Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method

Transcription

1 Journal of Computational and Applied Mathematics 94 (6) Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method Taanori Ide a,, Masami Oada b a Q Promotion Department, Engineering Division, AISIN AW Co., Ltd., Anjo, Japan b Department of Mathematics, Toyo Metropolitan University, Hachioji, Japan Received 9 June 3; received in revised form August 5 Abstract Partial differential equations with possibly discontinuous coefficients play an important part in engineering, physics and ecology. In this paper, we will study nonlinear partial differential equations with variable coefficients arising from population models. Generally speaing, it is difficult to analyze the behavior of nonlinear partial differential equations; therefore, we usually rely on the numerical approximation. Currently, there is an increasing interest in designing numerical schemes that preserve energy properties for differential equations. We will design the numerical schemes that preserve discrete energy property and show numerical experiments for a nonlinear partial differential equation with variable coefficients. 5 Elsevier B.V. All rights reserved. MSC: 35-4; 35K55; 37K5; 65M6 Keywords: Finite difference; The discrete variational derivative method; Partial differential equation with variable coefficients; Energy dissipation Corresponding author. Tel.: ; fax: addresses: i484_ide@aisin-aw.co.jp (T. Ide), moa@comp.metro-u.ac.jp (M. Oada) /$ - see front matter 5 Elsevier B.V. All rights reserved. doi:.6/j.cam.5.8.9

2 46 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Introduction The main purposes of this paper are as follows: () the validity of the discrete variational derivative method is shown concretely for a partial differential equation with variable coefficients; and () to investigate the numerical solutions of a nonlinear partial differential equation with variable coefficients arising from the population model []. These days, many wors exist concerning the design of structure-preserving numerical schemes for ordinary differential equations [3,,,4,5,8,,4,3,33,34] (and their references) and partial differential equations [4 6,,6 8,3,8,3,3] (and their references). Recently, Furihata and Mori proposed one systematic procedure of designing a finite-difference scheme for partial differential equations [8,9]. This method is the discretization of the continuous variational derivative method. In this paper, we call this method the discrete variational derivative method to avoid confusion with the term of the discrete variational principle [33]. One of the most remarable features of the finite difference scheme which is derived by this method is that it preserves energy properties such as the energy dissipated, energy conservation and so on [3]. The other remarable feature of this method is that the relationship between the finite difference scheme and the boundary condition is satisfactorily clear. Matsuo and others expanded the discrete variational derivative method to a complex-valued problem []. They also expanded this method to design spatially accurate schemes under periodic conditions []. The variable coefficients of our target equation involve the discontinuous case. Therefore, we thin that the behavior of the solution rapidly changes around the neighborhood of the discontinuous point. Rannacher has already studied the finite element solution of a diffusion problem with irregularities in the initial or boundary data [9]. In this paper, we will design the finite-difference scheme for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method and show numerical experiments. The contents of this paper are as follows: in Section, we will ine the discrete symbols and their calculus employed throughout this paper. In Section 3, we will describe the continuous variational derivative and the discrete variational derivative [8,9]. In Section 4, we will show our target partial differential equation and its energy property. After that we will derive the finite-difference scheme and its discrete energy property. Also, we will prove the stability, the existence, the uniqueness and the convergence rate of the proposed finite difference scheme. Finally, we will demonstrate the numerical simulation.. Notations and preliminaries In this section, we ine the one-dimensional discrete operators that Furihata employed [8]. Let Ω =[α, β] and the mesh size with respect to x be ined by Δx = β α N. ()

3 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) We denote by f a numerical value supposed to be an approximation of f(α + Δx), =,,...,N f f(α + Δx), f ={f } N = RN+. ().. Discrete operators We ine the difference operators as follows: δ + f = f + f, (3) Δx δ f = f f, (4) Δx δ () f = f + f, (5) Δx δ () f = f + f + f (Δx). (6) We ine the shift operators as follows: s ± f = f ± : shift operator. (7) And the averaging operators are ined by using a shift operator as follows: μ ± = (s± + ): averaging operator. (8).. Summation by parts with the trapezoidal rule As the discretization of the integral, we adopt the trapezoidal rule N = that is ined by ( ) N f Δx N = f + f + f N Δx. (9) = = The following are the discretization of the integral by parts formula in usual calculus for two sequences: [ N N f (δ + g )Δx + (δ f (s + f )g Δx = g ) + (s f ] N )g, () = = N (δ + f )(δ + g ) + (δ f )(δ g ) N () Δx + (δ f )g Δx = = [ (δ + = f )(μ + g ) + (δ f )(μ g ] N ), () where f ={f } N =, g ={g } N = RN+ and g is an approximation of g(a + Δx). = =

