Optimal convergence of a second order low-regularity integrator for the KdV equation
aa r X i v : . [ m a t h . NA ] A ug OPTIMAL CONVERGENCE OF A SECOND ORDER LOW-REGULARITYINTEGRATOR FOR THE KDV EQUATION
YIFEI WU AND XIAOFEI ZHAO
Abstract.
In this paper, we establish the optimal convergence result of a second order exponential-type integrator from (136, Numer. Math., 2017) for solving the KdV equation under rough initialdata. The scheme is explicit and efficient to implement. By rigorous error analysis, we show thatthe scheme provides the second order accuracy in H γ for initial data in H γ +4 for any γ ≥ Keywords:
KdV equation, rough data, low-regularity method, second order accuracy, errorestimates, exponential-type integrator
AMS Subject Classification: Introduction
During the past few years, to solve efficiently problems with rough initial data, the so-calledlow-regularity numerical integrators have been proposed for some dispersive models. Compared tothe classical numerical discretizations such as the finite difference methods or standard exponentialintegrators [10] or splitting schemes [24], the low-regularity methods require less regularity of thesolution to reach their optimal convergence rates. For example, for the cubic nonlinear Schr¨odingerequation, the first order convergence in H r has been achieved under only H r +1 -data [17, 25], andfor the one-dimensional quadratic nonlinear Schr¨odinger equation [17, 25] or the nonlinear Diracequations [30], only H r -data is needed. In this work, we are concerned with the Korteweg-de Vries(KdV) equation, which is a classical mathematical model for the waves on shallow water surfaces,under the rough initial data on a torus: ∂ t u ( t, x ) + ∂ x u ( t, x ) = 12 ∂ x ( u ( t, x )) , t > , x ∈ T ,u (0 , x ) = u ( x ) , x ∈ T , (1.1)where T = (0 , π ), u = u ( t, x ) : R + × T → R is the unknown and u ∈ H s ( T ) with some 0 ≤ s < ∞ is a given initial data. For the numerical studies of the KdV equation (1.1) under smooth enoughinitial data cases, we refer to [7, 14, 16, 20, 22, 23, 28, 31, 32]. However, in practice the initial datamay not be ideally smooth due to multiply reasons such as measurements or noise [9]. Analytically,the global well-posedness of (1.1) on the torus under rough data has already been established in[4, 18]. That is, for any u ∈ H s ( T ) , s ≥ − and any positive time T >
0, there exists a uniquesolution of (1.1) in a certain Banach space of functions X ⊂ C ([0 , T ]; H s ( T )) as established in [4]by PDE methods, and [18] extended the result to s ≥ − v := e ∂ x t u and the Duhamel’s formula at t n = nτ with τ > v ( t n + τ, x ) = v ( t n , x ) + 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x v ( t n + s, x ) (cid:17) ds, (1.2)[11] proposed an exponential-type numerical scheme by letting v ( t n + s, x ) ≈ v ( t n , x ), and thenthe integration for s was found exactly and explicitly in the physical space. Comparing to classical first order methods, this strategy gives rise to a first order numerical scheme that allows the lowregularity requirement for solving the KdV equation (1.1): k u ( t n , · ) − u n k H . τ, up to some finite time for u ∈ H as was proved in [11], where u n denotes the numerical solutionat t n . That is to say for solving the KdV, in order to reach the first order accuracy, the regularityrequirement of such exponential-type low-regularity integrator is the boundedness of two additionalspatial derivatives of the solution. This requirement is to some extend essential for the directintegration methods due to the Burgers-type nonlinearity in the KdV equation [3].To push the convergence rate of such low-regularity integrators to the second order, one naturalway is to use the idea of Picard iteration, i.e. using the first order scheme to approximate thesolution in the nonlinearity in the Duhamel’s formula (1.2) and then integrate exactly for s in theFourier space. Obviously by doing this, the Burgers-nonlinearity will add one more loss of derivative.What makes it worse after the iteration is that, the quadratic nonlinearity in the KdV equationwill generate four different modes in the Fourier space, which after the time integration can notbe translated back to an explicit function in the physical space, and the scheme will then have todeal with a heavy convolution in the Fourier space. A practical strategy as a compensation forcomputational efficiency is to drop some Fourier modes before the time integration, in order to getback to the physical space, which consequently further asks for more regularity of the solution forobtaining the second order convergence rate. See the very recent effort on the nonlinear Schr¨odingerequation for designing such second order low-regularity integrators in [17]. Together with the lossfrom the Burgers-nonlinearity, one major concern of the Picard iteration approach is that the secondorder convergence rate is achieved in H γ -norm for solutions in as least H γ +4 space. See a preciseconstruction in such approach for the KdV equation in the recent work [3].In this work, instead of the Picard iteration, we consider the construction for a second orderlow-regularity integrator (LRI) for the KdV (1.1) by simply the Taylor’s expansion in (1.2) as v ( t n + s, x ) ≈ v ( t n , x ) + s∂ t v ( t n , x ) . This strategy has been outlined in [11], and it leads to an explicit and efficient scheme. We areaiming to establish the optimal convergence result of this second order LRI scheme by showing that k u ( t n , · ) − u n k H γ . τ , up to some fixed time for u ∈ H γ +4 for any γ ≥
0. The proof relies on a key fact that ∂ t v = 32 e t∂ x ∂ x (cid:16) e − t∂ x ∂ x v (cid:17) + 13 e t∂ x ∂ x (cid:16) e − t∂ x v (cid:17) , and some tools from the harmonic analysis to overcome the absence of algebraic property of H γ when 0 ≤ γ ≤ . As shown by our theoretical estimate, this simplified strategy (compared withPicard iteration) is able to get the desired second order accuracy with only four additional boundedspatial derivatives, which is better than what was conjectured in [11], and it matches with the bestresult that one could expect from the direct Picard iteration as we have explained above and fromthe recent investigation in [3]. We shall show by numerical results that the regularity requirementfor this second order LRI is sharp. Compared with other classical second order numerical methodsin the literature, for example the Strang splitting method (with exact solutions at the Burgers’ step)[12, 13] that needs u ∈ H γ +5 for the second order convergence rate in H γ for γ ≥
1, the presentedLRI saves one spatial derivative and further reduces the requirement of the regularity. Moreover,one should note that the exact solutions at the Burgers’ step in the Strang splitting scheme requirea nonlinear solver which makes the practical scheme costly. Hence, the presented LRI is particularlymore efficient for solving the KdV (1.1) under rough data, which will be illustrated through numericalexperiments in the end.The rest of the paper is organized as follows. In Section 2, we present the detailed scheme of LRIfrom [11] and give its main convergence theorem. In Section 3, we give the proof of the convergenceresult. Numerical confirmations are reported in Section 4 and conclusions are drawn in Section 5.
