A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME FOR THE EXTENDED-FISHER-KOLMOGOROV EQUATION. Tlili Kadri and Khaled Omrani

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1 Bull. Korean Math. Soc. 0 (0), No. 0, pp. 0 pissn: / eissn: A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME FOR THE EXTENDED-FISHER-KOLMOGOROV EQUATION Tlili Kadri and Khaled Omrani Abstract. In this paper, a nonlinear high-order difference scheme is proposed to solve the Extended-Fisher-Kolmogorov equation. The existence, uniqueness of difference solution and priori estimates are obtained. Furthermore, the convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete L -norm. Some numerical examples are given in order to validate the theoretical results.. Introduction In this article, we consider the following periodic initial value problem of the Extended Fisher-Kolmogorov (EFK) equation u (.) t + γ 4 u x 4 u + f(u) = 0, x Ω, t (0, T ], x subject to the initial condition (.) u(x, 0) = u 0 (x), x Ω, and periodic condition (.3) u(x + L, t) = u(x, t), x Ω, t (0, T ], where f(u) = u 3 u, Ω = (0, ), T > 0, γ is a positive constant and u 0 is a given L-periodic function. When γ = 0 in (.), we obtain the standard Fisher- Kolmogorov (FK) equation. However, by adding a stabilizing fourth-order derivative term to the Fisher-Kolmogorov equation, Coullet et al. [5] proposed (.) and called as the EFK equation. Problem (.) arises in a variety of applications such as pattern formation in bi-stable systems [6], propagation of domain walls in liquid crystals [7], travelling waves in reaction diffusion system [, 3] and mezoscopic model of a phase transition in a binary system near Lipschitz point []. For computational studies, L. J. Tarcius Doss and A. Received December 3, 06; Accepted June 4, Mathematics Subject Classification. 65M06, 65M, 65M5. Key words and phrases. extended Fisher-Kolmogorov equation, difference scheme, existence, uniqueness, high-order convergence. c 0 Korean Mathematical Society

2 T. KADRI AND K. OMRANI P. Nandini has proposed a H -Galerkin mixed Finite Element Method for the EFK equation in []. Related to fourth and quintic order evolution equations, a second order difference scheme for the Sivashinsky equation is analyzed by Omrani in [6, 7], for Rosenau equation by Omrani [8], for Cahn-Hilliard equation by Khiari et al. [], for Kuramoto-Sivashinsky equation by Akrivis [], for Rosenau-RLW equation by Pan et al [9, 0], for Rosenau-Kawahara equation by D. He et al. [9, 0]. In [3], Khiari and Omrani have proposed a nonlinear finite difference scheme for the EFK equation (.)-(.3). This scheme is second-order accurate in space. For many application problems, it is desirable to use high-order numerical algorithms to compute accurate solutions. To obtain satisfactory higher order numerical results with reasonable computational cost, there have been attempts to develop higher order compact (HOC) schemes [7, 8, 4, 5, 5]. However, because the discretization of nonlinear term in compact scheme is more complicated than that in second-order one, a priori estimate in the discrete L -norm is hard to be obtained, so the unconditional convergence of any compact difference scheme for nonlinear equation is difficult to be proved. In this article, we establish a new high-order difference scheme for the EFK equation and prove that the scheme is convergent with the convergence of O(h 4 + k ) in the discrete L -norm. The remainder of the article is arranged as follows. In Section, a nonlinear difference scheme is derived. In Section 3, existence, a priori estimations and uniqueness for numerical solutions are shown. The L -convergence for the difference scheme is proved in Section 4. In the last section, some numerical experiments are presented to support our theoretical results. Throughout this article, C denotes a generic positive constant which is independent of the discretisation parameters h and k, but may have different values at different places.. High-order conservative difference scheme In this section, we propose a two-level nonlinear Crank-Nicolson-type finite difference scheme for the problem (.)-(.3). For convenience, the following notations are used. For a positive integer N, let time-step k = T N, tn = nk, n = 0,,..., N. For a positive integer M, let space-step h = L M, x i = ih, i = 0,,..., M, and let R M per = {V = (V i ) i Z / V i R and V i+m = V i, i Z}. We define the difference operators for U n R M per as (Ui n ) x = U i+ n U i n, (Ui n ) x = U i n Ui n, (Ui n )ˆx = U i+ n U i n, h h h U n+ i = (U n+ i + U n i ), t U n i = k (U n+ i U n i ).