4 48 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Remar. The discrete operators with respect to t are ined in the same manner with f n corresponding to f(nδt), n =,,, The discrete variational derivative method In this section, we consider the inition of the discrete variational derivative [8,9]. We recall the continuous variational derivative method briefly [7]. Firstly, we ine the energy as follows: J [u, b] = β α G(u, u x,b)dx, () where u = u(x) and b = b(x). In this paper, we call G(u, u x,b)the energy function. And we assume that the energy function taes on the form G(u, u x,b)= m f l (u)g l (u x )h l (b), (3) l= where m Z + and f l,g l,h l : R R are differential functions. By the usual variational argument, we have β [ ] δg G β J [u + δu, b] J [u, b] δu dx + δu, (4) α δu u x α where the variational derivative δg/δu is ined by δg = G δu u ( ) G. (5) x u x Before giving the inition of the discrete variational derivative, we ine U ={U } N = RN+, (6) V ={V } N = RN+, (7) B ={B } N = RN+. (8) The discrete energy is ined by N J d [U, B]= Gd (U, B) Δx, U ={U } N = RN+, B ={B } N = RN+, (9) = where G d (U, B) corresponds to an approximation to G(u, u x,b)at x = Δx. We assume that G d (U, B) taes on the following form: m G d (U, B) = f l (U )g l + (δ + U )gl (δ U )h l (B ), () l=

5 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) where g l +,gl : R R are differential functions. The discrete energy G d (U, B) should satisfy the following condition: lim f l(u )g + Δx l (δ + U )gl (δ U )h l (B ) = f l (u)g l (u x )h l (b) x=δx. () Then we ine the discrete variational derivative for U,V, B, R N+ as follows: where δg d = δ(u,v, B) W ± l (U,V, B) = ( m l= df l d(u,v ) g l + (δ + U )gl (δ U ) + g l + (δ + V )gl (δ V ) h l (B ) ), () δ + W l (U,V, B) δ W + l (U,V, B) ( ) ( fl (U ) + f l (V ) gl (δ U ) + gl (δ V ) ) dg ± l d(δ ± U, δ ± V ) h l(b ) (3) and f(b) f(a) df : a = b, = b a (4) d(a, b) df da : a = b. We apply the summation by parts formula () to the difference J d [U, B] J d [V, B] to obtain N [ ] δg d G N d J d [U, B] J d [V, B]= (U V )Δx +, (5) δ(u,v, B) (U,V, B) = where G d = m (W l + (U,V, B) (U,V, B) s + (U V ) + Wl (U,V, B) s (U V ) l= + (s + W l (U,V, B) + s W l + (U,V, B) )(U V )). (6) 4. Nonlinear partial differential equation with variable coefficients In this section, we derive the finite-difference scheme of a nonlinear partial differential equation with variable coefficients involving a discontinuous case which arose from the population model of the form u t =du xx +u (b(x) u), where u represents the population density, d is the constant of the diffusion rate and b(x) describes the growth rate of the living being. If the growth rate b(x) is positive, b(x) indicates favorable habitats, i.e. b(x) represents the ease of living in the environment surrounding the living being. If the growth rate b(x) is negative, b(x) indicates unfavorable habitats, i.e. b(x) represents the living being as being surrounded by a harsh environment. Cantrell and Cosner have already considered the partial differential equation of the form u t = du xx + u(b(x) u) analytically [], but the nonlinear term of this partial differential equation is different from our target equation.

6 43 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) The partial differential equation with variable coefficient and its energy property We consider the nonlinear partial differential equation with a variable coefficient as the following initial-boundary value problem: u (x, t) = δg t δu, δg δu = d u x (x, t) (u(x, t)) (b(x) u(x, t)) in Ω,t>, u (x, t) = x on Ω,t>, u(x, ) = φ(x), (7) which corresponds, in our formulation, to the energy function G(u(x, t), u x (x, t), b(x)) = d ( ) u (x, t) b(x) x 3 (u(x, t))3 + 4 (u(x, t))4. (8) In this paper, we have assumed that Ω =[α, β] and d> is a small constant and b(x) is bounded. The total energy of this problem is given by J(t)= β α G(u(x, t), u x (x, t), b(x)) dx. (9) Remar. The energy function (8) has a double-well potential and is spatially inhomogeneous. Such ind of problems, for example the inhomogeneous Allen Cahn equation, appear in phase-transition problems ([5,6] and references therein). The purpose of such ind of problems is to investigate () the location of the transition layer and () the multiplicity of transition layer. In this paper, we will investigate these problems numerically in Section 5. The total energy of our target equation (7) has the following energy property [8]. Theorem 3 (Energy-dissipated property). Let u be a solution of (7). Then, we have the following inequality which indicates the dissipated property of total energy. Namely, d dt J(t)= d dt β α G(u(x, t), u x (x, t), b(x)) dx. (3)