OW-REGULARITY INTEGRATOR FOR KDV EQUATION 3 Numerical scheme
In this section, we present the detailed numerical scheme of the second order exponential-typeintegrator as outlined in [11] and give its main convergence theorem. In the following for simplicity,we shall assume that the zero-mode/average of the initial value of the (1.1) is zero. Otherwise, wemay consider ˜ u := u (cid:16) t, x − d ( u ) t (cid:17) − d ( u ) instead, and one can note that ˜ u also obeys the same KdV equation of (1.1) with initial data˜ u := u − d ( u ) . Here we denote d ( u ) l for l ∈ Z as the Fourier coefficients of u .With the twisted variable v ( t, x ) := e t∂ x u ( t, x ) , t ≥ , x ∈ T , (2.1)the KdV equation (1.1) becomes: ∂ t v ( t, x ) = 12 e t∂ x ∂ x (cid:16) e − t∂ x v ( t, x ) (cid:17) , t ≥ , x ∈ T . (2.2)For n ≥ t = t n + s , plugging v ( t n + s, x ) ≈ v ( t n , x ) + s∂ t v ( t n , x ) into the Duhamel’s formula(1.2), we get v ( t n +1 , x ) ≈ v ( t n , x ) + 12 Z τ e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x (cid:0) v ( t n , x ) + s∂ t v ( t n , x ) (cid:1)i ds ≈ v ( t n , x ) + 12 I ( t n , x ) + I ( t n , x ) , (2.3)where I ( t n , x ) := Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x v ( t n , x ) (cid:17) ds, (2.4a) I ( t n , x ) := Z τ e ( t n + s ) ∂ x ∂ x h s (cid:16) e − ( t n + s ) ∂ x v ( t n , x ) (cid:17) (cid:16) e − ( t n + s ) ∂ x ∂ t v ( t n , x ) (cid:17)i ds. (2.4b)By calculation in the Fourier frequency space and noting the key relation( k + k ) − k − k = 3( k + k ) k k , the terms in I and I can be exactly integrated in the physical space. We refer the readers to [11]for those detailed calculations. The zero Fourier mode of I or I is clearly zero, and for the othermodes, here we directly write down their explicitly formulas in the physical space: I ( t n , x ) = X l =0 d ( J ) l ( t n )e ilx , I ( t n , x ) = X l =0 d ( J ) l ( t n )e ilx , where d ( J ) l and d ( J ) l denote respectively the Fourier coefficients of functions J and J J ( t n , x ) := 13 e t n +1 ∂ x (cid:16) e − t n +1 ∂ x ∂ − x v ( t n , x ) (cid:17) −
13 e t n ∂ x (cid:16) e − t n ∂ x ∂ − x v ( t n , x ) (cid:17) , and J ( t n , x ) := τ t n +1 ∂ x (cid:16) e − t n +1 ∂ x ∂ − x v ( t n , x ) (cid:17) (cid:16) e − t n +1 ∂ x ∂ − x ∂ t v ( t n , x ) (cid:17) −
19 e t n +1 ∂ x ∂ − x (cid:16) e − t n +1 ∂ x ∂ − x v ( t n , x ) (cid:17) (cid:16) e − t n +1 ∂ x ∂ − x ∂ t v ( t n , x ) (cid:17) + 19 e t n ∂ x ∂ − x (cid:16) e − t n ∂ x ∂ − x v ( t n , x ) (cid:17) (cid:16) e − t n ∂ x ∂ − x ∂ t v ( t n , x ) (cid:17) . Here the operator ∂ − mx ( m ∈ N ) is defined for function f ( x ) ∈ L ( T ) as ∂ − mx f ( x ) := X l =0 ( il ) − m b f l e ilx , x ∈ T . Noting (2.2), we substitute ∂ t v ( t n , x ) = 12 e t n ∂ x ∂ x (cid:16) e − t n ∂ x v ( t n , x ) (cid:17) into J ( t n , x ), then the approximation (2.3) forms an update from t n to t n +1 for v ( t, x ), and byreverting the change of variable (2.1) we get the scheme for u ( t, x ).The detailed second order low-regularity integrator (LRI) for solving the KdV equation (1.1)then reads: denote u n = u n ( x ) ≈ u ( t n , x ) as the numerical solution for n ≥ I nj = I nj ( x ) ≈ e − t n +1 ∂ x I j ( t n , x ) for j = 1 , u = u ( x ), then u n +1 = e − τ∂ x u n + 12 I n + I n , n = 0 , , . . . , (2.5)where I n = X l =0 d ( J n ) l e ilx , I n = X l =0 d ( J n ) l e ilx , and J n := 13 (cid:16) e − τ∂ x ∂ − x u n (cid:17) −
13 e − τ∂ x (cid:0) ∂ − x u n (cid:1) ,J n := τ (cid:16) e − τ∂ x ∂ − x u n (cid:17) (cid:16) e − τ∂ x ( u n ) − \ (( u n ) ) (cid:17) − ∂ − x (cid:16) e − τ∂ x ∂ − x u n (cid:17) (cid:16) e − τ∂ x ∂ − x ( u n ) (cid:17) + 118 e − τ∂ x ∂ − x (cid:0) ∂ − x u n (cid:1) (cid:0) ∂ − x ( u n ) (cid:1) . The above LRI (2.5) is explicit and preserves the mass of the KdV equation (1.1) at the discretelevel, i.e. Z T u n ( x ) dx ≡ Z T u ( x ) dx, n = 0 , , . . . . In practice, the Fourier series in the scheme are truncated to an integer
N > O ( N log N ) at each time level, and it has no CFL-type conditions.Now, we state the convergence theorem of the presented (semi-discretized) LRI method (2.5) asthe main result of the paper and the proof is given in the next section. Theorem 2.1.
Let u n be the numerical solution of the KdV (1.1) obtained from the LRI scheme(2.5) up to some fixed time T > . Under assumption that u ∈ H γ +4 ( T ) for some γ ≥ , thereexists constants τ , C > such that for any < τ ≤ τ , we have k u ( t n , · ) − u n k H γ ≤ Cτ , n = 0 , . . . , Tτ , (2.6) where the constants τ and C depend only on T and k u k L ∞ ((0 ,T ); H γ +4 ) . We shall show later by numerical results that the estimate and the regularity assumption in theabove convergence theorem for LRI are sharp. The convergence result of LRI is indeed better thanconjectured in [11]. 3.
Convergence analysis
In this section, we give the rigorous proof of the main result. To do so, we shall firstly introducesome tools from harmonic analysis in subsection 3.1, and then establish the stability result and thelocal error estimate, respectively in subsection 3.4 and subsection 3.5. The final proof of Theorem2.1 is given in subsection 3.6.3.1.
Some notations and tools.
