Finite Element Method for a Space-Fractional Anti-Diffusive Equation
FFINITE ELEMENT METHOD FOR A SPACE-FRACTIONAL ANTI-DIFFUSIVEEQUATION
AFAF BOUHARGUANEA
BSTRACT . The numerical solution of a nonlinear and space-fractional anti-diffusive equation usedto model dune morphodynamics is considered. Spatial discretization is effected using a finite elementmethod whereas the Crank-Nicolson scheme is used for temporal discretization. The fully discretescheme is analyzed to determine stability condition and also to obtain error estimates for the approxi-mate solution. Numerical examples are presented to illustrate convergence results.
1. I
NTRODUCTION
We consider the Fowler equation [7](1.1) ∂ t u ( t, x ) + ∂ x (cid:18) u (cid:19) ( t, x ) − ∂ xx u ( t, x ) + I [ u ]( t, x ) = 0 , x ∈ R , t > , where I is a nonlocal operator defined as follows: for any Schwartz function ϕ ∈ S ( R ) and any x ∈ R ,(1.2) I [ ϕ ]( x ) := (cid:90) + ∞ | ξ | − ϕ (cid:48)(cid:48) ( x − ξ ) dξ. The Fowler equation was introduced to model the formation and dynamics of sand structures suchas dunes and ripples [7]. This equation is valid for a river flow over an erodible bottom u ( t, x ) withslow variation. Its originality resides in the nonlocal term, wich is anti-dissipative, and can been seenas a fractional Laplacian of order / . Indeed, it has been proved in [2] that F ( I [ ϕ ])( ξ ) = − π Γ( 23 ) (cid:32) − i sgn ( ξ ) √ (cid:33) | ξ | / F ( ϕ )( ξ ) , where Γ is the gamma function and F denotes the Fourier transform.Therefore, this term has a deregularizing effect on the initial data but the instabilities produced bythe nonlocal term are controled by the diffusion operator − ∂ x which ensures the existence and theuniqueness of a smooth solution. We then always assume that there exists a sufficiently regularsolution u ( t, x ) .The use of Fourier transform is a natural way to study this equation but it also can be useful toconsider the following formula:for all r > and all ϕ ∈ S ( R ) , I [ ϕ ]( x ) = I [ ϕ ]( x ) + I [ ϕ ]( x ) , (1.3) Key words and phrases.
Fractional anti-diffusive operator, finite element method, Crank-Nicolson scheme, stability,error analysis. a r X i v : . [ m a t h . NA ] A ug A. BOUHARGUANE with I [ ϕ ]( x ) = (cid:90) r | ξ | − / ϕ (cid:48)(cid:48) ( x − ξ ) dξ and I [ ϕ ]( x ) = − (cid:90) ∞ r | ξ | − / ϕ (cid:48) ( x − ξ ) dξ + ϕ (cid:48) ( x − r ) r − / . Several numerical approaches have been suggested in the literature to overcome the equations withnonlocal operator. Droniou used a general class of difference methods for fractional conservationlaws [5], Zheng and Roop proposed a finite element method to solve a space-fractional advectionequations [14], [11]. Liu proposed a numerical solution for the fractional fokkerplanck equation [9].Meerschaert studied finite difference approximations of fractional advection dispersion flow equa-tion [10]. Fix presented a least squares finite-element approximations of a fractional order differentialequation [6]. Xu applied the discontinuous Galerkin method to fractional convection diffusion equa-tions with a fractional Laplacian of order λ ∈ (1 , [13] and, recently Guan investigated stabitlityand error estimates for θ schemes for finite element discretization of the space-time fractional diffu-sion equations [8].To solve the Fowler equation (1.1) some numerical experiments have been performed using mainlyfinite difference method and split-step Fourier method [3].We propose here to use the standard Galerkin method for the space approximation and a Crank-Nicolson scheme for the time discretization, which is a more simple way to improve approximationsand to model complex geometries.For T > , L > , we seek a function u defined on R × [0 , T ] , 2L-periodic in the second variableand satisfying(1.4) ∂ t u ( t, x ) + ∂ x (cid:18) u − ∂ x u + J [ u ] (cid:19) ( t, x ) = 0 , x ∈ R , t ∈ (0 , T ) ,u (0 , x ) = u ( x ) , x ∈ R , where u is a given 2L-periodic function and(1.5) J [ ϕ ]( x ) := (cid:90) + ∞ | ξ | − ϕ (cid:48) ( x − ξ ) dξ. To prove the convergence of the numerical scheme we use the standard material on the finite elementmethod for parabolic problems [12]. However, the analysis of the variational solution to the Fowlerequation is more complicated than the usual parabolic equations because the fractional differentialoperator is not local and is anti-diffusive.In this paper we analyze the discretization of (1.4) by a Crank-Nicolson method in time combinedwith the standard Garlerkin-finite element method in space. Our main result consists in prove thefollowing error estimate: || u ( t n , · ) − U n || ≤ C (∆ t + h k ) , where k is the optimal spatial rate of convergence in L , ∆ t = T /N is the time step, t n = n ∆ t, n =0 , · · · , N and h is the spatial discretization. U , · · · , U N are the approximations of the solutions atdifferent times. EM FOR THE FOWLER EQUATION 3