7 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) The implicit finite-difference scheme for the target equation and its discrete energy property 4... A derivation of the discrete variational derivative In this subsection, we will calculate the discrete variational derivative for (7). We compute the solution U (n) for a discretized equation of (7), which is expected to approximate the solution u(x, t) to (7) at points (Δx,nΔt), U (n) u(δx,nδt), =,,,...,N,N +, n=,,,..., (3) B is expected to approximate b(x) at point Δx B = b(δx), =,,...,N (3) and φ is expected to approximate φ(x) at point Δx φ = φ(δx). (33) The discrete boundary conditions are U (n) = U (n), U(n) N+ = U (n) N, n=,,.... (34) The above conditions correspond to the Neumann condition of our target equation u x = u x=α x =. (35) x=β Condition (34) satisfies δ () U (n) = = δ () U (n) A natural discretization of the energy function is G d (U (n), B) = d =N =. (36) (δ + U (n) ) + (δ U (n) ) 3 B (U (n) ) 3 + (n) (U ) 4, 4 U ={U } N = RN+, B ={B } N = RN+. (37) Remar 4. We cannot ine δ U (n) = and δ + U (n) =N in Eq. (37). To ine δ U (n) = and δ + U (n) =N throughout this paper, we employ the following initions: U (n) = U (n), U(n) N+ = U (n) N, n=,,.... (38)

8 43 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Then the discretization of the total energy is given as follows: J d [U (n), B]= N Gd (U (n), B) Δx, U (n) ={U (n) } N = RN+, B ={B } N = RN+. (39) = Then the discrete variational derivative is calculated as in the following proposition. Proposition 5 (The discrete variational derivative for the target equation). We calculate the discrete variational derivative for the target equation by means of the discrete variational derivative method under the discrete Neumann condition. Then, we have δg d δ(u (n+), U (n) = d, B) δ() + 4 (n+) (U {(U (n+) + U (n) ) 3 + (U (n+) ) U (n) ) 3 B {(U (n+) ) + U (n+) + U (n+) (U (n) U (n) + (U (n) ) } ) + (U (n) ) 3 }. (4) Proof. According to the inition of (), we obtain the discrete variational derivative (4) An implicit scheme for target equation From Proposition 4, we obtain the implicit finite difference scheme for (7): U (n+) U (n) Δt δg d = δ(u (n+), U (n),, B) where δg d δ(u (n+), U (n) = d, B) δ() δ () U (n) =, =,N, (n+) (U + U (n) + (n+) {(U ) 3 + (U (n+) ) U (n) 4 =,...,N, ) 3 B {(U (n+) ) + U (n+) + U (n+) (U (n) U (n) ) + (U (n) ) 3 }, + (U (n) ) } U () = φ, =,...,N. (4) The next theorem is the energy dissipation property for the discretized energy function. Theorem 6 (Discrete energy-dissipated property). Let U (n) be a solution of (4). Then the total energy J d (U (n) ) is dissipation with respect to the time step n. Namely, Δt {J d[u (n+), B] J d [U (n), B]} = N {Gd (U (n+), B) Δt G d (U (n), B) }Δx, = U (n) ={U (n) } N = RN+, B ={B } N = RN+. (4)

9 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Proof. N N {Gd (U (n+), B) Δt G d (U (n) δg d (U (n+) U (n) ), B) }Δx = δ(u (n+), U (n) Δx, B) = = Δt N ( ) δg d = δ(u (n+), U (n) Δx., B) 4.3. Stability of the implicit finite difference scheme We prove the stability for the implicit finite difference (4) by the discrete Sobolev lemma. We ine the discrete Sobolev norm as f d (,) = N (f ) Δx + = = N (δ + f ) + (δ f ) Δx, = f = (f ) N++l = l R N++l ; l Z. (43) Then we have the following lemma [9]. Lemma 7 (Discrete Sobolev lemma). ( max f max, ) L f d (,), (44) N L where L = b a. We recall the discrete energy dissipation property for the implicit finite difference scheme (4). The discrete dissipation property is as follows (4): N Gd (U (n+), B) Δx = N Gd (U (n), B) Δx. = Lemma 8. We have the following inequality for the finite-difference scheme (4): where [ U (n) d (,) N Gd min(, d ) (U (), B) Δx + L + 4 = N = D Δx ], (45) D = B 4 + B + B (B B + 8) + 8. (46)

10 434 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Proof. From the energy-dissipated property (4), we have the following inequality: N Gd (U (), B) Δx = = N Gd (U (n), B) Δx = N = N = { { 4 B 3 (n) (U ) 3 + (n) (U ) 4 + d 4 (U (n) ) + d (δ + U (n) ) + (δ U (n) ) (δ + U (n) ) + (δ U (n) } ) Δx {B 4 + B B (B + 8) B + 8 } } Δx ( since 4 x4 p 3 x3 x { } ) p 4 + p p(p + 8) p 4 + 8,p ( min, d ) N U (n) d (,) Then we have following inequality: = 4 {B 4 + B B (B + 8) B + 8 } Δx. U (n) d (,) min(, d ) N + 4 = [ N Gd (U (), B) Δx = {B 4 + B B (B + 8) B + 8 } Δx From these two lemmas, we have the following theorem. ]. Theorem 9 (Stability of the finite difference (4)). We have the following inequality which indicates the stability of the finite-difference scheme (4). max N U (n) max(l, L ) [ N Gd min(, d ) (U (), B) Δx + L + N 4 = = D Δx ]. (47)

