Zi (T ) = Ui (τ ) < Wi (T ). Go Back We deduce from Lemma 2.2 that Zi (t) ≤ Wi (t), that is to say Full Screen (3.1) Ui (t + τ − T ) ≤ Wi (t) for t ≥ T, Close which leads us to the result. The lemma below gives a lower bound of the solution for the semidiscrete prob- lem. 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 . Discretizations of a Semilinear Parabolic Equation A direct calculation yields Nabongo Diabate and Théodore K. Boni 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 vol. 9, iss. 2, art. 33, 2008 dt = t−λ−1 (−λ(C0 − ε) − (λ + 1)t−1 + a(C0 − ε + t−1 )p ) Title Page because λp = λ + 1. From the mean value theorem, we have Contents (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 Page 12 of 26 dVI 2b Go Back − δ 2 VI + a|VI |p−1 VI + |VI |q−1 VI dt h Full Screen 2b = t−λ−1 µ(ε) − (λ + 1)t−1 + χi t−1 + t−qλ+λ+1 (C0 − ε + t−1 )q . Close h Obviously −qλ + λ + 1 < 0. Also, since µ(0) = 0 and µ (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, 0 ≤ i ≤ I − 1, t ∈ [T, +∞), dt 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, 0 ≤ i ≤ I − 1, t ≥ 1, dt Discretizations of a Semilinear Parabolic Equation dXI 2b Nabongo Diabate and − δ 2 XI + a|XI |p−1 XI + |XI |q−1 XI < 0, t ≥ 1, Théodore K. Boni dt h vol. 9, iss. 2, art. 33, 2008 Xi (1) = Vi (τ ) < Ui (1). We deduce from Lemma 2.2 that Title Page (3.2) Ui (t) ≥ Vi (t + τ − 1) for t ≥ 1, Contents 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 Page 13 of 26 Ui (t) Ui (t) Go Back (C0 − ε) ≤ lim inf ≤ lim sup ≤ (C0 + ε), t→∞ tλ t→∞ tλ Full Screen and we have the desired result. Close 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 Discretizations of a the initial condition at (2.3) satisﬁes Semilinear Parabolic Equation Nabongo Diabate and 0 (4.1) Uh − uh (0) ∞ = o(1) as h → 0, Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 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 0 Title Page (4.2) max Uh (t) − uh (t) ∞ = O( Uh − uh (0) ∞ + h2 ) as h → 0. 0≤t≤T Contents Proof. Let K > 0 and L be such that 2 uxxx ∞ K uxxxx ∞ K ≤ , ≤ , u ∞ ≤ K, ap(K + 1)p−1 ≤ L, 3 2 12 2 Page 14 of 26 (4.3) 2q(K + 1)q−1 ≤ L. Go Back The problem (2.1) – (2.3) has for each h, a unique solution Uh ∈ C 1 ([0, Tqh ), RI+1 ). Full Screen Let t(h) the greatest value of t > 0 such that Close (4.4) Uh (t) − uh (t) ∞ < 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 Uh (t) ∞ ≤ u(x, t) ∞ + Uh (t) − uh (t) ∞ f or t ∈ (0, t∗ (h)), which implies that (4.5) Uh (t) ∞ ≤ 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)), d h2 Discretizations of a ei (t) − δ 2 ei (t) = uxxxx (xi , t) − apξip−1 ei (t), Semilinear Parabolic Equation dt 12 Nabongo Diabate and Théodore K. Boni d 2 q−1 2h2 h2 p−1 eI (t) − δ 2 eI (t) = qθI eI + uxxx (xI , t) + uxxxx (xI , t) − apξI eI (t), vol. 9, iss. 2, art. 33, 2008 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 Title Page d Contents (4.6) ei (t) − δ 2 ei (t) ≤ L|ei (t)| + Kh2 , 0 ≤ i ≤ I − 1, dt deI (t) (2eI−1 (t) − 2eI (t)) L|eI (t)| (4.7) − 2 ≤ + L|eI (t)| + Kh2 . Page 15 of 26 dt h h Consider the function Go Back 2 0 z(x, t) = e((M +1)t+Cx ) ( Uh − uh (0) ∞ + Qh2 ) Full