We also prove that our numerical scheme is stable if the following condition is satisfied: C ∆ th + C ∆ th / ≤ , where C , C are two positive constants independent of ∆ t and h .It is clear that our analysis can easily be extended to the case where the nonlocal term I is replacedwith a Fourier multiplier homogeneous of degree λ ∈ ]1 , and not only λ = 4 / . It also canreplaced with the Riemann-Liouvillle integral. Indeed, for causal functions, our nonlocal term is, upto a multplicative constant, a Riemann-Liouville operator defined as follows: d / ϕdx ( x ) = 1Γ(2 / (cid:90) + ∞ | ξ | − / ϕ (cid:48)(cid:48) ( x − ξ ) dξ. The rest of this paper is construct as follows. In the next section we give the preliminary knowl-edge regarding the fractional operator and some technical Lemmas. We also introduce a projectionoperator and derive some error estimates which will play an important role in the sequel. The errorestimate for the Galerkin-finite element method to solve the problem (1.4) is studied in Section 3. Insection 4, we derive error estimates and prove existence and uniqueness of the fully discrete approx-imations. We also give a stability result.We finally perform some numerical experiments to confirm the theoretical results in section 5.1.1.
Notations. • We denote by C ( c , c , ... ) a generic positive constant, strictly positive, which depends onparameters c , c , · · ·• For m ∈ N , let H mper be the periodic Sobolev space of order m , consisting of the L − periodicelements of H mloc ( R ) . We denote by || · || m the norm over a period in H mper , by || · || the normin L ( − L, L ) , and by ( · , · ) the inner product in L ( − L, L ) . • We denote by C n ( ϕ ) the Fourier coefficient of ϕ defined by: for all n ∈ Z C n ( ϕ ) = 12 L (cid:90) L − L ϕ ( x ) e − i nL x dx
2. P
RELIMINARIES
In this section, we give the variational formulation of the problem (1.4) and we introduce a pro-jection operator. We derive some estimates wich will be useful in the next sections.We shall discretize (1.4) in space by the Galerkin method. To this effect, let − L = x < x < · · · < x N = L be a partition of [ − L, L ] and h := max j ( x j +1 − x j ) .For integer r ≥ , let S rh denote a space of continuously differentiable, 2L-periodic functions ofdegree r − in which approximations to the solution u ( t, · ) (1.4) will be sought for t ∈ [0 , T ] .We assume that this family is a finite-dimensional subspaces of H per such that, for some integer r ≥ and small h ,(2.1) inf χ ∈ S rh {|| v − χ || + h ||∇ ( v − χ ) ||} ≤ Ch s || v || s , for ≤ s ≤ r, where v ∈ H sper (cf. e.g [1] and references therein ). A. BOUHARGUANE
Note that since the pratical implementation of the scheme requires to make some truncations includ-ing the integral operator J , we replace (cid:82) + ∞ with (cid:82) L in (1.5).A variational form of the problem is:(2.2) ( u t , v ) + ( uu x , v ) + ( u x , v (cid:48) ) − ( J [ u ] , v (cid:48) ) = 0 ∀ v ∈ H per , ∀ t ∈ (0 , T ) . Proposition 2.1 ( L -estimate) . Let u the solution of the variational form (2.2) . Then, for all t ∈ [0 , T ] , || u ( t, · ) || ≤ e w t || u || , where w is a positive constant.Proof. Taking v = u ( t, · ) in (2.2), we obtain by periodicity(2.3) ddt || u ( t, · ) || + ( u x − J [ u ] , u x ) = 0 . Using the Fourier analysis, we have C n ( u x ) = iπ nL C n ( u ) and since J [ u ] = ψ ∗ u x , with ψ ( x ) = x − / χ (0 , ∞ ) , then C n ( J [ u ]) = C n ( ψ ) C n ( u x ) . But since C n ( ψ ) = 12 L (cid:90) L x − / e − iπ nL x dx = 12 L / π / n − / (cid:90) πn e − iu u / du then, ( u x − J [ u ] , u x ) = + ∞ (cid:88) n = −∞ [( πnL ) − ( πnL ) / L (cid:90) πn e − iu u / du ] | C n ( u ) | ≥ + ∞ (cid:88) n = −∞ [( πnL ) − | ( πnL ) / L (cid:90) πn e − iu u / du | ] | C n ( u ) | Since (cid:90) ∞ cos( u ) u / du = 12 Γ( 23 ) , and (cid:90) ∞ sin( u ) u / du = √
32 Γ( 23 ) it follows that | L (cid:90) πn e − iu u / du | ≤ C, where C is a positive constant. Therefore by Plancherel’s formula, ( u x − J [ u ] , u x ) ≥ + ∞ (cid:88) n = −∞ [( πnL ) − ( πnL ) / C )] | C n ( u ) | ≥ − w || u ( t, · ) || , where − w = min n [( πnL ) − ( πnL ) / C ] ≤ . Finally, using (2.3), we obtain || u ( t, · ) || ≤ e w t || u || . The proof of this proposition is now complete. (cid:3)
EM FOR THE FOWLER EQUATION 5
Remark . Following the same lines as the proof of the Proposition 2.1, we have that: ∀ ν > , ∃ α > such that ( νu x − J [ u ] , u x ) ≥ − α || u || . Lemma 2.3.