For convenience, we introduce some notations and definitions,some of which are employed from [4]. We use A . B or B & A to denote the statement that A ≤ CB for some absolute constant C > τ or n , and we denote A ∼ B for A . B . A . We define ( dξ ) to be the normalized counting measureon Z such that Z a ( ξ ) ( dξ ) = X ξ ∈ Z a ( ξ ) . OW-REGULARITY INTEGRATOR FOR KDV EQUATION 5
The Fourier transform of a function f on T is defined by F ( f )( ξ ) = b f ( ξ ) = 12 π Z T e − ixξ f ( x ) dx, and thus the Fourier inversion formula f ( x ) = Z e ixξ b f ( ξ ) ( dξ ) . Then the following usual properties of the Fourier transform hold: k f k L ( T ) = (cid:13)(cid:13) b f (cid:13)(cid:13) L (( dξ )) (Plancherel); h f, g i = Z T f ( x ) g ( x ) dx = Z b f ( ξ ) b g ( ξ ) ( dξ ) (Parseval); d ( f g )( ξ ) = Z b f ( ξ − ξ ) b g ( ξ ) ( dξ ) (Convolution) . The Sobolev space H s ( T ) for s ≥ (cid:13)(cid:13) f (cid:13)(cid:13) H s ( T ) = (cid:13)(cid:13) J s f (cid:13)(cid:13) L ( T ) = (cid:13)(cid:13)(cid:13) h ξ i s b f ( ξ ) (cid:13)(cid:13)(cid:13) L (( dξ )) , where we denote the operator J s = (1 − ∂ xx ) s , and h·i = (1 + | · | ) / . As a tool to overcome the absence of the algebraic property of H s when s ≤ , we will frequentlycall the following Kato-Ponce inequality, where a general form was proved in [15] originally and animportant progress in the endpoint case was made in [2, 19] very recently. Lemma 3.1. (Kato-Ponce inequality) The following inequalities hold: (i)
For any γ ≥ , γ > , f, g ∈ H γ ∩ H γ , then k J γ ( f g ) k . k f k H γ k g k H γ + k f k H γ k g k H γ . In particular, if γ > , then k J γ ( f g ) k . k f k H γ k g k H γ . (ii) For any γ ≥ , γ > , f ∈ H γ + γ , g ∈ H γ , then k J γ ( f g ) k . k f k H γ + γ k g k H γ . Moreover, we will need the following specific commutator estimate. Here the commutator isdefined as [
A, B ] = AB − BA . Lemma 3.2.
Let f, g be the Schwartz functions. If ≤ γ ≤ , then the following inequality holdsfor any γ > : (cid:13)(cid:13) [ J γ , f ] ∂ x g (cid:13)(cid:13) L ≤ C k f k H γ k g k H γ . Furthermore, if γ > , then (cid:13)(cid:13) [ J γ , f ] ∂ x g (cid:13)(cid:13) L ≤ C (cid:16) k f k H γ + γ k g k H + k f k H γ k g k H γ (cid:17) , or (cid:13)(cid:13) [ J γ , f ] ∂ x g (cid:13)(cid:13) L ≤ C (cid:16) k f k H γ k g k H γ + k f k H γ k g k H γ (cid:17) . Proof.
Taking the Fourier transform on [ J γ , f ] ∂ x g , we get F (cid:16) [ J γ , f ] ∂ x g (cid:17) ( ξ ) = i Z ξ = ξ + ξ (cid:0) h ξ i γ − h ξ i γ (cid:1) ξ b f ( ξ ) b g ( ξ ) ( dξ ) . We assume that b f and b g are positive, otherwise one may replace them by | b f | and | b g | . We could alsoassume that ξ = 0 and ξ = 0, otherwise the term in the above integral vanishes. DenoteΩ = (cid:26) ( ξ, ξ , ξ ) : ξ = ξ + ξ , | ξ | ≤ | ξ | (cid:27) , Ω = (cid:26) ( ξ, ξ , ξ ) : ξ = ξ + ξ , | ξ | > | ξ | (cid:27) . Y. WU AND X. ZHAO
Then by Plancherel’s identity, (cid:13)(cid:13) [ J γ , f ] ∂ x g (cid:13)(cid:13) L ≤ (cid:13)(cid:13)(cid:13) Z Ω (cid:0) h ξ i γ − h ξ i γ (cid:1) ξ b f ( ξ ) b g ( ξ ) ( dξ ) (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) Z Ω (cid:0) h ξ i γ − h ξ i γ (cid:1) ξ b f ( ξ ) b g ( ξ ) ( dξ ) (cid:13)(cid:13)(cid:13) L . In Ω : | ξ | ≤ | ξ | . Then, | ξ | ≤ | ξ | ≤ | ξ | . When 0 ≤ γ ≤ (cid:12)(cid:12) h ξ i γ − h ξ i γ (cid:12)(cid:12) | ξ | . h ξ i| ξ | γ . Hence we have (cid:12)(cid:12)(cid:12) Z Ω (cid:0) h ξ i γ − h ξ i γ (cid:1) ξ b f ( ξ ) b g ( ξ ) ( dξ ) (cid:12)(cid:12)(cid:12) . Z ξ = ξ + ξ h ξ i| ξ | γ b f ( ξ ) b g ( ξ ) ( dξ ) = F (cid:16) h∇i f · |∇| s g (cid:17) ( ξ ) . Hence, by Plancherel’s identity and Sobolev’s inequality, (cid:13)(cid:13)(cid:13) Z Ω (cid:0) h ξ i γ − h ξ i γ (cid:1) ξ b f ( ξ ) b g ( ξ ) ( dξ ) (cid:13)(cid:13)(cid:13) L ≤ (cid:13)(cid:13)(cid:13) F (cid:16) h∇i f · |∇| s g (cid:17) ( ξ ) (cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13) h∇i f · |∇| s g (cid:13)(cid:13) L . (cid:13)(cid:13) h∇i f (cid:13)(cid:13) L ∞ (cid:13)(cid:13) |∇| s g (cid:13)(cid:13) L . (cid:13)(cid:13) h∇i γ f (cid:13)(cid:13) L (cid:13)(cid:13) |∇| s g (cid:13)(cid:13) L . When γ >
1, we have (cid:12)(cid:12) h ξ i γ − h ξ i γ (cid:12)(cid:12) | ξ | . h ξ i γ | ξ | . Then similarly, (cid:13)(cid:13)(cid:13) Z Ω (cid:0) h ξ i γ − h ξ i γ (cid:1) ξ b f ( ξ ) b g ( ξ ) ( dξ ) (cid:13)(cid:13)(cid:13) L . min n(cid:13)(cid:13) h∇i γ + γ f (cid:13)(cid:13) L (cid:13)(cid:13) ∇ g (cid:13)(cid:13) L , (cid:13)(cid:13) h∇i γ f (cid:13)(cid:13) L (cid:13)(cid:13) h∇i γ ∇ g (cid:13)(cid:13) L o , by L ∞ L -H¨older’s or L L ∞ -H¨older’s inequality. These give the desired result in Case 1.In Ω : | ξ | > | ξ | . Then (cid:12)(cid:12) h ξ i γ − h ξ i γ (cid:12)(cid:12) | ξ | . h ξ i| ξ | γ . Hence, similar as above, we obtain (cid:13)(cid:13)(cid:13) Z Ω (cid:0) h ξ i γ − h ξ i γ (cid:1) ξ b f ( ξ ) b g ( ξ ) ( dξ ) (cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13) h∇i γ f (cid:13)(cid:13) L (cid:13)(cid:13) |∇| s g (cid:13)(cid:13) L . This gives the desired result in Ω , and the lemma is proved. (cid:3) Remark . In the following of the section, we shall just adopt a weaker version of the estimatesfrom Lemma 3.2: For any γ ≥ e γ > max { γ + , } , (cid:13)(cid:13) [ J γ , f ] ∂ x g (cid:13)(cid:13) L . k f k H e γ k g k H γ , (3.1)and for any γ > , (cid:13)(cid:13) [ J γ , f ] ∂ x g (cid:13)(cid:13) L . k f k H γ k g k H γ . (3.2)Based on the above inequalities, we can deduce some estimates as follows, which will be used toobtain the a prior estimate of the numerical solution. Lemma 3.4.