Screen Close 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), 20 z(x, 0) = eCx ( Uh − uh (0) ∞ + Qh). By a semidiscretization of the above problem, we may choose M, C, Q large enough that d (4.8) z(xi , t) > δ 2 z(xi , t) + L|z(xi , t)| + Kh2 , 0 ≤ i ≤ I − 1, dt d L Discretizations of a (4.9) z(xI , t) > δ 2 z(xI , t) + |z(xI , t)| + L|z(xI , t)| + Kh2 , Semilinear Parabolic Equation dt h Nabongo Diabate and Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 (4.10) z(xi , 0) > ei (0), 0 ≤ i ≤ I. It follows from Lemma 3.4 that Title Page z(xi , t) > ei (t) f or t ∈ (0, t∗ (h)), 0 ≤ i ≤ I. Contents By the same way, we also prove that z(xi , t) > −ei (t) f or t ∈ (0, t∗ (h)), 0 ≤ i ≤ I, Page 16 of 26 which implies that Go Back Uh (t) − uh (t) ∞ 0 ≤ e(M t+C) ( Uh − uh (0) ∞ + Qh2 ), t ∈ (0, t∗ (h)). Full Screen Let us show that t∗ (h) = T . Suppose that T > t(h). From (4.4), we obtain Close (M T +C) 0 2 (4.11) 1 = Uh (t(h)) − uh (t(h)) ∞ ≤e ( Uh − uh (0) ∞ + Qh ). 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. 5. Full Discretizations In this section, we study the asymptotic behavior, using full discrete schemes (ex- plicit and implicit) of (1.1) – (1.3). Firstly, we approximate the solution u(x, t) of (n) n n n (1.1) – (1.3) by the solution Uh = (U0 , U1 , . . . , UI )T of the following explicit scheme Discretizations of a (n+1) (n) Ui − Ui (n) (n) p−1 (n+1) Semilinear Parabolic Equation (5.1) = δ 2 Ui −a Ui Ui , 0 ≤ i ≤ I − 1, Nabongo Diabate and ∆t Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 (n+1) (n) UI − UI (n) (n) p−1 (n+1) 2b (n) q−1 (n+1) (5.2) = δ 2 UI − a UI UI − U UI , ∆t h I Title Page Contents (0) (5.3) Ui = φi > 0, 0 ≤ i ≤ I, 2 where n ≥ 0, ∆t ≤ h . We need the following lemma which is a discrete form of 2 the maximum principle for ordinary differential equations. Page 17 of 26 Lemma 5.1. Let f ∈ C 1 (R) and let an and bn be two bounded sequences such that Go Back an+1 − an bn+1 − bn Full Screen (5.4) + f (an ) ≥ + f (bn ), n ≥ 0, ∆t ∆t Close (5.5) a0 ≥ b 0 . Then we have an ≥ bn , n ≥ 0 for h small enough. Proof. Let Zn = an − bn . We get Zn+1 − Zn (5.6) + f (ξn )Zn ≥ 0, ∆t where ξn is an intermediate value between an and bn . Obviously (5.7) Zn+1 ≥ Zn (1 − ∆tf (ξn )). Discretizations of a Semilinear Parabolic Equation Nabongo Diabate and Since an and bn are bounded and f ∈ C 1 (R), there exists a positive M such that Théodore K. Boni |f (ξn )| ≤ M . Let j be the ﬁrst integer such that Zj < 0. From (5.5), j ≥ 0. We vol. 9, iss. 2, art. 33, 2008 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. Title Page Now, we may state the following. Contents (n) Theorem 5.2. Let Uh be the solution of (5.1) – (5.3). We have Uh ≥ 0 and (n) 1 Uh ≤ 1 , ∞ 1−p p−1 Page 18 of 26 (0) Uh + A(p − 1)n∆t ∞ Go Back a where A = (0) p−1 . Full Screen 1+a∆t Uh ∞ Close Proof. A straightforward calculation yields ∆t (n) 2∆t (n) (n) (n+1) U h2 i+1 + 1− h2 Ui + Ui−1 (5.8) Ui = p−1 , 1 ≤ i ≤ I − 1, (n) 1 + a∆t Ui 2∆t (n) 2∆t (n) (n+1) h2 U1 + 1− h2 U0 (5.9) U0 = p−1 , (n) 1 + a∆t U0 2∆t (n) 2∆t (n) Discretizations of a (n+1) h2 UI−1 + 1− h2 UI Semilinear Parabolic Equation (5.10) UI = p−1 q−1 . Nabongo Diabate and (n) b (n) 1 + a∆t UI + 2 h ∆t UI Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 Since 1 − 2 ∆t h2 is nonnegative, using a recursive argument, it is easy to see that (n) (n) (n) Uh ≥ 0. Let i0 be such that Ui0 = Uh . From (5.8), we get Title Page ∞ Contents ∆t (n) 2∆t (n) (n) U h2 i0 +1 + 1− h2 Uh + Ui0 −1 (n+1) ∞ Uh ≤ p−1 if 1 ≤ i0 ≤ I − 1. ∞ (n) 1 + a∆t Uh ∞ Applying the triangle inequality and the fact that 1 − 2∆t is nonnegative, we arrive Page 19 of 26 h2 at Go Back (n) Uh Full Screen (n+1) ∞ (5.11) Uh ≤ p−1 . ∞ (n) Close 1 + a∆t Uh ∞ We obtain the same estimation if i0 = 0 or i0 = I. The inequality (5.11) implies that (n+1) (n) (n) (0) Uh ≤ Uh and by iterating, we obtain Uh ≤ Uh . From ∞ ∞ ∞ ∞ (5.11), we also observe that p (n+1) (n) Uh − U (n) ∞ a Uh ∞ ∞ ≤− p−1 . ∆t (n) 1 + a∆t Uh ∞ (n) (0) Using the fact that Uh ≤ Uh , we have Discretizations of a ∞ ∞ Semilinear Parabolic Equation Nabongo Diabate and (n+1) (n) Uh − Uh p Théodore K. Boni ∞ ∞ (n) ≤ −A Uh . vol. 9, iss. 2, art. 33, 2008 ∆t ∞ We introduce the function α(t) which is deﬁned as follows Title Page 1 α(t) = 1 . Contents 1−p p−1 (0) Uh + A(p − 1)t ∞ We remark that α(t) obeys the following differential equation (0) Page 20 of 26 α (t) = −Aαp (t), α(0) = Uh . ∞ Go Back Using a Taylor’s expansion, we have Full Screen (∆t)2 α(tn+1 ) = α(tn ) + ∆tα (tn ) + α (t˜ ), n Close 2 where t˜ is an intermediate value between tn and tn+1 . It is not hard to see that α(t) n is a convex function. Therefore, we obtain α(tn+1 ) − α(tn ) ≥ −Aαp (tn ). ∆t (n) From Lemma 5.1, we get Uh ≤ α(tn ), which ensures that ∞ (n) 1 Uh ≤ 1 , ∞ 1−p p−1 (0) Uh + A(p − 1)n∆t ∞ Discretizations of a and we have the desired result. Semilinear Parabolic Equation Nabongo Diabate and Remark 2. The estimate of Theorem 5.2 is the discrete form of the one given in (1.4) Théodore K. Boni for the continuous problem. vol. 9, iss. 2, art. 33, 2008 Now, we approximate the solution u(x, t) of problem (1.1) – (1.3) by the solution (n) Uh of the following implicit scheme Title Page (n+1) (n) Ui − Ui (n+1) (n) p−1 (n+1) Contents (5.12) = δ 2 Ui − Ui Ui , 0 ≤ i ≤ I − 1, ∆t (n+1) (n) UI − UI (n+1) (n) p−1 (n+1) 2b (n) p−1 (n+1) (5.13) = δ 2 UI − a UI UI − U UI , Page 21 of 26 ∆t h I Go Back (0) Full Screen (5.14) Ui = φi > 0, 0 ≤ i ≤ I, Close 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: (n) 2∆t (n+1) ∆t (n) p−1 (n+1) U0 =− 2 U1 + 1 + 2 2 + a∆t U0 U0 , h h (n) ∆t (n+1) ∆t (n) p−1 (n+1) ∆t (n+1) Ui =− Ui−1 + 1 + 2 2 + a∆t Ui Ui − U , 1 ≤ i ≤ I−1, h2 h h2 i+1 (n) 2∆t (n+1) ∆t (n) p−1 2b (n) q−1 (n+1) UI =− UI−1 + 1 + 2 2 + a∆t UI + ∆t UI UI , h2 h h which gives the following linear system Discretizations of a (n+1) (n) A(n) Uh = Uh Semilinear Parabolic Equation Nabongo Diabate and Théodore K. Boni where A(n) is the tridiagonal matrix deﬁned as follows vol. 9, iss. 2, art. 33, 2008 d0 −2∆t 0 0 ··· 0 0 h2 −∆t −∆t h2 d1 h2 0 ··· 0 0 Title Page −∆t 0 d2 −∆t 0 ··· 0 h2 h2 (n) . . .. .. .. . . Contents A = . . . . . . . . . . . , −∆t −∆t 0 0 ··· h2 dI−2 h2 0 0 0 0 · · · −∆t dI−1 h2 −∆t h2 −2∆t 0 0 0 ··· 0 h2 dI Page 22 of 