Let ϕ ∈ H / per . Then (2.4) ||J [ ϕ || ≤ C || ϕ || / . Proof.
From Fourier analysis and using computations from the Proposition 2.1, we have ||J [ ϕ || = (cid:88) n | C n ( J ) | = (cid:88) n | C n ( ψ ) C n ( ϕ ) | ≤ C (cid:88) n n / | C n ( ϕ ) | , = C (cid:88) n (cid:18) n n (cid:19) / (1 + n ) / | C n ( ϕ ) | , ≤ C (cid:88) n (1 + n ) / | C n ( ϕ ) | , = C || ϕ || / . (cid:3) Lemma 2.4 (Bilinear form) . Let u, v ∈ H per . Then, it exists λ > such that the bilinear form a ( u, v ) = ( u (cid:48) , v (cid:48) ) − ( J [ u ] , v (cid:48) ) + λ ( u, v ) is continuous and coercive.Proof. Using Lemma 2.3, we can easily see that a is continuous. Let us now check the coercivity.Fom Remark 2.2, it exists α > such that for all v ∈ H per a ( v, v ) = 12 || v x || + ( 12 v x − J [ v ] , v x ) + λ || v || , ≥ || v x || + ( λ − α ) || v || , Therefore, for(2.5) λ > α ,a is coervice. (cid:3) Lemma 2.5 (Projection) . We define the projection operator P : H per → S rh by (2.6) ( v (cid:48) − ( P v ) (cid:48) , χ (cid:48) ) − ( J [ v ] − J [ P v ] , χ (cid:48) ) + λ ( v − P v, χ ) = 0 , ∀ χ ∈ S rh , where λ satisfies the condition (2.5) . Then for all ≤ s ≤ r and for all v ∈ H sper , we have || ( v − P v ) (cid:48) || ≤ Ch s − || v || s || v − P v || ≤ Ch s || v || s A. BOUHARGUANE
Proof.
1. Arguing as the proof of the Cea’s Lemma and from (2.1), we get for all v ∈ H sper (2.7) || v − P v || ≤ Ch s − || v || s . Indeed, using the bilinear form a defined in Lemma 2.4 we have a ( v − P v, v − P v ) = a ( v − P v, v − χ + χ − P v ) = a ( v − P v, v − χ ) ∀ χ ∈ S rh , and from the coecivity and continuity properties, we get C || v − P v || ≤ || ( v − P v ) (cid:48) || || ( v − χ ) (cid:48) || + ||J [ v − P v ] || || ( v − χ ) (cid:48) || + λ || v − P v || || v − χ ||≤ C || v − P v || (cid:0) || ( v − χ ) (cid:48) || + || ( v − χ ) (cid:48) || + λ || v − χ || (cid:1) Therefore, || v − P v || ≤ inf χ ∈ S h || ( v − χ ) (cid:48) || , and using finally the property of S rh (2.1), we obtain || v − P v || ≤ C h s − || v || s , ∀ v ∈ H sper .