The following inequalities hold: (i)
For any γ ≥ , γ > , f ∈ H γ , g ∈ H γ + γ +1 , then (cid:10) J γ ∂ x ( f g ) , J γ f (cid:11) . k f k H γ k g k H γ + γ . (ii) For any γ > , f ∈ H γ , then (cid:10) J γ ∂ x (cid:0) f (cid:1) , J γ f (cid:11) . k f k H γ . Proof. (i) Directly, we have (cid:10) J γ ∂ x ( f g ) , J γ f (cid:11) = (cid:10) J γ ∂ x f · g, J γ f (cid:11) + (cid:10) J γ (cid:0) f · ∂ x g (cid:1) , J γ f (cid:11) + (cid:10)(cid:2) J γ , g (cid:3) ∂ x f, J γ f (cid:11) . For the first term on the right-hand side, by using integration-by-parts, it is equal to − Z ∂ x g (cid:12)(cid:12) J γ f (cid:12)(cid:12) dx. Therefore, we have the estimate (cid:12)(cid:12)(cid:10) J γ ∂ x f · g, J γ f (cid:11)(cid:12)(cid:12) . (cid:13)(cid:13) g (cid:13)(cid:13) H γ (cid:13)(cid:13) f (cid:13)(cid:13) H γ , OW-REGULARITY INTEGRATOR FOR KDV EQUATION 7 for any γ > . For the second term, by Lemma 3.1 (ii), we have (cid:12)(cid:12)(cid:10) J γ (cid:0) f · ∂ x g (cid:1) , J γ f (cid:11)(cid:12)(cid:12) . (cid:13)(cid:13) f · ∂ x g (cid:13)(cid:13) H γ (cid:13)(cid:13) f (cid:13)(cid:13) H γ . (cid:13)(cid:13) g (cid:13)(cid:13) H γ + γ k f k H γ . For the third term, by (3.1) we have (cid:12)(cid:12)(cid:10)(cid:2) J γ , g (cid:3) ∂ x f, J γ f (cid:11)(cid:12)(cid:12) . (cid:13)(cid:13)(cid:2) J γ , g (cid:3) ∂ x f (cid:13)(cid:13) L (cid:13)(cid:13) f (cid:13)(cid:13) H γ . (cid:13)(cid:13) f (cid:13)(cid:13) H γ (cid:13)(cid:13) g (cid:13)(cid:13) H γ + γ . Combining the three estimates above, we get the estimate in (i).(ii) We use the similar argument to write (cid:10) J γ ∂ x (cid:0) f (cid:1) , J γ f (cid:11) = 2 (cid:10) J γ ∂ x f · f, J γ f (cid:11) + 2 (cid:10)(cid:2) J γ , f (cid:3) ∂ x f, J γ f (cid:11) , and then for the first term on the right-hand side, we get for any γ > , (cid:12)(cid:12)(cid:10) J γ ∂ x f · f, J γ f (cid:11)(cid:12)(cid:12) . k f k H γ k f k H γ . By choosing γ properly, we have k f k H γ . k f k H γ . For the second term, by applying (3.2)instead, we get (cid:12)(cid:12)(cid:10)(cid:2) J γ , f (cid:3) ∂ x f, J γ f (cid:11)(cid:12)(cid:12) . (cid:13)(cid:13) f (cid:13)(cid:13) H γ , and hence, we obtain the estimate in (ii). (cid:3) Problem reduction.
Now, we start to illustrate the proof of the convergence theorem. Forthe simplicity of notations, we shall omit the space variable x in the functions, and we define v n := e t n ∂ x u n and v nt := 12 e t n ∂ x ∂ x (cid:16) e − t n ∂ x v n (cid:17) , (3.3)where u n is the numerical solution from the LRI scheme (2.5). Noting that the scheme (2.5) isobtained by exactly integrating (2.3), so we have v n +1 = v n + 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x v n (cid:17) ds + Z τ s e ( t n + s ) ∂ x ∂ x h(cid:16) e − ( t n + s ) ∂ x v n (cid:17) (cid:16) e − ( t n + s ) ∂ x v nt (cid:17)i ds. (3.4)Since the operator e t∂ x in the change of variable (2.1) is unitary, so to prove (2.6), it is sufficient toprove k v ( t n ) − v n k H γ ≤ Cτ , n = 0 , , . . . , Tτ . To do this, we subtract (3.4) from the exact Duhamel’s formula (1.2) to get: v ( t n +1 ) − v n +1 = L n + Φ n ( v ( t n )) − Φ n ( v n ) , (3.5)where we define the local error term as L n := 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x v ( t n + s ) (cid:17) ds − Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x v ( t n ) (cid:17) ds − Z τ s e ( t n + s ) ∂ x ∂ x h(cid:16) e − ( t n + s ) ∂ x v ( t n ) (cid:17) (cid:16) e − ( t n + s ) ∂ x ∂ t v ( t n ) (cid:17)i ds, (3.6)and the numerical propagator asΦ n ( v ) := v + 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x v (cid:17) ds + 12 Z τ s e ( t n + s ) ∂ x ∂ x (cid:20)(cid:16) e − ( t n + s ) ∂ x v (cid:17) (cid:18) e − s∂ x ∂ x (cid:16) e − t n ∂ x v (cid:17) (cid:19)(cid:21) ds. (3.7)Hence, to obtain a Gronwall type inequality, it reduces to control L n and Φ (cid:0) v ( t n ) (cid:1) − Φ (cid:0) v n (cid:1) , whichare regarded as the local error estimate and the stability in the following.By directly calculations, we have the following key facts. Y. WU AND X. ZHAO
Lemma 3.5.