3 A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME 3 We introduce a discrete inner product as M (U n, V n ) h = h Ui n V n i= i, U n, V n R M per. The discrete L -norm h, H seminorm,h, H seminorm,h, L p norm p,h and the discrete maximum-norm,h are defined, respectively as U n h = M (U n, U n ) h, U n,h = h (Ui n ) x ), U n,h = h M [ M (Ui n ) x x, U n p,h = h Ui n p] p, p, i= U n,h = max U i n. i M Denote Hper(Ω) the periodic Sobolev space of order. We discretize problem (.)-(.3) by the following finite difference scheme: [ ] [ ] t Ui n 5 + γ 3 (U n+ i ) xx x x 3 (U n+ 4 i ) x xˆxˆx 3 (U n+ i ) x x 3 (U n+ i )ˆxˆx (.) + f(u n+ i ) = 0, i =,..., M, n =,..., N, (.) U n i+m = U n i, i =,..., M, n =,..., N, i= (.3) U 0 i = u 0 (x i ), i =,..., M. In the next, we collect some auxiliary results. Using summation by parts and the discrete boundary condition, we can easily check the following Lemma. i= Lemma. For any grid functions U, V R M per, we have (.4) (U x, V ) h = (U, V x ) h, (.5) (Uˆx, V ) h = (U, Vˆx ) h, (.6) (U x x, V ) h = (U x, V x ) h, (.7) (Uˆxˆx, V ) h = (U, Vˆxˆx ) h. Therefore, we may easily seen that the following relations hold, for U R M per: (.8) (U x x, U) h = U x h, (.9) (Uˆxˆx, U) h = Uˆx h, (.0) (U xx x x, U) h = U xx h, (.) (U x xˆxˆx, U) h = U xˆx h.

4 4 T. KADRI AND K. OMRANI Lemma. For any U R M per, there is (.) Uˆx h U x h, (.3) U xˆx h U xx h. Proof. For U R M per, we have from (.6) (.4) U x h = U x h. Using the periodic boundary condition (.) condition, we get M ( Uˆx Uj+ U ) j h = h h = h h j= M ( Uj+ U j + U j U j h h j= ) M ( Uj+ U ) j M ( Uj U j + h h h j= = U x h + U x h. Therefore, (.) follows immediately from (.4). For U R M per, we have U xˆx h = h = h h M ( Uj+ U j+ U j + U ) j h j= j= M ( Uj+ U j+ + U j + U j+ U j + U j ) h h j= M ( Uj+ U j+ + U ) j M ( Uj+ U j + U ) j + h h h j= Further from (.6) = U xx h + U x x h. (.5) U x x h = U xx h, and (.3) follows. 3.. Existence j= 3. Existence and uniqueness The existence of the approximate solution of the system (.)-(.3) is proved by the use of the Browder theorem [4]. )

5 A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME 5 Lemma 3. Let (H, (, ) h ) be a finite dimensional inner product space, h be the associated norm, suppose that g : H H is continuous and there exists λ > 0 such that (g(x), x) h > 0 for all x H with x h = λ. Then, there exists x H such that g(x ) = 0 and x h λ. Theorem. The numerical solution of discrete scheme (.)-(.3) exists. Proof. In order to prove the theorem by the mathematical induction, we assume that U 0, U,..., U n exists for n < N. Let g be a function defined by (3.) g(v ) = V U n + 5kγ 3 V xx x x kγ 3 V x xˆxˆx 4k 3 V x x + k 3 Vˆxˆx + kf(v ). Then g is obviously continuous. Taking the inner product of (3.) by V and using Lemma, we have (g(v ), V ) h = V h (U n, V ) h + 5kγ 3 V xx h kγ 3 V xˆx h Applying Lemma, we get + 4k 3 V x h k 3 Vˆx h + k(f(v ), V ) h. (g(v ), V ) h V h U n h V h + kγ V xx h + k V x h + k V 4 4,h k V h. Therefore for γ > 0, (g(v ), V ) h [( k ] ) V h U n h V h. For k < and V h = k U n h +, obviously (g(v ), V ) h > 0 and via Lemma 3 we deduce the existence of V R M per such that g(v ) = 0. If we take U n+ = V U n, then U n+ satisfies the equation (.). 3.. Priori estimates Lemma 4 (Discrete Gronwall inequality [6]). Assume {G n /n 0} is nonnegative sequences and satisfies G 0 A, n G n A + Bk G i n =,,..., where A and B are non negative constants. Then G satisfies i=0 G n Ae Bnk, n = 0,,,.... For the difference solution of scheme (.)-(.3), we have the following priori bound. Theorem. Let U n be the solution of the difference scheme (.)-(.3). Assume that u 0 L (Ω). Then, there exists a positive constant C independent of h and k such that (3.) U n h C.