2. To estimate || v − P v || we consider the auxiliary problem a ( ψ, ϕ ) = ( v − P v, ϕ ) . Then, for χ ∈ S rh , we have from continuity of a , assumption (2.1) and estimate (2.7) || v − P v || = a ( ψ − χ, v − P v ) ≤ C inf χ ∈ S rh || ψ − χ || || v − P v || = ˜ Ch || ψ || || v − P v || ≤ Ch s || ψ || || v || s . Now using the decomposition (1.3) of I , we get ||I [ ψ ]] ≤ r / || ψ (cid:48)(cid:48) || + C ( r ) || ψ (cid:48) || . Taking r sufficiently small and using the coercivity, we obtain the regularity estimate || ψ || ≤ C || v − P|| which yields || v − P v || ≤ Ch s || v || s , ∀ v ∈ H sper . This completes the proof of this Lemma. (cid:3)
3. D
ISCRETIZATION WITH RESPECT TO THE SPACE VARIABLE
Motivated by (2.2) we define the semidiscrete approximation u h ( t, · ) ∈ S rh , t ∈ (0 , T ) , to u by(3.1) (cid:40) ( u ht , v h ) + ( u h u hx , v (cid:48) h ) + ( u hx , v (cid:48) h ) − ( J [ u h ] , v (cid:48) h ) = 0 , ∀ v h ∈ S rh , t ∈ (0 , T ) u h (0 , x ) = u h ( x ) , where u h ∈ S rh is an approximation of u and u h is such that(3.2) || u h − u || ≤ Ch r − . The semidiscrete approximation has the following property(3.3) || u h ( t, · ) || ≤ e w t || u h || , t ∈ (0 , T ) . EM FOR THE FOWLER EQUATION 7
This inequality can be proved in the same way as Proposition 2.1. Now since S rh is finite-dimensionalwe have(3.4) max t ∈ (0 ,T ) || u h ( t, · ) || ∞ ≤ C ( h ) . Then, regarding the equation (3.1) as a system of ODE, we deduce existence and uniqueness of thesemidiscrete approximation u h . Theorem 3.1.
Let the solution u of (1.4) sufficiently smooth, and let (3.2) hold. Then (3.5) max t ∈ [0 ,T ] || u ( t, · ) − u h ( t, · ) || ≤ Ch r − , where C = C ( u ) is a positive constant.Proof. Let u − u h = u − P u + P u − u h = ρ + V , where P is the operator projection defined in (2.6). By Lemma 2.5, we have max t ∈ [0 ,T ] || ρ ( t, · ) || ≤ Ch r Thus, it remains to estimate ||V ( t, · ) || . ( V t , χ ) + a ( V , χ ) = ( P u t − ( u h ) t , χ ) + a ( P u, χ ) − a ( u h , χ )= ( P u t , χ ) + a ( P u, χ ) − (( u h ) t , χ ) − a ( u h , χ ) but since, ∀ χ ∈ S rh a ( P u, χ ) = a ( u, χ ) . then ( V t , χ ) + a ( V , χ ) = ( P u t , χ ) + a ( u, χ ) + ( u h ( u h ) x , χ ) − λ ( u h , χ )= − ( ρ t , χ ) − ( uu x − u h ( u h ) x , χ ) + λ ( u − u h , χ )= − ( ρ t , χ ) + λ ( ρ, χ ) + λ ( V , χ ) − ( uu x − u h ( u h ) x , χ ) , i.e. ( V t , χ ) + ( V x , χ (cid:48) ) − ( J [ V ] , χ (cid:48) ) = − ( ρ t , χ ) + λ ( ρ, χ ) − ( uu x − u h ( u h ) x , χ ) . Taking χ = V , we obtain ddt ||V ( t, · ) || + ||V x ( t, · ) || − ( J [ V ] , V x ) = − ( ρ t , V ) + λ ( ρ, V ) − ( uu x − u h ( u h ) x , V )= − ( ρ t , V ) + λ ( ρ, V ) − ( u ( u − u h ) x , V ) − ( u hx ( u − u h ) , V ) Therefore, we have ddt ||V ( t, · ) || + ( 12 V x − J [ V ] , V x ) + 12 ||V x ( t, · ) || ≤ || ρ t || ||V|| + λ || ρ || ||V|| + C {|| ρ || + || ρ x || + ||V|| + ||V x ||} ||V|| , ≤ ||V x || + ˜ C (cid:0) || ρ || + || ρ x || + || ρ t || + ||V|| (cid:1) . A. BOUHARGUANE
Since || ρ || ≤ Ch r , || ρ t || ≤ Ch r and || ρ x || ≤ Ch r − (see Lemma 2.5) we have ddt ||V ( t, · ) || − w ||V ( t, · ) || ≤ ˜ Ch r − + C (cid:48) ||V ( t, · ) || . Therefore, we obtain ddt ||V ( t, · ) || ≤ C h r − + C ||V ( t, · ) || , and Gronwall’s lemma yields max t ∈ [0 ,T ] ||V ( t, · ) || ≤ ch r − , which concludes the proof of this theorem. (cid:3)
4. C
RANK -N ICOLSON DISCRETIZATION