The following equalities hold:(i) Let v be the solution of (2.2) , then ∂ t v ( t ) = 32 e t∂ x ∂ x (cid:16) e − t∂ x ∂ x v ( t ) (cid:17) + 13 e t∂ x ∂ x (cid:16) e − t∂ x v ( t ) (cid:17) , t ≥ . (ii) Let f, g ∈ L with b f (0) = b g (0) = 0 , then for any t n ≥ , Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds = 13 e t n +1 ∂ x (cid:16) e − t n +1 ∂ x ∂ − x f · e − t n +1 ∂ x ∂ − x g (cid:17) −
13 e t n ∂ x (cid:16) e − t n ∂ x ∂ − x f · e − t n ∂ x ∂ − x g (cid:17) ; Moreover for k = 1 , , . . . , Z τ s k e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds = τ k t n +1 ∂ x (cid:16) e − t n +1 ∂ x ∂ − x f · e − t n +1 ∂ x ∂ − x g (cid:17) − k Z τ s k − e ( t n + s ) ∂ x (cid:16) e − ( t n + s ) ∂ x ∂ − x f · e − ( t n + s ) ∂ x ∂ − x g (cid:17) ds. Proof. (i) Noting that ∂ t b v ( t, ξ ) = 12 iξ Z ξ = ξ + ξ e − it ( ξ − ξ − ξ ) b v ( ξ ) b v ( ξ ) ( dξ ) , t ≥ , (3.8)and ξ − ξ − ξ = 3 ξξ ξ , we have that for any t ≥ ∂ t b v ( t, ξ ) = 32 ξ Z ξ = ξ + ξ e − it ( ξ − ξ − ξ ) ξ ξ b v ( ξ ) b v ( ξ ) ( dξ )+ 12 iξ Z ξ = ξ + ξ e − it ( ξ − ξ − ξ ) ∂ t (cid:0)b v ( ξ ) b v ( ξ ) (cid:1) ( dξ ) . From (3.8) and symmetry, we get ∂ t b v ( t, ξ ) = 32 ξ Z ξ = ξ + ξ e − it ( ξ − ξ − ξ ) ξ ξ b v ( ξ ) b v ( ξ ) ( dξ )+ iξ Z ξ = ξ + ξ + ξ i ( ξ + ξ ) e − it ( ξ − ξ − ξ − ξ ) b v ( ξ ) b v ( ξ ) b v ( ξ ) ( dξ )( dξ ) . By symmetry again, the second term is equal to − ξ Z ξ = ξ + ξ + ξ ( ξ + ξ + ξ ) e − it ( ξ − ξ − ξ − ξ ) b v ( ξ ) b v ( ξ ) b v ( ξ ) ( dξ )( dξ )= − ξ Z ξ = ξ + ξ + ξ e − it ( ξ − ξ − ξ − ξ ) b v ( ξ ) b v ( ξ ) b v ( ξ ) ( dξ )( dξ ) . Hence, we obtain that ∂ t b v ( t, ξ ) = 32 ξ Z ξ = ξ + ξ e − it ( ξ − ξ − ξ ) ξ ξ b v ( ξ ) b v ( ξ ) ( dξ ) − ξ Z ξ = ξ + ξ + ξ e − it ( ξ − ξ − ξ − ξ ) b v ( ξ ) b v ( ξ ) b v ( ξ ) ( dξ )( dξ ) . This proves the equality in (i) by the inverse Fourier transform.
OW-REGULARITY INTEGRATOR FOR KDV EQUATION 9 (ii) For k ≥
0, by taking the Fourier transform we get for any t n ≥ F (cid:18)Z τ s k e ( t n + s ) ∂ x ∂ x (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1) ds (cid:19) ( ξ )= iξ Z τ Z ξ = ξ + ξ s k e − i ( t n + s ) ( ξ − ξ − ξ ) b f ( ξ ) b g ( ξ ) ( dξ ) ds. Note that for k = 0, Z τ e − i ( t n + s ) ( ξ − ξ − ξ ) ds = − iξξ ξ e − it n +1 ( ξ − ξ − ξ ) + 13 iξξ ξ e − it n ( ξ − ξ − ξ ) , and for k ≥ Z τ s k e − i ( t n + s ) ( ξ − ξ − ξ ) ds = − τ k iξξ ξ e − it n +1 ( ξ − ξ − ξ ) + k iξξ ξ Z τ s k − e − i ( t n + s ) ( ξ − ξ − ξ ) ds. Then by the above formulas, we find F (cid:18)Z τ e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g i ds (cid:19) ( ξ )= − Z ξ = ξ + ξ ξ ξ e − it n +1 ( ξ − ξ − ξ ) b f ( ξ ) b g ( ξ ) ( dξ )+ Z ξ = ξ + ξ ξ ξ e − it n ( ξ − ξ − ξ ) b f ( ξ ) b g ( ξ ) ( dξ ) , and for k ≥ F (cid:18)Z τ s k e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g i ds (cid:19) ( ξ )= − τ k Z ξ = ξ + ξ ξ ξ e − it n +1 ( ξ − ξ − ξ ) b f ( ξ ) b g ( ξ ) ( dξ )+ k Z τ Z ξ = ξ + ξ s k − ξ ξ e − i ( t n + s ) ( ξ − ξ − ξ ) b f ( ξ ) b g ( ξ ) ( dξ ) ds, which give the two equalities in (ii) by the inverse Fourier transform. (cid:3) Some consequences of the above formulas together with the Kato-Ponce inequality are the fol-lowing two lemmas, which will be used for the proof of the boundedness of the numerical solution.
Lemma 3.6.
Let f ∈ H γ , g ∈ H γ + γ with b f (0) = b g (0) = 0 for γ ≥ , γ > , then the followinginequality holds for any t n ≥ : (cid:13)(cid:13)(cid:13) Z τ e ( t n + s ) ∂ x ∂ x (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1) ds (cid:13)(cid:13)(cid:13) H γ . √ τ k f k H γ k g k H γ + γ . Moreover, if γ > , then (cid:13)(cid:13)(cid:13) Z τ e ( t n + s ) ∂ x ∂ x (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1) ds (cid:13)(cid:13)(cid:13) H γ . √ τ k f k H γ k g k H γ . (3.9) Proof.
