WANG JUE

(College of Science,Harbin Engineering University,Harbin,150001)

Dynamics and Long Time Convergence of the Extended Fisher-Kolmogorov Equation under Numerical Discretization*

WANG JUE

(College of Science,Harbin Engineering University,Harbin,150001)

Communicated by Ma Fu-ming

We present a numerical study of the long time behavior of approximation solution to the Extended Fisher–Kolmogorov equation with periodic boundary conditions.The unique solvability of numerical solution is shown.It is proved that there exists a global attractor of the discrete dynamical system.Furthermore,we obtain the long-time stability and convergence of the dif f erence scheme and the upper semicontinuity d(Ah,τ,A)→0.Our results show that the dif f erence scheme can ef f ectively simulate the inf i nite dimensional dynamical systems.

Extended Fisher–Kolmogorov equation,f i nite dif f erence method,global attractor,long time stability and convergence

1 Introduction

The Extended Fisher-Kolmogorov(EFK)equation is given by

with the boundary condition

and the initial condition

where β＞0,0＜L＜+∞and Ω=(0,L)is a bounded domain in R with boundary?Ω, and u0is a given L-periodic function.

When β=0 in(1.1),the standard Fisher-Kolmogorov equation was obtained(see[1–2]).Adding a stabilizing fourth order derivative term to the standard Fisher-Kolmogorovequation,the equation(1.1)is proposed and called as Extended Fisher-Kolmogorov equation (see[3–6]).

The equation(1.1)occurs in a variety of applications such as pattern formation in bistable systems,propagation of domain walls in liquid crystals,travelling waves in reaction di ff usion systems and mezoscopic model of a phase transition in a binary system near the Lipschitz point(see[4,7–9]).In particular,in the phase transitions near critical points (Lipschitz points),the higher order gradient terms in the free energy functional can no longer be neglected and the fourth order derivative becomes important.

There have been a number of papers in the literature dealing with equations similar to the equation(1.1)(see[10–13]).In recent years,attention has been focused on the connection between fi nite-dimensional dynamical system theory and the long-time behavior of solutions of a priori in fi nite-dimensional dynamical systems described by partial di ff erential equations. In particular,the techniques have been developed to establish this connection in a rigorous and quantitative way by showing how the dimension of the global attractor may be estimated for some dissipative partial di ff erential equations(see[14–17]).The long-time behavior of the solutions to(1.1)is studied theoretically in[18].

For the long-time computation of partial di ff erential equations,the error estimate are important in both space and time directions.Simo and Armero[19]pointed out that the fi rst order scheme with long time stability and convergence is more e ff ective than the second order scheme.Recently,some useful results about equivalence of equi-attraction and continuous convergence of attractors in di ff erent spaces have been given in[20].To compute a trajectory numerically,long-time computation generally su ff ers from error accumulation at the unavoidable exponential rates.A numerical trajectory eventually leaves the exact trajectory and no longer shows any information about the original trajectory.On the other hand,for dissipative system such as the EFK equation,if the discretization schemes are appropriately selected,the numerical trajectory is expected to approach a discrete attractor and it eventually enters and stays in a small neighborhood of the attractor.

For this reason,we consider the error estimates for a global attractor.Existence of attractors for the dissipative systems is proved.The remainder of this paper is organized as follows.In Section 2,we describe a new fi nite di ff erence scheme for the EFK equation and prove that the di ff erence scheme is uniquely solvable.In Section 3,we derive the priori error estimates for numerical solution to obtain the existence of a global attractor.In Section 4, we discuss the long time stability and convergence of the di ff erence scheme and the upper semicontinuity d(Ah,τ,A)→0.

2 Finite Dif f erence Scheme and Unique Solvability of Dif f erence Approximation

Let h=L/J be the uniform step size in the spatial direction for a positive integer J.Let τ denote the uniform step size in the temporal direction.Denote=V(xi,tn)for tn=nτ,n=0,1,···and

We de fi ne the di ff erence operator for a function Vi∈,respectively,as

Furthermore,we de fi ne operator?tVnas2

We now introduce the discrete L-inner product and the associated norm by

The discrete Hk-seminorm|·|k,h,Hk-norm ‖·‖k,hand L∞-norm ‖·‖∞,hare de fi ned, respectively,as

Let Ωh={ih;0≤i≤J}.It is convenient to let(Ωh)and(Ωh)(k≥1)denote the normed vector spaces,respectively,as

Thanks to the periodicity of the discrete function V∈(Ωh),we have

Throughout this paper,we denote ci＞0 as a generic constant independent of step sizes h.To obtain some important results,we introduce the following lemmas.