11 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Existence and uniqueness of the solution for the implicit finite difference scheme The purpose of this subsection is to show the existence and uniqueness of the solution for the proposed (4). The existence of the solution for the finite-difference scheme means that we can compute successively nth solution U (n) for n =,,...for a given initial value {U () ; =,,..., N}. Therefore, it suffices to show the existence of the sequence {U (n+) ; =,,...,N} satisfying (4). We rewrite the finitedifference scheme. Firstly, we put v = U (n). (48) Then (4) is reduced to (n+) (U v ) = d Δt δ() where Q d (U (n+),v ) (n+) (U + v ) + Q d (U (n+),v ), (49) = B (n+) {(U ) + U (n+) v + (v ) } 3 (n+) {(U ) 3 + (U (n+) ) v + U (n+) v 4 + v3 }. (5) In order to prove the existence of the solution, we rewrite (49) as follows: ( Δt d ) ( δ() U (n+) = Δt + d ) δ() v + Q d (U (n+),v ). (5) Note that (5) is expressed as follows: ( Δt I d D ) U = where U, v R N+ are ined by ( Δt I + d D ) v + Q d (U, v), (5) U = (U,U,...,U N ), (53) v = (v,v,...,v N ) (54) and I R N+ is the unit matrix and D is D = (Δx) , (55) under the boundary condition as in (4).

12 436 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) In (5), we replace the right-hand side of U (n+) ) ( Δt I d D where u R N+ denotes by u ( =,,...,N). Then we obtain ( U = Δt I + d ) D v + Q d (u, v), (56) u = (u,u,...,u N ). (57) Now let T v : u U be a mapping ined by U = ( Δt I d D ) {( Δt I + d D )v + Q d (u, v). Our problem is thus reduced to show that T v has a unique fixed point, i.e. a solution of (5). We can prove the following theorems for the finite-difference (4). Let K be ined by where K ={u R N+ u 4M}, (58) M = v, (59) v = N (v ). (6) = Furthermore, we assume that B is bounded. We ine c as c = max B, =,,...,N. (6) Then we have the following theorem. Theorem (Existence and uniqueness of the solution to (4)). If Δt M (7c + 57M), (6) then the mapping T v : K K, where T v is ined above has a unique fixed point in the closed ball K. Remar. Eq. (6) is independent of Δx. Proof. We have to show that T v has a fixed point under these two conditions. () T v is well-ined for any v. () T v satisfies three conditions of the Brouwer fixed-point theorem. (B) K( R N+ ) is a bounded convex set. (B) T v is mapping K K. (B3) T v is continuous.

13 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Firstly, we prove (). The matrix expression of mapping T v is ( Δt I d ) ( D U = Δt I + d ) D v + Q d (u, v), where matrix D is denoted by (55), whose eigenvectors are ( ) cos N π ( ) cos N π ( ) cos x = N π,hsip.em =,...,N, (63). ( ) cos π (N ) N ( ) cos N π (N) where corresponding eigenvalues are λ = { ( ) } (Δx) cos N π, =,...,N, (64) i.e. D x = λ x, =,,...,N. (65) Therefore, we obtain the inequality for the eigenvalues of Δt I D Δt d { ( ) } (Δx) cos N π Δt (66) which implies that Δt I d D is invertible. Therefore, the mapping T v is well-ined. Secondly, we prove (). (B) is trivial. Let us prove (B). We diagonalize the matrix D by using the eigenvectors of (63). Then where D = XΛX, (67) X = (x, x,...,x N ), (68) Λ = diag(λ ). (69)

14 438 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Then the matrix expression of proposed scheme is given by ( U = X Δt I d ) ( Λ Δt I + d ) ( Λ X v + X Δt I d ) Λ X Q d (u, v) (7) from which it follows that ( U X Δt I d ) ( Λ Δt I + d ) Λ X v ( + X Δt I d ) Λ X Q d (u, v), (7) where the matrix norm is ined by Ax A = sup, x = x x R N+, (7) and thus diag(λ ) = max λ. (73) By using (73), we can estimate (7) as follows: Δt U X max λ Δt d λ X v + X max Δt d λ X Q d (u, v). (74) Here we note that Δt max λ Δt d λ, (75) max Δt d λ Δt. (76) Maing use of these two inequalities (75) and (76), we have U X X v + X Δt X Q d (u, v). (77) From assumption (59), we obtain U X X M + X Δt X Q d (u, v). (78) Now we can estimate Q d (u, v) by assumption (58). We divide Q d (u, v) as follows: Q d (u, v) = Q d (u, v) + Q d (u, v), (79)

15 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) where we ine Q d (u, v) and Q d (u, v) as Q d (u, v) = B 3 (u + u v + v ), (8) Q d (u, v) = 4 (u3 + u v + u v + u3 ). (8) We can estimate Q d (u, v) and Q d (u, v) as follows: Q d (u, v) 7M c, (8) 493 Q d (u, v) 8 M3. (83) From (8) and (83) Q d (u, v) Q d (u, v) + Q d (u, v) ( ) M c + 8 M. (84) We substitute (84) into (78). Then we have ( ) U X X M + X Δt X M c + 8 M. (85) Now we use the following estimates for X, X [7]: X N, (86) X N. (87) Maing use of these two inequalities (86) and (87), we can compute { ( )} U + ΔtM c + 8 M M. (88) Therefore we see that T v is a mapping K K, ifδt satisfies Δt ( ), (89) M 493 7c + 8 M because of the inition of K (58). And thus we have proved the property (B). Thirdly, we prove the continuity of the mapping T v. This is the property of (B3).