From Lemma 3.5-(ii) and integration-by-parts, we get for any t n ≥ γ ≥ (cid:13)(cid:13)(cid:13) Z τ e ( t n + s ) ∂ x ∂ x (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1) ds (cid:13)(cid:13)(cid:13) H γ (3.10)= 13 Z τ D J γ e ( t n + s ) ∂ x (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1) , e t n ∂ x ∂ x J γ (cid:0) e − t n ∂ x ∂ − x f · e − t n ∂ x ∂ − x g (cid:1)E ds − Z τ D J γ e ( t n + s ) ∂ x (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1) , e t n +1 ∂ x ∂ x J γ (cid:0) e − t n +1 ∂ x ∂ − x f · e − t n +1 ∂ x ∂ − x g (cid:1)E ds. By Cauchy-Schwarz’s inequality, (cid:12)(cid:12) (3.10) (cid:12)(cid:12) . Z τ (cid:13)(cid:13)(cid:13) J γ (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1)(cid:13)(cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) ∂ x J γ (cid:0) e − t n ∂ x ∂ − x f · e − t n ∂ x ∂ − x g (cid:1)(cid:13)(cid:13)(cid:13) L ds + Z τ (cid:13)(cid:13)(cid:13) J γ (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1)(cid:13)(cid:13)(cid:13) L (cid:13)(cid:13)(cid:13) ∂ x J γ (cid:0) e − t n +1 ∂ x ∂ − x f · e − t n +1 ∂ x ∂ − x g (cid:1)(cid:13)(cid:13)(cid:13) L ds. For simplicity, we shall only present the estimate of the second term on the right-hand side of theabove inequality, and the first term can be treated in the same way. By Lemma 3.1 (ii), we have forany γ ≥ (cid:13)(cid:13)(cid:13) J γ (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1)(cid:13)(cid:13)(cid:13) L . k f k H γ k g k H γ + γ ;Or by Lemma 3.1 (i), when γ > , (cid:13)(cid:13)(cid:13) J γ (cid:0) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:1)(cid:13)(cid:13)(cid:13) L . k f k H γ k g k H γ . Similarly, we have for any γ ≥ , γ > , (cid:13)(cid:13)(cid:13) ∂ x J γ (cid:0) e − t n +1 ∂ x ∂ − x f · e − t n +1 ∂ x ∂ − x g (cid:1)(cid:13)(cid:13)(cid:13) L . k f k H γ k g k H γ + γ , and for γ > , (cid:13)(cid:13)(cid:13) ∂ x J γ (cid:0) e − t n +1 ∂ x ∂ − x f · e − t n +1 ∂ x ∂ − x g (cid:1)(cid:13)(cid:13)(cid:13) L . k f k H γ k g k H γ . Therefore, in total we find that for γ ≥ , γ > , (cid:12)(cid:12) (3.10) (cid:12)(cid:12) . τ k f k H γ k g k H γ + γ ;Or when γ > , (cid:12)(cid:12) (3.10) (cid:12)(cid:12) . τ k f k H γ k g k H γ . This finishes the proof of the lemma. (cid:3)
Moreover, we have
Lemma 3.7.
The following estimates hold: (i)
Let f ∈ H γ , f ∈ H γ + γ and f ∈ H γ + γ for γ ≥ , γ > with b f j (0) = 0 for j = 1 , , ,then for any t n ≥ , t ∈ R and k ≥ , (cid:13)(cid:13)(cid:13) Z τ s k e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t + s ) ∂ x ∂ x (cid:0) f f (cid:1) · e − ( t n + s ) ∂ x f (cid:17) ds (cid:13)(cid:13)(cid:13) H γ . τ k k f k H γ k f k H γ + γ k f k H γ + γ − . (ii) Let f j ∈ H γ for γ > with b f j (0) = 0 for j = 1 , , , then for any t n ≥ , t ∈ R and k ≥ , (cid:13)(cid:13)(cid:13) Z τ s k e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t + s ) ∂ x ∂ x (cid:0) f f (cid:1) · e − ( t n + s ) ∂ x f (cid:17) ds (cid:13)(cid:13)(cid:13) H γ . τ k k f k H γ k f k H γ k f k H γ − . Proof. (i) From Lemma 3.5-(ii), we find for k ≥ Z τ s k e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t + s ) ∂ x ∂ x (cid:0) f f (cid:1) · e − ( t n + s ) ∂ x f (cid:17) ds = τ k t n +1 ∂ x (cid:16) e − ( t + τ ) ∂ x (cid:0) f f (cid:1) · e − ( t n + τ ) ∂ x ∂ − x f (cid:17) − k Z τ s k − e ( t n + s ) ∂ x (cid:16) e − ( t + s ) ∂ x (cid:0) f f (cid:1) · e − ( t n + s ) ∂ x ∂ − x f (cid:17) ds. OW-REGULARITY INTEGRATOR FOR KDV EQUATION 11
By H¨older’s inequality and Sobolev’s inequality, we have for any γ ≥ , γ > , (cid:13)(cid:13)(cid:13) Z τ s k e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t + s ) ∂ x ∂ x (cid:0) f f (cid:1) · e − ( t n + s ) ∂ x f (cid:17) ds (cid:13)(cid:13)(cid:13) H γ . τ k (cid:16) k f k H γ k f k H γ k ∂ − x f k H γ + k f k L k f k H γ + γ k ∂ − x f k H γ + k f k L k f k H γ k ∂ − x f k H γ + γ (cid:17) . τ k k f k H γ k f k H γ + γ k f k H γ + γ − . (ii) By similar arguments as above but with the different H¨older’s inequality, we have that forany γ > , (cid:13)(cid:13)(cid:13) Z τ s k e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t + s ) ∂ x ∂ x (cid:0) f f (cid:1) · e − ( t n + s ) ∂ x f (cid:17) ds (cid:13)(cid:13)(cid:13) H γ . τ k (cid:16) k f k H γ k f k H γ k ∂ − x f k H γ + k f k H γ k f k H γ k ∂ − x f k H γ + k f k H γ k f k H γ k ∂ − x f k H γ (cid:17) . τ k k f k H γ k f k H γ k f k H γ − , where we used the fact γ > in the last step. (cid:3) A priori estimate.
With the prepared lemmas before, we can obtain the a priori estimateof the numerical solution v n which will be a key for the stability proof later. It is done here byestablishing a weaker convergence rate of the scheme as in [21] together with estimates from theKato-Ponce inequaltiy. Lemma 3.8. (A priori estimate of v n ) For any γ > , if v ∈ H γ +2 , then there exist constants τ > and C > , such that for any < τ ≤ τ we have k v n k H γ ≤ C, n = 0 , , . . . , Tτ , where τ and C depend only on T and k v k L ∞ ((0 ,T ); H γ ) .Proof. The proof goes in the manner of bootstrap argument by assuming that v n ∈ H γ for some0 ≤ n ≤ Tτ . Taking the difference between (3.4) and the exact Duhamel’s formula (1.2), we have v n +1 − v ( t n +1 ) = v n − v ( t n ) + L + L , n = 0 , . . . , Tτ − , where we denote L = 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:20)(cid:16) e − ( t n + s ) ∂ x v n (cid:17) − (cid:16) e − ( t n + s ) ∂ x v ( t n + s ) (cid:17) (cid:21) ds,L = Z τ s e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x v n · e − ( t n + s ) ∂ x v nt i ds. Thus, we get that (cid:13)(cid:13) v n +1 − v ( t n +1 ) (cid:13)(cid:13) H γ ≤ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + 2 (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) + 2 (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) + 2 (cid:13)(cid:13) L (cid:13)(cid:13) H γ + 2 (cid:13)(cid:13) L (cid:13)(cid:13) H γ . (3.11)In the following, we shall give estimate of the right-hand side of (3.11) term by term.Firstly, we decompose L into two parts as L = 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:20)(cid:16) e − ( t n + s ) ∂ x v n (cid:17) − (cid:16) e − ( t n + s ) ∂ x v ( t n ) (cid:17) (cid:21) ds + 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:20)(cid:16) e − ( t n + s ) ∂ x v ( t n ) (cid:17) − (cid:16) e − ( t n + s ) ∂ x v ( t n + s ) (cid:17) (cid:21) ds =: L + L . (3.12) Then we write (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) = (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) + (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) . For