Lemma 2.1[21]For V,W ∈,there holds

Lemma 2.2[21]For V∈(Ωh),we have

and

for 0≤k≤n,where K1and K2are constants independent of h and the discrete function V.

Lemma 2.3[21]For a function V∈(Ωh),the following inequality holds:

According to Lemma 2.3 and(2.1),we have a lemma as follows.

Lemma 2.5 For s∈R,there hold

We propose a new di ff erence scheme for the solution of the problem(1.1)–(1.3)as follows:

Below,we prove the solvability of the discrete system(2.5)–(2.7).

Theorem 2.1 The di ff erence scheme(2.5)–(2.7)is uniquely solvable.

where 0≤λ≤1.It de fi nes a mapping φ=Tλ(φ)of(Ωh)into itself.Obviously,the mapping Tλ(φ)is continuous for any φ∈(Ωh).Since a di ff erence solution is a fi xed point of T1,it only needs to prove the existence of the fi xed point of T1,i.e.,it is sufficient to prove the uniform boundedness for the mapping Tλwith respect to the parameter 0≤λ≤1 by the Leray-Schauder fi xed point theorem.Taking an inner product of(2.8)with φ and using Lemma 2.1,we obtain

By(2.3)and(2.9),we get

Let Un+1and Vn+1be the solutions of the discrete system(2.5)-(2.6)with initial conditions U0and V0,respectively.Then εn+1=Un+1-Vn+1satis fi es that

Computing the inner product of(2.10)with εn+1,and using Lemma 2.1,we obtain

It follows from(2.4),(2.13)and the mean value theorem that

3 Global Attractor of Discrete Dynamical System

In this section,we consider the existence of a global attractor for the semigroupassociated with the discrete system(2.5)–(2.7).Then the semigroupacting onfor every n≥0 is de fi ned by

To obtain the existence of a global attractor,we introduce the following lemmas.

Lemma 3.1 Suppose that u0is smooth enough.Then the solution of the di ff erence scheme(2.5),(2.6)and(2.7)is estimated as follows:

where α=2(βc1+c1·c2).Furthermore,there exists a constantsuch that the ball

is a bounded absorbing set in(Ωh)under the semigroup

Proof.Taking an inner product of(2.5)with Un+1,we have

An application of Lemma 2.3,Lemma 2.4 and(2.3)yields

Let α=2(βc1·c2+c1).Then,it follows from(3.1)that

This completes the proof.

Lemma 3.2 Suppose that u0is smooth enough.Then the solution of the di ff erence scheme(2.5),(2.6)and(2.7)is estimated as follows:

where

Furthermore,there exists a constant such that the ball

is a bounded absorbing set in(Ωh)under the semigroupProof. Taking an inner product of(2.5)withby Lemma 2.1,we have

Using Young's inequality,we obtain

It follows from Lemmas 2.4 and 3.1 that

Let Thus,(3.3)yields

Hence,we obtain from Lemma 2.4 that

This completes the proof.

According to the Lemma 3.1,Lemma 3.2 and(2.2),we have

Theorem 3.1 Assume that u0is sufficiently regular.Let Unbe the solution of the difference scheme(2.5)–(2.7).Then,there exists a positive constant C independent of h and of τ such that

Obviously,the family of operatorssatisfy the semigroup properties

By the above estimates,there exists a bounded setwhich is absorbing in(Ωh)under {(Sh.τ)}n≥0.Using Theorem 1.1 in[22],we obtain the following theorem.

Theorem 3.2 Suppose that the conditions of Lemma 3.2 are satis fi ed.Then the discrete dynamical system associated with the fi nite di ff erence scheme(2.5)–(2.7)possesses a global attractor Ah,τin(Ωh),and

4 Long Time Convergence and Stability

To prove long-time convergence,we make discussions for di ff erent cases.

(i)f is a smooth function satisfying

Then,we can obtain Proposition 4.1 as follows.