We investigate the following second-order in time fully discrete finite element method for (1.4).Let N ∈ N , ∆ t := TN and t n := n ∆ t, n = 0 , · · · , N. For u ( t, · ) ∈ L ( − L, L ) and t ∈ [0 , T ] , let U n := u ( t n , · ) , ∂U n = U n +1 − U n ∆ t , and U n +1 / := U n + U n +1 . The Crank-Nicolson approximations U n ∈ S rh to u ( t n , · ) are given by ∀ n = 0 , · · · , N − , (4.1) (cid:40) ( ∂U n , χ ) + ( U n +1 / U n +1 / x , χ ) + ( U n +1 / x , χ (cid:48) ) − ( J [ U n +1 / ] , χ (cid:48) ) = 0 , ∀ χ ∈ S rh U := u h In this section, we prove the existence of the Crank-Nicolson approximations U , · · · , U N , derivethe error estimate and show uniqueness of the Crank-Nicolson approximations. We also give astability result for this scheme.The proof of the existence of the Crank-Nicolson approximations (4.1) is based on the followingvariant of the Brouwer fixed-point theorem: Lemma 4.1 ( Browder, [4]) . Let ( H, ( · , · ) H ) be a finite-dimensional inner product space and denoteby || · || H the induced norm. Suppose that g : H → H is continuous and there exists an α > suchthat ( g ( x ) , x ) H > for all x ∈ H with || x || H = α. Then there exists x ∗ ∈ H such that g ( x ∗ ) = 0 and || x ∗ || ≤ α. Proposition 4.2 (Existence) . For ∆ t > sufficiently small, there exists a solution U n ∈ S rh satisfy-ing (4.1) .Proof. We prove the existence of U , · · · , U N by induction.Assume that U , · · · , U n , for n < N exist and let g : S rh → S rh be defined by ( g ( V ) , χ ) = 2( V − U n , χ ) + ∆ t ( V V (cid:48) , χ ) + ∆ t ( V (cid:48) , χ (cid:48) ) − ∆ t ( J [ V ] , χ (cid:48) ) , ∀ V, χ ∈ S rh . We can easily see that this mapping is continuous. Moreover, taking χ = V we have ( g ( V ) , V ) = 2( V − U n , V ) + ∆ t || V (cid:48) || − ∆ t ( J [ V ] , V (cid:48) ) , EM FOR THE FOWLER EQUATION 9 and using Remark 2.2 (which is still valable in S rh ), we obtain ( g ( V ) , V ) ≥ || V || (cid:26) (1 − α ∆ t || V || − || U n || (cid:27) , ∀ V ∈ S rh . Therefore, assuming ∆ t < α and for V = − α ∆ t U n + 1 , we obtain ( g ( V ) , V ) > . The existenceof a V ∗ ∈ S h such that g ( V ∗ ) = 0 follows from Lemma 4.1. Finally, U n +1 := 2 V ∗ − U n satisfies(4.1). (cid:3) Uniqueness is less obvious, we need first to show an error estimate to get it. We will show it afterthe main theorem.The time discretization being semi-implicit, we need a stability condition to ensure the validity ofthe computations. We then prove that the numerical process (4.1) is stable in the following sense:
Definition 4.3 (C-stability) . A numerical scheme is C-stable for the norm || · || if for all
T > , thereexists a constant K ( T ) > independent of the time and space steps ∆ t, h such that for all initialdata U (4.2) || U n || ≤ K ( T ) || U || , ∀ ≤ n ≤ T ∆ t . Proposition 4.4 (Stability ) . Under the appropriate regularity assumptions, it exists two positiveconstants C , C independent of ∆ t, h , and dependent of initial data, such that, if (4.3) C ∆ th + C ∆ th / ≤ , then the numerical scheme is C-stable.Proof. Taking χ = U n +1 in (4.1), we obtain ( U n +1 − U n ∆ t , U n +1 ) + ( U n +1 / U n +1 / x , U n +1 ) + ( U n +1 / x , U n +1 x ) − ( J [ U n +1 / ] , U n +1 x ) = 0 (4.4)But(4.5) ( U n +1 − U n , U n +1 ) = 12 || U n +1 || − || U n || + 12 || U n +1 − U n || , and − ( U n +1 / x , U n +1 x ) + ( J [ U n +1 / ] , U n +1 x ) = 12 ( U n +1 x − U nx , U n +1 x ) −