the first part, we have2 (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) = Z τ (cid:28) J γ e − ( t n + s ) ∂ x ( v n − v ( t n )) , J γ ∂ x (cid:20)(cid:16) e − ( t n + s ) ∂ x ( v n − v ( t n )) (cid:17) (cid:21)(cid:29) ds (3.13a)+ 2 Z τ D J γ e − ( t n + s ) ∂ x ( v n − v ( t n )) , J γ ∂ x h e − ( t n + s ) ∂ x (cid:0) v n − v ( t n ) (cid:1) · e − ( t n + s ) ∂ x v ( t n ) i E ds. (3.13b)For (3.13a), using Lemma 3.4-(ii), we get that for any γ > , (cid:12)(cid:12) (3.13a) (cid:12)(cid:12) . τ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ . For (3.13b), using Lemma 3.4-(i) instead, we get that for any γ ≥ , γ > , (cid:12)(cid:12) (3.13b) (cid:12)(cid:12) . τ k v ( t n ) k H γ + γ k v n − v ( t n ) k H γ , and then for any γ > , by properly choosing the γ, γ and the assumption of the lemma with γ = γ + γ + 1, we have k v ( t n ) k H γ + γ . . Hence, in total we obtain that (cid:12)(cid:12)(cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11)(cid:12)(cid:12) ≤ Cτ (cid:16)(cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:17) , (3.14)where the constant C > k v k L ∞ (( t n ,t n +1 ); H γ γ ) .For (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) , we claim that (cid:13)(cid:13) L (cid:13)(cid:13) H γ . τ k v k L ∞ (( t n ,t n +1 ); H γ ) . (3.15)Indeed, using Lemma 3.1 (i), we get that (cid:13)(cid:13) L (cid:13)(cid:13) H γ ≤ Z τ (cid:13)(cid:13)(cid:13) J γ ∂ x h e − ( t n + s ) ∂ x (cid:0) v ( t n ) − v ( t n + s ) (cid:1) · e − ( t n + s ) ∂ x (cid:0) v ( t n ) + v ( t n + s ) (cid:1)i (cid:13)(cid:13)(cid:13) L ds . Z τ (cid:13)(cid:13)(cid:13) J γ +1 h e − ( t n + s ) ∂ x (cid:0) v ( t n ) − v ( t n + s ) (cid:1) · e − ( t n + s ) ∂ x (cid:0) v ( t n ) + v ( t n + s ) (cid:1)i (cid:13)(cid:13)(cid:13) L ds . τ (cid:13)(cid:13) v ( t n + s ) − v ( t n ) (cid:13)(cid:13) L ∞ ((0 ,τ ); H γ ) k v k L ∞ (( t n ,t n +1 ); H γ ) . Note that (cid:13)(cid:13) v ( t n + s ) − v ( t n ) (cid:13)(cid:13) L ∞ ((0 ,τ ); H γ ) = (cid:13)(cid:13)(cid:13) Z s ∂ t v ( t n + t ) dt (cid:13)(cid:13)(cid:13) L ∞ ((0 ,τ ); H γ ) ≤ τ (cid:13)(cid:13) ∂ t v ( t ) (cid:13)(cid:13) L ∞ (( t n ,t n +1 ); H γ ) . Now we need the estimate on ∂ t v ( t n ). From the definition (2.2), using Lemma 3.1 (i), we have thatfor any γ ≥ k ∂ t v ( t n ) k H γ . (cid:13)(cid:13)(cid:0) e − t n ∂ x v ( t n ) (cid:1) (cid:13)(cid:13) H γ +1 . (cid:13)(cid:13) v ( t n ) (cid:13)(cid:13) H γ +1 . (3.16)Using (3.16), we have (cid:13)(cid:13) v ( t n + s ) − v ( t n ) (cid:13)(cid:13) L ∞ ((0 ,τ ); H γ ) ≤ τ k v k L ∞ (( t n ,t n +1 ); H γ ) . Hence, we obtain (3.15) and then we get (cid:12)(cid:12)(cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11)(cid:12)(cid:12) . τ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ k v k L ∞ (( t n ,t n +1 ); H γ ) . The last estimate together with (3.14) and Cauchy-Schwartz’s inequality, we establish that (cid:12)(cid:12)(cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11)(cid:12)(cid:12) ≤ Cτ (cid:16)(cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:17) + Cτ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ ≤ Cτ (cid:16)(cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:17) + Cτ , (3.17) OW-REGULARITY INTEGRATOR FOR KDV EQUATION 13 where the constant
C > k v k L ∞ (( t n ,t n +1 ); H γ ) .Now we consider k L j k H γ for j = 1 , L , from (3.12) and (3.15), we only need toconsider L . Indeed, from (3.9), we have (cid:13)(cid:13) L (cid:13)(cid:13) H γ . √ τ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:13)(cid:13) v n + v ( t n ) (cid:13)(cid:13) H γ . √ τ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + √ τ (cid:13)(cid:13) v ( t n ) (cid:13)(cid:13) H γ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ . The above estimate together with (3.15) give (cid:13)(cid:13) L (cid:13)(cid:13) H γ ≤ C √ τ (cid:16)(cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:17) + Cτ , (3.18)where the constant C > k v k L ∞ (( t n ,t n +1 ); H γ ) .For L , we write L = Z τ s e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x v n · e − ( t n + s ) ∂ x v nt i ds = Z τ s e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x v n · e − ( t n + s ) ∂ x v nt − e − ( t n + s ) ∂ x v ( t n ) · e − ( t n + s ) ∂ x ∂ t v ( t n ) i ds (3.19a)+ Z τ s e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x v ( t n ) · e − ( t n + s ) ∂ x ∂ t v ( t n ) i ds. (3.19b)For (3.19a), from (2.2) and (3.3), and Lemma 3.7-(ii), we get (cid:13)(cid:13) (3.19a) (cid:13)(cid:13) H γ . τ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:16)(cid:13)(cid:13) v n (cid:13)(cid:13) H γ + (cid:13)(cid:13) v ( t n ) (cid:13)(cid:13) H γ (cid:17) . τ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + τ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:13)(cid:13) v ( t n ) (cid:13)(cid:13) H γ . For (3.19b), by Lemma 3.1 (i), we get (cid:13)(cid:13) (3.19b) (cid:13)(cid:13) H γ . τ (cid:13)(cid:13)(cid:13) ∂ x h e − ( t n + s ) ∂ x v ( t n ) · e − ( t n + s ) ∂ x ∂ t v ( t n ) i(cid:13)(cid:13)(cid:13) H γ . τ k v ( t n ) k H γ k ∂ t v ( t n ) k H γ . Then using (3.16), we get (cid:13)(cid:13) (3.19b) (cid:13)(cid:13) H γ . τ (cid:13)(cid:13) v ( t n ) (cid:13)(cid:13) H γ . Combining with these two estimates yields (cid:13)(cid:13) L (cid:13)(cid:13) H γ ≤ Cτ (cid:16)(cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:17) + Cτ , (3.20)and thus by H¨older’s and Cauchy-Schwartz’s inequalities, (cid:10) J γ ( v n − v ( t n )) , J γ L (cid:11) ≤ Cτ (cid:16)(cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:17) + Cτ , (3.21)where the constant C > k v k L ∞ (( t n ,t n +1 ); H γ ) .Now inserting the estimates (3.17), (3.18), (3.20) and (3.21) into (3.11), we obtain that (cid:13)(cid:13) v n +1 − v ( t n +1 ) (cid:13)(cid:13) H γ ≤ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + Cτ (cid:16)(cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ (cid:17) + Cτ . This implies that (cid:13)(cid:13) v n +1 − v ( t n +1 ) (cid:13)(cid:13) H γ ≤ (1 + Cτ ) (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + Cτ (cid:13)(cid:13) v n − v ( t n ) (cid:13)(cid:13) H γ + Cτ . Noting v = v (0) and by Gronwall’s inequality, we obtain that (cid:13)(cid:13) v n +1 − v ( t n +1 ) (cid:13)(cid:13) H γ ≤ Cτ n +1 X j =0 (cid:0) Cτ (cid:1) j ≤ Cτ (cid:0) Cτ (cid:1) Tτ ≤ Cτ e CT , for 0 ≤ n ≤ T /τ −
1. This proves the claimed result of the lemma when 0 < τ ≤ τ for some τ depending on T and k v k L ∞ ((0 ,T ); H γ ) . (cid:3) Stability.