Proposition 4.1 Suppose that f is a smooth function satisfying(4.1)and the solution u(x,t)of(1.1)–(1.3)is sufficiently regular.Then,the solution of the di ff erence scheme (2.5),(2.6)and(2.7)converges to the solution of the problem(1.1)–(1.3)in the discrete(Ωh)-norm and the rate of convergence is O(τ+h2).

Proof. Making use of Taylor's expansion,we fi nd

and the constant M is independent of τ and h.

Subtracting(4.2)from(2.5),we fi nd

Taking in(4.6)the inner product with En+1and using Lemma 2.1,we obtain

An application of the mean value theorem and(4.1)yield

Using Young's inequality,Lemmas 2.3 and 2.4,we obtain

Then,we get

Combining(4.5),(4.8)and(4.10),we fi nd

which completes the proof of Proposition 4.1.

(ii)f is a smooth function satisfying

In the following Proposition 4.2,our argument is based on some hypothesis as follows.

First,we consider the corresponding stationary problem

For any positive function ω∈C(),we consider the eigenvalue problem with weight

Letμ1[ω]denote its smallest eigenvalue.Recall thatμ1[ω]can be characterized as

We assume that(4.13)-(4.14)has a classical solution u,which is linearized stable in the sense that,for some real number δ,

In order to show the convergence of the fi nite di ff erence approximate solutions,the following lemma is needed.

Lemma 4.1 Suppose that f is a smooth function satisfying(4.12)and the solution u(x,t) of(1.1)–(1.3)is sufficiently regular.For each B1,B2＞0 and 0＜δ＜1,assume that

(1)u is a solution of(4.13)-(4.14)with

(2)u0∈(Ω)is such that u(t)→u in(Ω)as t→∞;

(3)u0and the corresponding solution u of(1.1)–(1.3)satisfy the bounds

Then,there is a number δ1such that

where

We are now ready for the proof of Proposition 4.2.

Proposition 4.2 Let N∈Z+.Under the assumptions of Lemma 4.2,for τ sufficiently small,the solution of the di ff erence scheme(2.5),(2.6)and(2.7)converges to the solution of the problem(1.1)–(1.3)in the discrete(Ωh)-norm and the rate of convergence is O(τ+h2).

Proof. An application of the mean value theorem and Theorem 3.3,(4.9)yields2

Let κ=6C+3.Then(4.5),(4.8)and(4.18)yield

For 0＜n≤N,it follows from(4.19)that

We now turn to the long-time estimate.Using(4.9)and Lemma 4.2,we obtain

Using Young's inequality,from Lemmas 2.3 and 2.4 we obtain

It implies that

Combining(4.8)and(4.22),we fi nd

Thus from(4.23),we get

Combining(4.20)and(4.23),we fi nd

This completes the proof.

Theorem 4.1 Under the conditions of the Proposition 4.1 or Proposition 4.2,the solution of the di ff erence scheme(2.5),(2.6)and(2.7)converges to the solution of the problem(1.1)–(1.3)in the discrete(Ωh)-norm and the rate of convergence is O(τ+h2).

Below,we can similarly prove stability of the di ff erence solution.

Theorem 4.2 Under the conditions of the Theorem 4.1,the solution of the problem(2.5), (2.6)and(2.7)is long-time stable for initial data in the discrete(Ωh)-norm.

Now,we discuss the upper semiconsciousness of approximate attractor Ah,τ.

Theorem 4.3 Suppose that the following conditions are satis fi ed:

(1)is a family of closed subspaces of a Banach space H,satisfying thatis dense in H;

(2){Sη(t):Hη→Hη}t≥0are linear semi-group of operator,Aη?Hηand A?H are the global attractors of Sη(t)and S(t),respectively;

(3)For every compact interval I?(0,+∞),

Then Aηconverges to A in the sense of semi-distance:

where

Theorem 4.4 Suppose that the conditions of Theorem 4.1 are satisf i ed.Then we have dist(Ah,τ,A)→0, as τ→0,h→0.

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tion:65M06,37L99,35B40,35Q55,65M20

A

1674-5647(2013)01-0051-10

*Received date:Aug.24,2010.

The NSF(10871055)of China and the Fundamental Research Funds(HEUCFL20111102) for the Central Universities.