12 ( J [ U n +1 ] − J [ U n ] , U n +1 x ) − ( U n +1 x , U n +1 x ) + ( J [ U n +1 ] , U n +1 x ) ≤ || U n +1 x − U nx || + 14 ||J [ U n +1 ] − J [ U n ] || − ( 14 U n +1 x − J [ U n +1 ] , U n +1 x ) − || U n +1 x || ≤ || U n +1 x − U nx || + 14 ||J [ U n +1 ] − J [ U n ] || + α || U n +1 || − || U n +1 x || , where α > . From Lemma 2.3 and from inverse inequatity, we have(4.6) || ( u h ) x || ≤ Ch || u h || , ||J [ u h ] || ≤ Ch / || u h || ∀ u h ∈ S rh , then − ( U n +1 / x , U n +1 x ) + ( J [ U n +1 / ] , U n +1 x ) ≤ C h || U n +1 − U n || + C h / || U n +1 − U n || + α || U n +1 || − || U n +1 x || Let study now the nonlinear term. − U n +1 / U n +1 / x , U n +1 ) = (( U n +1 − U n )( U n +1 x − U nx ) , U n +1 ) − U n U nx , U n +1 )= (( U n +1 − U n )( U n +1 x − U nx ) , U n +1 ) + ( U n U n +1 x , U n )= (( U n +1 − U n )( U n +1 x − U nx ) , U n +1 ) + ( U n U n +1 x , U n − U n +1 )+(( U n − U n +1 ) U n +1 x , U n +1 ) , by the boundedness of U n +1 and U n , we obtain − ( U n +1 / U n +1 / x , U n +1 ) ≤ C || U n +1 − U n || || U n +1 x − U nx || + ˜ C || U n +1 x || || U n − U n +1 || . (4.7)Therefore, using (4.4), (4.5), (4.6) and (4.7), we get (1 − α ∆ t ) || U n +1 || − || U n || + (1 − C ∆ th − C ∆ th / ) || U n +1 − U n || ≤ C ∆ t || U n +1 − U n || . Under the condition − C ∆ th − C ∆ th / ≥ , namely C ∆ th + C ∆ th / ≤ , we have || U n +1 || ≤ (1 + C ∆ t ) || U n || ≤ e CT || U || , which shows that the numerical scheme is C-stable. (cid:3) The main result of this papier is given in the following theorem:
Theorem 4.5 (Error estimate ) . Let the solution u of (1.4) be sufficiently smooth, U , · · · , U N satisfy (4.1) and (3.2) hold. Then, for ∆ t sufficiently small, we have (4.8) max ≤ n ≤ N || u n − U n || ≤ C (∆ t + h r − ) , where C = C ( u ) is a positive constant.Proof. Let W n := P u ( t n , · ) , ρ n := u n − W n and V n := W n − U n . Then u n − U n = ρ n + V n . Using Lemma 2.5, we have max ≤ n ≤ N || ρ n || ≤ Ch r . EM FOR THE FOWLER EQUATION 11
Let us now estimate ||V n || . ( ∂ V n , χ ) + a ( V n +1 / , χ ) = ( ∂W n , χ ) + a ( W n +1 / , χ ) − ( ∂U n , χ ) − a ( U n +1 / , χ ) and since a ( W n +1 / , χ ) = a ( u n +1 / , χ ) and ( ∂U n , χ ) + a ( U n +1 / , χ ) = − ( U n +1 / U n +1 / x , χ ) + λ ( U n +1 / , χ ) then ( ∂ V n , χ ) + a ( V n +1 / , χ ) = ( ∂W n , χ ) + a ( u n +1 / , χ ) + ( U n +1 / U n +1 / x , χ ) − λ ( U n +1 / , χ )= ( ∂W n , χ ) − ( u n +1 / t , χ ) − ( u n +1 / u n +1 / x , χ ) + λ ( u n +1 / , χ ) + ( U n +1 / U n +1 / x , χ ) − λ ( U n +1 / , χ )= ( w + w + w , χ ) + λ ( ρ n +1 / , χ ) + λ ( V n +1 / , χ ) (4.9)with w := ∂W n − ∂u n , w := ∂u n − u n +1 / t and w := U n +1 / U n +1 / x − u n +1 / u n +1 / x . We havethat || w || ≤ Ch r . Let us study w . We have ∆ t w = u n +1 − u n − ∆ tu n +1 / t = 12 (cid:90) t n +1 / t n ( s − t n ) u t ( s ) ds + 12 (cid:90) t n +1 t n +1 / ( s − t n +1 ) u t ( s ) ds ≤ C ∆ t (cid:90) t n +1 t n || u t ( s ) || ds. Let us study w : Since w = U n +1 / U n +1 / x − u n +1 / u n +1 / x = U n +1 / ( U n +1 / x − u n +1 / x ) + u n +1 / x ( U n +1 / − u n +1 / ) then || w || ≤ || U n +1 / || ∞ || U n +1 / x − u n +1 / x || + || u n +1 / x || ∞ || U n +1 / − u n +1 / || But, U n +1 / x − u n +1 / x = ρ n +1 / x + V n +1 / x .Now, taking χ = V n +1 / in (4.9), we get ( ∂ V n , V n +1 / ) + ( V n +1 / x , V n +1 / x ) − ( J [ V n +1 / ] , V n +1 / ) + λ ( V n +1 / , V n +1 / ) =( w , V n +1 / ) + ( w , V n +1 / ) + ( w , V n +1 / ) + λ ( ρ n +1 / , V n +1 / ) + λ ( V n +1 / , V n +1 / ) and since ( ∂ V n , V n +1 / ) = 12∆ t ||V n +1 || − t ||V n || , then we have ||V n +1 || − ||V n || + 2∆ t (cid:16) ||V n +1 / x || − ( J [ V n +1 / ] , V n +1 / x ) + λ ||V n +1 / || (cid:17) ≤ t || w ||||V n +1 / || +2∆ t (cid:16) || w || ||V n +1 / || + || w || ||V n +1 / || (cid:17) + 2∆ t λ || ρ n +1 / || ||V n +1 / || + 2∆ tλ ||V n +1 / || . Using Lemma 2.5 and