16 44 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) We put U = T v u, U = T v u for two vectors u and u whose components are given, respectively, by U = (U,U,...,U N ), (9) U = (U,U,...,U N ), (9) u = (u,u,...,u N ), (9) u = (u,u,...,u N ). (93) Similar to the estimate for (B), we have U U Δt Q d (u, v) Q d (u, v) Δt( Q d (u, v) Q d (u, v) + Q d (u, v) Q d (u, v) ). (94) We can estimate Q d (u, v) Q d (u, v) and Q d (u, v) Q d (u, v) as follows: Q d (u, v) Q d (u, v) 3cM u u, (95) Q d (u, v) Q d (u, v) 57M u u. (96) 4 From these inequalities (95) and (96), we obtain ( U U 6M c + 9 ) 4 M Δt u u, (97) which means that the mapping T v is continuous. We have proved the property of (B3). Finally, we prove the uniqueness of the solution of the proposed implicit scheme. In fact, estimate (97) gives us the condition of contraction mapping, Δt < 6M ( c + 9 (98) 4 M). Comparing (89) and (98), we have arrived at the conclusion Convergence analysis for the proposed scheme (4) We show the error estimates for the numerical solution U (n) of our proposed (4). We first ine the error of the numerical solution as e (n) = U (n) u(δx,nδt), =,,,...,N,N +, n=,,,..., (99) where u(x, t) is the solution of (7). As an extension of u(x, t), we ine u( Δx,t) = u(δx,t), () u((n + )Δx,t) = u((n )Δx,t). ()

17 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) In this paper, we measure the error estimate discrete L -norm ined by f d (,) = N (f ) Δx, f = (f ) N+l = l RN++l ; l Z. () = Furthermore, we ine time difference operators as follows: δ n f (n) = f (n+ ) f (n ), (3) Δt δ n f (n) = f (n+ ) f (n) + f (n ) (, (4) Δt) s n f (n) = f (n+ ) + f (n ). (5) Lemma. We have the following inequality for the error e (n) = (e (n) )N+ =. { } e (n+) d (,) Δt e(n) d (,) { } e (n+) d (,) + e(n) d (,) + + ζ(n+ ) where + 4 (B ; U (n+) ; U (n) ) (n+ ) (B ; U (n+) ; U (n) ) = B ( U (n+) ζ(n+ ) d (,) ) + U (n+) 3 d (,) d (,) + 4 (n+) ψ(u ; U (n) ) ψ (n+ ) U (n) d (,), (6) + (U (n) ), (7) (n+ ) = b(x)u (x,t)=(δx,(n+ )Δt), (8) ψ(u (n+) ; U (n) ) = (U (n+) ) 3 + (U (n+) ) U (n) 4 + U (n+) (U (n) ) + (U (n) ) 3, (9) ψ (n+ ) = u 3 (x,t)=(δx,(n+ )Δt), () ζ (n+ ), = {( ζ (n+ ), = d {( ) t δ n u}, () (x,t)=(δx,(n+ )Δt) δ () s n x ) u} (x,t)=(δx,(n+ )Δt). ()

18 44 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Proof. e (n+) Then we have e (n+) e (n) Δt e (n) Δt We ine F (n+ ) = (n+) (U Δt = U (n) ) {u(δx,(n + )Δt) u(δx,nδt)} Δt δg d δ(u (n+), U (n), B) δ n u (x,t)=(δx,(n+ )Δt). δg d = δ(u (n+), U (n) + δg, B) δu + u ( ( Δx, n + ) Δt t as F (n+ ) δg d = δ(u (n+), U (n) + δg, B) δu From the above inition and (), we obtain e (n+) e (n) Δt ( ( Δx, n + ) ) Δt,b(x) ) δ n u (x,t)=(δx,(n+ )Δt). (3) ( ( Δx, n + ) ) Δt,b(x). (4) = F (n+ ) + ζ (n+ ),. (5) We recall the variational derivative of the target equation and the discrete variational derivative (4); we rewrite F (n+ ) as F (n+ ) Furthermore, we ine = (B ; U (n+) ; U (n) ) (n+ ) + ψ (n+ ) + dδ () e (n+) + e (n) G (n+ ) = (B ; U (n+) ; U (n) ) (n+ ) Then we have F (n+ ) = G (n+ ) = + dδ () From (5) and (8), we have ( ) N (n+) e = e (n) Δt e (n+) (e (n+) + e (n) + e (n) )Δx N { G (n+ ) + ζ (n+ ), + ζ (n+ ), + d δ() = (n+) ψ(u ; U (n) ) + ζ (n+ ),. (6) (n+) ψ(u ; U (n) ) + ψ (n+ ). (7) + ζ (n+ ),. (8) (e(n+) } + e (n) ) (e (n+) + e (n) )Δx. (9)