6 6 T. KADRI AND K. OMRANI Proof. Taking in (.) the inner product with U n+ and using Lemma, we obtain: t U n h + 5γ 3 U n+ xx h γ 3 U n+ xˆx h U n+ x h 3 U n+ ˆx = U n+ 4 4,h + U n+ h. By lemma we get Therefore for γ > 0 This yields when k 3, which gives t U n h + γ U n+ xx h + U n+ x h U n+ h. k ( U n+ h U n h) ( U n+ h + U n h). ( k) U n+ h ( + k) U n h, U n+ h ( + 3k) U n h, 0 n N. Summing the above inequality from 0 to n, we have It follows from Lemma 4 that This completes the proof. n U n h U 0 h + 3k U l h. l=0 U n h e 3T U 0 h, 0 n N. Next we use the following lemmas (see [6]). Lemma 5. For v R M per, we have for p [ ] (3.3) v p,h C v q v q,h + v h, where q = p h and C is a positive constant independent of p and h. Lemma 6. For any x 0, y 0, there is (3.4) (x + y) 6 5 (x 6 + y 6 ). Theorem 3. Assume that u 0 Hper(Ω). Then, the solution of the difference scheme (.)-(.3) is estimated as follows (3.5) U n,h C. h

7 A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME 7 Proof. Taking in (.) the inner product with t U n, and using Schwarz inequality, we obtain t U n h + 5γ 3 t U n xx h γ 3 t U n xˆx h t U n x h 3 t U ṋ x h = ( (U n+ ) 3 + U n+, t U n ) h U n+ 6 6,h + t U n h + U n+ h + t U n h. It follows from Theorem that (3.6) 5γ 3 t U n xx h γ 3 t U n xˆx h t U n x h 3 t U ṋ x h U n+ 6 6,h + C. Applying Lemma 5 with p = 6, we obtain (3.7) U n+ 6,h C[ U n+ 3 From Lemma 6, we have It follows from (3.) that h U n+ x 3 h + U n+ h ]. U n+ 6 6,h 5 C 6 [ U n+ 4 h U n+ x h + U n+ 6 h ]. (3.8) U n+ 6 6.h C U n+ x h + C, where C is a generic constant depending on U 0 h and T. Using (3.6) and (3.8), we have for γ > 0 (3.9) 5γ 3 t Uxx n h γ 3 t Uxˆx n h t Ux n h 3 t Ux ṋ h C U n+ h + C x C( U n+ x h + γ U n+ xx h) + C. Let B n = 5γ 3 U xx n h γ 3 U xˆx n h U x n h 3 U x ṋ h and summing up (3.9) from 0 to n, we obtain n (3.0) B n B 0 + Ck ( Ux l h + γ Uxx l h) + C(T ). It follows from Lemma that l=0 (3.) B n γ U n xx h + U n x h. Then using (3.0) and (3.), we have (3.) γ U n xx h + U n x h B 0 + Ck n ( Ux l h + γ Uxx l h) + C(T ). Letting now G n = γ Uxx n h + U x n h and using the above inequality we find n (3.3) G n B 0 + Ck G l + C(T ). l=0 l=0