Remark 2.2 we have ||V n +1 || − ||V n || ≤ ∆ tC ( u )( h r − + ∆ t + ||V n +1 / || ) Since ||V n +1 / || = ||V n +1 || + ||V n || + 2( V n +1 , V n ) then for ∆ t sufficiently small and using thediscrete Gronwall lemma, we get max ≤ n ≤ N ||V n || ≤ c ( u )(∆ t + h r − ) , which concludes the proof. (cid:3) Remark . We return to the question of uniqueness of the solution of (4.1). We showthat this holds for ∆ t, h sufficently small when the solution of the continuous problem is smooth andwhen (3.2) holds.Let U n and V n be two solutions of (4.1) with U n − given. Letting E n := U n − V n , we obtain bysubtraction ( ∂E n , χ ) + ( E n +1 / x , X (cid:48) ) − ( J [ E n +1 / ] , X (cid:48) ) = ( E n +1 / E n +1 / x , χ ) + ( U n +1 / E n +1 / , χ (cid:48) ) ∀ χ ∈ S rh . Taking χ = E n +1 / we obtain by periodicity t ( || E n +1 || − || E n || ) + || E n +1 / x || − ( J [ E n +1 / ] , E n +1 / ) == ( U n +1 / E n +1 / x , E n +1 / x ) ≤ || U n +1 / || ∞ || E n +1 / || + 12 || E n +1 / x || ≤
12 ( || W n +1 / || ∞ + ||V n +1 / || ∞ ) || E n +1 / || + 12 || E n +1 / x || . Using Remark 2.2, Theorem 4.5 and since the following inverse inequality holds(4.10) || χ || ∞ ≤ Ch − / || χ || , ∀ χ ∈ S rh , we obtain t ( || E n +1 || − || E n || ) ≤ C (1 + ∆ t + h r − ) || E n +1 / || ≤ C (1 + h − / ∆ t + h − / h r − )( || E n || + || E n +1 || ) Therefore if we assume E n = 0 , we get for ∆ t h − / and ∆ t h r − / sufficiently small E n +1 = 0 . We deduce uniqueness of the Crank-Nicolson approximations.5. N
UMERICAL EXPERIMENTS
We conclude this paper by presenting some experimental results obtained using numerical scheme(4.1) with Crank-Nicolson method for the time disretization and the Garlerkin method for differentpolynomial orders. In our numerical experiments we have imposed a zero Dirichlet boundary con-dition on the whole exterior domain {| x | > } and we have confined the nonlocal operator J to thedomain Ω = {| x | ≤ } . This means we have computed the value of U n +1 by using only the values U n ( x i ) with x i ∈ Ω .For all the numerical tests, the stability condition stated in Proposition 4.4 is satisfied. EM FOR THE FOWLER EQUATION 13 F IGURE
1. Example 1: r = 2 , T = 0 . and N = 640 In order to magnify the effect of the nonlocal term, we add a small viscous coefficient ε in the Fowlerequation(5.1) ∂ t u ( t, x ) + ∂ x (cid:18) u J [ u ] (cid:19) ( t, x ) − ε∂ xx u = 0 , We consider the following two initial data:Example 1: u ( x ) = if x ≤ − . x + 2 . if − . < x ≤ − . . if − . < x ≤ . . − x if . < x ≤ . . if x > . Example 2: u ( x ) = e − x +0 . . The numerical results are presented in Figures 1 and 2. For the first example, we used linear el-ements for the Galerkin method ( r = 2 ) while we used a second order polynomial approximations ( r = 3) for the second example. In all plots, the solid line represents the initial datum while thedotted line the numerical solution at t = T. As we expect from the viscous Burgers equation theinitial data are propagated downstream but we observe here in addition an ”erosive process” behindthe bump due to the nonlocal term.The numerical rate of convergence for the solutions in Figures 1 and 2 are presented in Tables 1 and2.We have measured the L -error E h = || u h ( T, · ) − ˆ u e ( T, · ) || , F IGURE
2. Example 2: r = 3 , T = 0 . and N = 640 N error relative error order20 4.0759e-04 4.3296e-04 1.953240 1.0526e-04 1.1181e-04 1.917380 2.7867e-05 2.9601e-05 1.7207160 8.4546e-06 8.9808e-06 -T ABLE