19 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Then we have { } e (n+) d (,) Δt e(n) d (,) N (n+ = (G ) + ζ (n+ ), + ζ (n+ ), )(e (n+) + e (n) )Δx = N { } d + δ() (e(n+) + e (n) ) (e (n+) + e (n) )Δx = N (n+ = (G ) + ζ (n+ ), + ζ (n+ ), )(e (n+) + e (n) )Δx = d N {δ + (e(n+) + e (n) )} +{δ (e(n+) + e (n) )} Δx = N (n+ (G ) + ζ (n+ ), + ζ (n+ ), )(e (n+) + e (n) )Δx = N (n+) {(e ) + (e (n) ) }Δx = N (n+ N + (G ) ) (n+ N Δx + (ζ ) ) (n+ Δx + (ζ ) ) Δx. = = = Therefore we arrived at (6). Lemma 3. where (B ; U (n+) ; U (n) ) (n+ ) d (,) C C { e(n+) d (,) + e(n) d (,) } { C = max l n+ max N + C ζ(n+ ) (l) U 3 d (,) + 4C C ζ(n+ ) 4 d (,), (), sup x [a,b] } u(x, lδt), () { } C = max max B, sup b(x), () N x [a,b]

20 444 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) ζ (n+ ) 3, ={u(δx,(n + )Δt) u(δx,nδt)}, (3) ( ( ζ (n+ ) 4, = (s n )u Δx, n + ) ) Δt. (4) Proof. Let be 4 = I m,, (5) m= where I, = (B ; U (n+) ; U (n) ) (B ; u(δx,(n + )Δt); U (n) ), (6) I, = (B ; u(δx,(n + )Δt); U (n) ) (B ; u(δx,(n + )Δt); u(δx,nδt)), (7) I 3, = (B ; u(δx,(n+)δt); u(δx,nδt)) I 4, = Now, we can easily see that ( u(δx,(n + )Δt) + u(δx,nδt) ( u(δx,(n+)δt)+u(δx,nδt) ), (8) ) ( ( ( u Δx, n + ) )) Δt. (9) I, C C e (n+), (3) I, C C e (n), (3) I 3, C {u(δx,(n + )Δt) u(δx,nδt)}, (3) I 4, C C (s n )u(δx,(n + )Δt). (33) Then we have the following inequalities: I d (,) C C e(n+) d (,), (34) I d (,) C C e(n+) d (,), (35) I 3 d (,) C {u(,(n+ )Δt) u(,nδt)} d (,), (36) I 4 d (,) 4C C (s n )u(,nδt) d (,). (37)

21 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Substituting these inequalities into 4 (B ; U (n+) ; U (n) ) (n+ ) d (,) I m, d (,), (38) m= we obtain (). Lemma 4. ( ψ U (n+) Proof. Let ψ ψ be where ψ ψ = I, I, I 3, I 4, ) ; U (n) ψ (n+ ) d (,) 9 { e 4 C4 (n+) e + (n) d (,) + C 6 ζ(n+ ) 3 d (,) + 9C d (,) } ζ(n+ ) 4 d (,). (39) 4 I m,, (4) m= (n+) = ψ(u ; U (n) (n) ) ψ(u(δx,(n + )Δt); U ), (4) = ψ(u(δx,(n + )Δt); U (n) ) ψ(u(δx,(n + )Δt); u(δx,nδt)), (4) ( ) u(δx,(n + )Δt) + u(δx,nδt) = ψ(u(δx,(n + )Δt); u(δx,nδt)) ψ, (43) = ψ Now we can easily see that ( u(δx,(n + )Δt) + u(δx,nδt) ) ( ( ( ψ u Δx, n + ) )) Δt. (44) I, 3 C e(n+), (45) I, 3 C e(n), (46) I 3, C 4 {u(δx,(n + )Δt) u(δx,nδt)}, (47)

22 446 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) I 4, 3C (s n )u(δx,(n + )Δt). (48) Then we have the following inequalities: I d (,) 9 4 C4 e(n+) d (,), (49) I d (,) 9 4 C4 e(n+) d (,), (5) I 3 d (,) C 6 {u(,(n+ )Δt) u(,nδt)} d (,), (5) I 4 d (,) 9C (s n )u(,nδt) d (,). (5) Substituting these inequalities into 4 (n+) ψ(u ; U (n) ) ψ (n+ ) d (,) I m, d (,), (53) m= we obtain (39). Lemma 5. { Δt( + 4C C + 9C4 )} e(n+) d (,) e (n) d (,) + Δt{ ζ(n+ ) d (,) + ζ(n+ ) d (,) + ( 36 C + 4 C ) ζ(n+ ) 3 d (,) + (6C C + 36C ) ζ(n+ ) 4 d (,) }. (54) Proof. From (6), () and (39), we can easily verify the following inequality: { } e (n+) d (,) Δt e(n) d (,) } ( + 4C C + 9C4 { e ) (n+) d (,) + e(n) d (,) + ζ(n+ ) d (,) + + ( 36 C + 4 ) C ζ (n+ ) 3 ζ(n+ ) d (,) d (,) + (6C C + 36C ) ζ (n+ ) 4 d (,).