26 with ∆t (n) Go Back di = 1 + 2 + a∆t|Ui |p−1 for 0≤i≤I −1 h2 Full Screen and ∆t (n) p−1 2b (n) q−1 Close dI = 1 + 2 + a∆t UI + ∆t UI . h2 h Let us remark that the tridiagonal matrix A(n) satisﬁes the following properties (n) (n) Aii > 0 and Aij < 0 i = j, (n) (n) Aii > Aij . i=j n (n) These properties imply that exists for any n and Uh ≥ 0 (see for instance [2]). Uh 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 Discretizations of a Semilinear Parabolic Equation (n) 1 Nabongo Diabate and Uh ≤ 1 , Théodore K. Boni ∞ 1−p p−1 (0) Uh + A(p − 1)n∆t vol. 9, iss. 2, art. 33, 2008 ∞ a where A = (0) p−1 . 1+a∆t Uh Title Page ∞ (n) Proof. We know that Uh ≥ 0 as we have seen above. Now, let us obtain the above Contents (n) (n) estimate to complete the proof. Let i0 be such that = Ui0 . Using the Uh ∞ equality (5.12), we have ∆t (n) (n+1) (n) ∆t (n) ∆t (n) 1 + 2 2 + a∆t Uh Uh ≤ Uh + 2 Ui0 −1 + 2 Ui0 +1 Page 23 of 26 h ∞ ∞ ∞ h h if 1 ≤ i0 ≤ I − 1. Go Back Applying the triangle inequality, we derive the following estimate Full Screen (n) Uh Close (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. 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 = h . In the following tables, in the rows, we give the 2 ﬁrst n when (n) n∆tUh − 1 < ε, Discretizations of a ∞ Semilinear Parabolic Equation Nabongo Diabate and the corresponding time T n = n∆t, the CPU time and the order(s) of method com- Théodore K. Boni puted from vol. 9, iss. 2, art. 33, 2008 log((T4h − T2h )/(T2h − Th )) s= . log(2) Title Page Table 1: (ε = 10−2 ) Numerical times, numbers of iterations, CPU times (seconds), Contents and orders of the approximations obtained with the implicit Euler method I Tn n CPU time s 16 674.0820 345129 103 - 32 674.2632 1.380890. 660 - Page 24 of 26 64 674.3085 5.523.934 6020 2.01 Go Back 128 674.3278 22095735 58290 1.24 256 674.4807 87383041 574823 2.99 Full Screen Close 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 Tn n CPU time s 16 674.3281 345.255 90 - 32 674.3452 1.381.058 720 - 64 674.3290 5.524.102 10820 0.08 Discretizations of a Semilinear Parabolic Equation 128 674.3187 22845950 323528 0.65 Nabongo Diabate and 256 674.3098 88237375 19457811 0.21 Théodore K. Boni vol. 9, iss. 2, art. 33, 2008 Title Page Contents Page 25 of 26 Go Back Full Screen Close 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 Discretizations of a parabolic and elliptic equation of second order with nonlinear boundary con- Semilinear Parabolic Equation Nabongo Diabate and ditions, Nonl. Anal. TMA, 45 (2001), 895–908. Théodore K. Boni [3] T.K. BONI, Extinction for discretizations of some semilinear parabolic equa- vol. 9, iss. 2, art. 33, 2008 tions, C.R.A.S, Serie I, 333 (2001), 795–800. [4] V.A. KONDRATIEV AND L. VÉRON, Asymptotic behaviour of solutions of Title Page some nonlinear parabolic or elliptic equation, Asymptotic Analysis, 14 (1997), Contents 117–156. [5] R.E. MICKENS, Relation between the time and space step-sizes in nonstan- dard ﬁnite difference schemes for the ﬁsher equation, Num. Methods. Part. Diff. Equat., 13 (1997), 51–55. 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