1. Example 1: Error, relative error and numerical rate of convergence forone order polynomial approximations ( r = 2 ). N denotes the number of elements. N error relative error order20 1.0381e-04 3.1458e-04 2.309740 2.0939e-05 6.3452e-05 2.079280 4.9551e-06 1.5015e-05 1.8057160 1.4174e-06 4.2952e-06 -T ABLE
2. Example 2: Error, relative error and numerical rate of convergence forsecond order polynomial approximations ( r = 3 ). N denotes the number of ele-ments.where ˆ u e is the numerical solution which has been computed using a very fine grid h = 2 / . Wealso have measured the relative error R h = (cid:18) || ˆ u e ( T, · ) || (cid:19) E h , EM FOR THE FOWLER EQUATION 15 and the approximation rate of convergence α h = (cid:18) (cid:19) (cid:0) log E h − log E h/ (cid:1) . We observe that the order of convergence is reached, confirming the theoretical results. Indeed, theexperimental rates of convergence are greather than one for the first numerical example ( r = 2 ) andfor the second example ( r = 3 ), the numerical rates of convergence are near to 2.R EFERENCES [1] G. D. A
KRIVIS , Finite element discretization of the Kuramoto-Sivashinsky equation , Numerical analysis and math-ematical modelling, 29 (1994), pp. 155 – 163.[2] N. A
LIBAUD , P. A
ZERAD , AND
D. I S ` EBE , A non-monotone nonlocal conservation law for dune morphodynamics ,Differential Integral Equations, 23 (2010), pp. 155–188.[3] A. B
OUHARGUANE AND
R. C
ARLES , Splitting methods for the nonlocal fowler equation , Mathematics of Compu-tation, 83 (2014).[4] F. E. B
ROWDER , Existence and uniqueness theorems for solutions of nonlinear boundary value problems , Proc.Sympos. Appl. Math., Vol. XVII, (1965), pp. 24–49.[5] J. D
RONIOU , A numerical method for fractal conservation laws , Mathematics of Computation, 79 (2010), pp. 95–124.[6] G. J. F
IX AND
J. P. R
OOF , Least squares finite-element solution of a fractional order two-point boundary valueproblem , Computers & Mathematics with Applications, 48 (2004), pp. 1017–1033.[7] A. C. F
OWLER , Dunes and drumlins , in Geomorphological fluid mechanics, A. Provenzale and N. Balmforth, eds.,vol. 211, Springer-Verlag, Berlin, 2001, pp. 430–454.[8] Q. G
UAN AND
M. G
UNZBURGER , θ -schemes for finite element discretization of the space-time fractional diffusionequations , Journal of Computational and Applied Mathematics, 288 (2015), pp. 264 – 273.[9] F. L IU , V. A NH , AND
I. T
URNER , Numerical solution of the space fractional fokkerplanck equation , Journal ofComputational and Applied Mathematics, 166 (2004), pp. 209 – 219. Proceedings of the International Conference onBoundary and Interior Layers - Computational and Asymptotic Methods.[10] M. M. M
EERSCHAERT AND
C. T
ADJERAN , Finite difference approximations for fractional advection-dispersionflow equations , Journal of Computational and Applied Mathematics, 172 (2004), pp. 65 – 77.[11] J. P. R
OOP , Computational aspects of fem approximation of fractional advection dispersion equations on boundeddomains in R , Journal of Computational and Applied Mathematics, 193 (2006), pp. 243 – 268.[12] V. T HOM ´ EE , Galerkin finite element methods for parabolic problems , Springer Series in Computational Mathematics,(2006).[13] Q. X
U AND
J. S. H
ESTHAVEN , Discontinuous Galerkin method for fractional convection-diffusion equations , SIAMJournal on Numerical Analysis, 52 (2014), pp. 405 – 423.[14] Y. Z
HENG , C. L I , AND
Z. Z
HAO , A note on the finite element method for the space-fractional advection diffusionequation , Computers & Mathematics with Applications, 59 (2010), pp. 1718–1726.I
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