23 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Then we obtain { e (n+) } e Δt (n) d (,) d (,) ( + 4C e C + 9C4 ) (n+) + d (,) + ( 36 C + 4 ) C ζ (n+ ) 3 Therefore (54) is calculated. Corollary 6. If d (,) ζ(n+ ) d (,) + + ( 6C C + ) 36C ζ (n+ ) 4 ζ(n+ ) d (,) d (,) Δt < ( + 4C C + (55) 9C4 ), we have the following inequalities: n { e (n) d (,) Δt (C 3 ) l ζ (n+ ) d (,) + ζ(n+ ) d (,) where C 3 = Proof. Let + l= ( 36 C + 4 C ) } ζ (n+ ) 3 d (,) + (6C C + 36C ) ζ(n+ ) 4 d (,), (56) Δt( + 4C C + (57) 9C4 ). α = ζ (n+ ) d (,) + ζ(n+ ) d (,) + ( 36 C + 4 C ) ζ(n+ ) 3 d (,) + (6C C + 36C4 ) ζ(n+ ) 4 d (,). (58) Then we have e (n) d (,) C 3( e (n ) d (,) + Δtα) C 3 (C 3 e (n ) d (,) + C 3Δtα) + C 3 Δtα (C 3 ) n e () d (,) + Δt n (C 3 ) l α, where e () disappears since the error of the initial data is ; hence, we obtain (56). l=.

24 448 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Theorem 7. We assume that the condition of the time mesh size satisfies (55). If the solution of continuous problem (7) is sufficiently smooth (u(,t) C 3 ([,T]) for any fixed x and u(x, ) C 6 ([a,b]) for any fixed t), then the solution of the finite-difference scheme (4) is convergent for the solution in the sense of discrete L -norm where the convergent rate is O(Δx + Δt ). Proof. Expanding a Taylor series, there is a constant C 4 satisfying the following inequalities: ζ(n+ ) C 4 (Δt + Δx ), (59) d (,) ζ(n+ ) d (,) ζ(n+ ) 3 d (,) ζ(n+ ) 4 d (,) C 4 (Δt + Δx ), (6) C 4 (Δt + Δx ), (6) C 4 (Δt + Δx ). (6) By using these inequalities, there is a constant C 5 satisfying e (n) d (,) n C 5 L(Δt + Δx ). (63) l= From the above inequality and (56), we obtain e (n) d (,) C5 LT exp[{ + ( + 4C C + 9C4 )}T ](Δt + Δx ). (64) 5. Numerical simulation Let us show the numerical experiments of the finite difference (4). In the numerical experiments, we compute (4) by the Newton method in the domain {(x, t) x, t 3} at diffusion coefficient d =.. We ine the time mesh size Δt = and the space mesh size Δx = 6, respectively. In this paper, we consider two cases of variable coefficients of b(x) and three different initial values. Case :, x 6, 4 b(x) =, 6 x 6, (65) 4, 6 x.

25 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Fig.. Initial data Fig.. Time step n = () φ(x) =.5(. cos(6πx))/. (66) () φ(x) = {.4, x 6,, 6 x. (67)

26 45 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Energy Time Fig. 3. Numerical energy with time Fig. 4. Initial data. (3) φ(x) = {, x 5 6,.4, 5 6 x. (68)

27 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Fig. 5. Time step n = Energy Time Fig. 6. Numerical energy with time. Case : b(x) =,,, x 6, 5 6 x 6, 5 6 x. (69)

28 45 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Fig. 7. Initial data Fig. 8. Time step n = () φ(x) =.5(. cos(6πx))/. (7) () φ(x) = {.4, x 6 6,, 6 6 x. (7)

29 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Energy Time Fig. 9. Numerical energy with time Fig.. Initial data. (3) φ(x) = {, x 54 6,.4, 54 6 x. (7) We computed numerical solution and total numerical energy (39). We firstly focus on one example that the variable coefficient b(x) is case (65) and the initial value is (66). Fig. shows the numerical

30 454 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Fig.. Time step n = Energy Time Fig.. Numerical energy with time. solution that we obtained by choosing Fig.. A transition layer exists around the discontinuous point. The discrete total energy theoretically decreases with time (Theorem 6). Fig. 3 shows the time dependency of the total numerical energy. This figure indicates that the total numerical energy decreases and agrees with the dissipation property (Figs. 3). Furthermore, we computed five other examples (Figs. 4 9). We can observe a transition layer around the discontinuous point in each cases. Figs. 6, 9,, 5 and 8 show

31 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Fig. 3. Initial data Fig. 4. Time step n = the time dependency of the total numerical energy. These figures indicate that the total numerical energy decreases and agrees with the dissipation property. Illustrations of numerical experiments of Case with initial value, and 3 are shown in Figs. 8.