8 8 T. KADRI AND K. OMRANI This yields n (3.4) ( Ck)G n C + Ck G l, l=0 where C is a positive constant depending on U 0 and T. Applying Lemma 4, we obtain (3.5) γ U n xx h + U n x h C(U 0, T ), as γ > 0, it follows that (3.6) U n x h C(U 0, T ). Using Lemma 5 with p = and (3.) and (3.6), we have U n,h C, where C = C(U 0, T ), which completes the proof Uniqueness The uniqueness of the solution of difference scheme (.) (.3) is proved in the next theorem. Theorem 4. For k small enough, the solution of the difference scheme (.)- (.3) is uniquely solvable. Proof. Assume that U n and V n both satisfy the scheme (.)-(.3), let θ n = U n V n, then we obtain (3.7) t θ n + 5γ 3 (θn+ )xx x x γ 3 (θn+ )x xˆxˆx 4 3 (θn+ )x x + 3 (θ n+ )ˆxˆx + f(u n+ ) f(v n+ ) = 0, i =,..., M, n =,..., N, (3.8) θ n i+m = θ n i, i =,..., M, n =,..., N, (3.9) θ 0 i = 0, i =,..., M. Taking in (3.7) the inner product with θ n+ (3.0) and using Lemma, we have t θ n h + 5γ 3 θn+ xx h γ 3 θn+ xˆx h θn+ x h 3 θn+ ˆx + (f(u n+ ) f(v n+ ), θ n+ )h = 0. By differentiability of f and (3.5) we find (3.) (f(u n+ ) f(v n+ ), θ n+ )h C θ n+ h. It follows from (3.0), (3.) and Lemma that t θ n h + γ θ n+ xx h + θ n+ x h C θ n+ h. h

9 This yields for γ > 0 Consequently, A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME 9 k ( θn+ h θ n h) C( θ n+ h + θ n h). ( kc) θ n+ h ( + kc) θ n h. For k <, applying Lemma 4 and (3.9), we obtain C θ n+ h = 0. This completes the proof of uniqueness. 4. Convergence According to Taylor expansion, we obtain the following Lemma Lemma 7. For any smooth function ψ we have (4.) (4.) 4 3 (ψ i) x x 3 (ψ i)ˆxˆx = d ψ dx (x i) + O(h 4 ), 5 3 (ψ i) xx x x 3 (ψ i) x xˆxˆx = d4 ψ dx 4 (x i) + O(h 4 ). Define the net function u n i = u(x i, t n ), i M, 0 n N. The truncation error of the scheme (.)-(.3) is 5 t u n i + γ[ (4.3) 3 (un+ i ) xx x x ] [ 3 (un+ 4 i ) x xˆxˆx 3 (un+ i ) x x ] 3 (un+ i )ˆxˆx + f(u n+ i ) = ri n. According to Taylor s expansion and Lemma 7, we obtain Lemma 8. Suppose that u(x, t) C 8,3 x,t ([0, L] [0, T ]), then the truncation error of the difference scheme (.)-(.3) satisfies (4.4) r n i = O(h 4 + k ) as k 0, h 0. The convergence of the difference scheme (.)-(.3) will be given in the following theorem. Theorem 5. Assume that the solution of (.)-(.3) u(x, t) C 8,3 x,t ([0, L] [0, T ]). Then the solution of the difference scheme (.)-(.3) converges to the solution of the problem (.)-(.3) in the discrete L -norm and the rate of convergence is the order of O(h 4 + k ), when h and k are sufficiently small. Proof. Denote c 0 = max 0 x L, 0 t T u(x, t).