32 456 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Energy Time Fig. 5. Numerical energy with time Fig. 6. Initial data. 6. Conclusion The validity of the discrete variational derivative method was shown concretely for a partial differential equation with variable coefficients. We proposed the implicit finite-difference scheme-preserving energy dissipation property for our target equation. This finite-difference scheme is implicit but stable in terms of

33 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Fig. 7. Time step n = Energy Time Fig. 8. Numerical energy with time. the discrete energy dissipation property. Furthermore, this finite-difference scheme has a unique solution and the convergence rate is O(Δx + Δt ) in the sense of the discrete L -norm. Acnowledgements The authors would lie to than Professors Daisue Furihata, Taayasu Matsuo, Tetsuya Ishiwata and Toshihide Ueno for their helpful suggestions and valuable advice. And the authors would also lie to than

34 458 T. Ide, M. Oada / Journal of Computational and Applied Mathematics 94 (6) Professor Kazuhiro Kurata for painting out this problem. The first author would lie to than Kazunaga Tanaa for simulating discussion and informing paper [6]. The first author also gives thans to Professors Peter J. Olver, Robert I. McLachlan, Brynjulf Owren, Hans Munthe Kaas, Elena Celledoni and AISIN AW Co., Ltd. This wor is supported by a Grant-in-Aid for Scientific Research (No. 3756) Ministry of Education, Culture, Science, and Technology, Japan. We especially than the referees for their detailed suggestions to improve the readability of the paper. References [] D.G. Aronson, H.F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve pulse propagation, Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, vol. 446, pp [] R.S. Cantrell, C. Conser, Diffusive logistic equations with ininite weights: population model in disrupted environments, Proc. Roy. Soc. Edinburgh Section A (3 4) (989) [3] A.J. Chorin, M.F. McCracen, T.J.R. Hughes, J.E. Marsden, Product formulas and numerical algorithms, Comm. Pure Appl. Math. 3 () (978) [4] Q. Du, R.A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 8 (5) (99) 3 3. [5] Z. Fei, L. Vázquez, Two energy conserving numerical schemes for the sine-gordon equation, Appl. Math. Comput. 45 (99) 7 3. [6] D.A. French, J.W. Schaeffer, Continuous finite element methods which preserve energy properties for nonlinear problems, Appl. Math. Comput. 39 (3) (99) [7] D. Furihata, General derivation of finite difference schemes by means of a discrete variation, Doctoral Thesis, University of Toyo, 997 (in Japanese). [8] D. Furihata, Finite difference schemes u ( ) α δg x δu that inherit energy conservation or dissipation property, J. Comput. t = Phys. 56 (999) 8 5. [9] D. Furihata, A stable and conservative finite difference scheme for the Cahn Hilliard equation, Numer. Math. 87 () [] D. Greenspan, Conservative numerical methods for ẍ = f(x), J. Comput. Phys. 4 (984) 8 4. [] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration Structure Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin,. [] C. Hirota, T. Ide, M. Oada, N. Fuuoa, Generalized energy integrals and energy conserving numerical schemes for partial differential equations, Japan J. Indust. Appl. Math. (4) [3] T. Ide, C. Hirota, M. Oada, Generalized energy integral for u t = δg δu, its finite difference schemes by means of the discrete variational method and an application to Fujita problem, Adv. Math. Sci. Appl. () () [4] A. Iserles, R.I. McLachlan, A. Zanna, Approximately preserving symmetries in the numerical integration of ordinary differential equations, European J. Appl. Math. (5) (999) [5] Y. Ishimori, Explicit energy conservative difference schemes for nonlinear dynamical systems with at most quadratic potentials, Phys. Lett. A 9 (5 6) (994) [6] S. Jimenez, P. Pascual, C. Aguirre, L. Vázquez, A panoramic view of some perturbed nonlinear wave equation, Internat. J. Bifurcation and Chaos 4 () (4) 4. [7] S. Li, L. Vu-Quoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein Gordon equation, SIAM J. Numer. Anal. 3 (995) [8] S. Maeda, Symplectic Runge Kutta methods from the viewpoint of symmetry, J. Math. Toushima Univ. 34 () 5. [9] T. Matsuo, Discrete variational method: its various extension and applications, Doctoral Thesis, University of Toyo,. [] T. Matsuo, D. Furihata, Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys. 7 () [] T. Matsuo, D. Furihata, M. Sugihara, M. Mori, Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method, Japan J. Indust. Appl. Math. 9 (3) () 3 33.

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