10 0 T. KADRI AND K. OMRANI Let e n i = u n i U i n, i M, 0 n N. Subtracting (.) from (4.4), we obtain [ ] [ ] t e n 5 i + γ 3 (4.5) (en+ i ) xx x x 3 (en+ 4 i ) x xˆxˆx 3 (en+ i ) x x 3 (en+ i )ˆxˆx + f(u n+ i ) f(u n+ i ) = ri n, i =,,..., M, n = 0,,... N. (4.6) e n i = e n i+m, i =,,..., M, n = 0,,..., N, (4.7) e 0 i = 0, i =,,..., M. Computing the inner product of (4.5) with e n+ (4.8) and using Lemma, we have t e n+ h + 5γ 3 en+ xx h γ 3 en+ xˆx h en+ x h 3 en+ ˆx = (f(u n+ ) f(u n+ ), e n+ )h + (r n, e n+ )h. It follows from Lemma that (4.9) t e n+ h + γ e n+ xx h + e n+ x h [ f(u n+ ) f(u n+ ) h + r n h ] e n+ h. For the nonlinear term f(u n+ ) f(u n+ ) h, we use the boundedness of U n,h and c 0 to find that (4.0) f(u n+ ) f(u n+ ) h C e n+ h. Substituting (4.0) in (4.9) and using Lemma 9, we obtain for γ > 0 k ( en+ h e n h) C( e n+ h + e n h) + (h 4 + k ). This implies that ( kc)) e n+ h ( + kc) e n h + k(h 4 + k ). For k sufficiently small, using (4.7) and an application of Lemma 4 yields the following inequality (4.) e n h C(h 4 + k ). Taking in (4.5) the inner product with t e n, and using Schwarz inequality, we have (4.) t e n h + 5γ 3 t e n xx h γ 3 t e n xˆx h t e n x h 3 t e ṋ x h = (f(u n+ ) f(u n+ ), t e n ) h + (r n, t e n ) h f(u n+ ) f(u n+ ) h + t e n h + r n h + t e n h. It follows from Lemma 8, (4.0) and (4.) that (4.3) 5γ 3 t e n xx h γ 3 t e n xˆx h t e n x h 3 t e ṋ x h C(h 4 + k ). h

11 A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME Letting A n = 5γ 3 en xx h γ 3 en xˆx h en x h 3 eṋ x h, and summing up (4.3) from 0 to n, we obtain (4.4) A n A 0 + Cnk(h 4 + k ). From (4.7), it follows that (4.5) A n CT (h 4 + k ). Again using Lemma, we obtain (4.6) γ e n xx h + e n x h A n C(T )(h 4 + k ), and hence, (4.7) e n x h C(T )(h 4 + k ). An application of Lemma 5 with p = and (4.) and (4.7) yields (4.8) e n,h C(h 4 + k ), where C = C(T ). This completes the proof. 5. Numerical results We give the numerical experiment that the exact solution of problem is known. Consider the following inhomogeneous periodic initial value problem of the EFK equation u (5.) t + γ 4 u x 4 u x + u3 u = g(x, t), x (0, ), t (0, T ], subject to the initial condition (5.) u(x, 0) = sin(πx), x (0, ), where [ g(x, y, t) = e t sin(πx) + 4π + 6γπ 4 + sin (πx)e t]. The analytic solution for Equations (5.) and (5.) is u(x, t) = e t sin(πx). We compute the numerical solutions to this problem by the difference scheme (.)-(.3) when γ = 0.0 and T =. Take h = M, t = T N. Denote (h, k), i M, 0 n N} the approximate solution. Suppose {U n i max 0 n N If O(k p ) is sufficiently small, we have max u(x i, t n ) Ui n (h, k) = O(h q ) + O(k p ). i M max u(x i, t N ) Ui N ( h i M, k) = O(hq ). Let E(h) = max u(x i, t N ) Ui N ( h, k), then i M ( E(h) ) log E( h ) q.

12 T. KADRI AND K. OMRANI Table. The maximum norm errors and spatial convergence order with fixed time step k = /0000. h E(h) q / / / / Table. The maximum norm errors and temporal convergence order with fixed space step h = /500. k F (k) p / / / / If O(h q ) is sufficiently small, we obtain max u(x i, t N ) Ui N (h, k i M ) = O(kp ). Denote F (k) = max u(x i, t N ) Ui N (h, k ), thus i M ( F (k) ) log F ( k ) p. In Table, the computation results are given for different space steps and when the time grid is fixed to be k = /0000. From this table, we conclude that the convergence order q in space is about 4, which is in accordance with the theoretical analysis. Next, we test the time convergence order of the scheme (.)-(.3). Fix spatial step h = /500. Table shows that the convergence order is about with respect to the temporal direction, which agrees with the prediction. Therefore, we conclude that the difference scheme (.)-(.3) is convergent with the convergence order of O(h 4 +k ) in maximum norm, which is in accordance Theorem 5.

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