Now we give stability result of the numerical propagator Φ n defined in (3.7) in thefollowing lemma. Lemma 3.9. (Stability) Let γ ≥ and v ∈ H γ +4 , then there exist some constant C, τ > , suchthat for any < τ ≤ τ , (cid:13)(cid:13) Φ n (cid:0) v ( t n ) (cid:1) − Φ n (cid:0) v n (cid:1)(cid:13)(cid:13) H γ ≤ (cid:0) Cτ (cid:1) k v ( t n ) − v n k H γ , n = 0 , , . . . , Tτ − , where the constants C, τ depend only on T and k v k L ∞ ((0 ,T ); H γ +4 ) .Proof. We denote f = v ( t n ) − v n , f = ∂ t v ( t n ) − v nt , g = v ( t n ) + v n , g = ∂ t v ( t n ) + v nt for0 ≤ n ≤ T /τ −
1, thenΦ n (cid:0) v ( t n ) (cid:1) − Φ n (cid:0) v n (cid:1) = f + 12 Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds + 12 Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds + 12 Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds. Hence, we have (cid:13)(cid:13) Φ (cid:0) v ( t n ) (cid:1) − Φ (cid:0) v n (cid:1)(cid:13)(cid:13) H γ ≤k f k H γ + D J γ Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds, J γ f E (3.22a)+ D J γ Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds, J γ f E (3.22b)+ D J γ Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds, J γ f E (3.22c)+ 34 (cid:13)(cid:13)(cid:13) Z τ e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds (cid:13)(cid:13)(cid:13) H γ (3.22d)+ 34 (cid:13)(cid:13)(cid:13) Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds (cid:13)(cid:13)(cid:13) H γ (3.22e)+ 34 (cid:13)(cid:13)(cid:13) Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds (cid:13)(cid:13)(cid:13) H γ . (3.22f)Now we estimate (3.22a)–(3.22f) term by term.We begin with estimate of (3.22a). Applying Lemma 3.4-(i), we get for any γ ≥ | (3.22a) | . τ (cid:13)(cid:13) f (cid:13)(cid:13) H γ (cid:13)(cid:13) g (cid:13)(cid:13) H γ +2 . From the a prior estimate in Lemma 3.8, we have that when 0 < τ ≤ τ , (cid:13)(cid:13) g (cid:13)(cid:13) H γ +2 ≤ C, (3.23)for some τ , C depend on T and k v k L ∞ ((0 ,T ); H γ +4 ) . Hence, we further obtain | (3.22a) | ≤ Cτ (cid:13)(cid:13) f (cid:13)(cid:13) H γ . (3.24)Now we estimate the terms (3.22b) and (3.22e) which can be done in the same manner. To dothis, by the formula g = 12 e t n ∂ x ∂ x h(cid:0) e − t n ∂ x v ( t n ) (cid:1) + (cid:0) e − t n ∂ x v n (cid:1) i , and Lemma 3.7-(i), we have for γ ≥ , γ > , (cid:13)(cid:13)(cid:13) Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds (cid:13)(cid:13)(cid:13) H γ . τ k f k H γ + γ − (cid:16)(cid:13)(cid:13) v ( t n ) (cid:13)(cid:13) H γ + γ + (cid:13)(cid:13) v n (cid:13)(cid:13) H γ + γ (cid:17) . OW-REGULARITY INTEGRATOR FOR KDV EQUATION 15
From Lemma 3.8, we further get (cid:13)(cid:13)(cid:13) Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds (cid:13)(cid:13)(cid:13) H γ ≤ Cτ k f k H γ , where the constant C > k v k L ∞ ((0 ,T ); H γ +4 ) . By this estimate, we get | (3.22b) | ≤ Cτ k f k H γ , (3.25a) | (3.22e) | ≤ Cτ k f k H γ . (3.25b)Next, we treat the terms (3.22c) and (3.22f) in the same manner. Using the relationship f = 12 e t n ∂ x ∂ x (cid:16) e − t n ∂ x f · e − t n ∂ x g (cid:17) , and Lemma 3.7-(i), we have for γ ≥ , γ > , (cid:13)(cid:13)(cid:13) Z τ s e ( t n + s ) ∂ x ∂ x (cid:16) e − ( t n + s ) ∂ x f · e − ( t n + s ) ∂ x g (cid:17) ds (cid:13)(cid:13)(cid:13) H γ . τ k f k H γ k g k H γ + γ . Using this estimate and (3.23), we get | (3.22c) | ≤ Cτ k f k H γ , (3.26a) | (3.22f) | ≤ Cτ k f k H γ . (3.26b)Now it is left to consider (3.22d). By Lemma 3.6 and (3.23), we get for γ ≥ , γ > | (3.22d) | ≤ Cτ (cid:13)(cid:13) f (cid:13)(cid:13) H γ (cid:13)(cid:13) g (cid:13)(cid:13) H γ + γ ≤ Cτ (cid:13)(cid:13) f (cid:13)(cid:13) H γ . (3.27)Combining the estimates (3.24)-(3.27), we conclude that (cid:13)(cid:13) Φ n (cid:0) v ( t n ) (cid:1) − Φ n (cid:0) v n (cid:1)(cid:13)(cid:13) H γ ≤ k f k H γ + Cτ (cid:13)(cid:13) f (cid:13)(cid:13) H γ , where C > T and k v k L ∞ ((0 ,T ); H γ +4 ) . Since √ Cτ ∼ Cτ when τ is small enough,we finish the proof of the lemma. (cid:3) Local error.
Next, we have the following optimal estimate for the local error term L n in (3.6). Lemma 3.10. (Local error estimate) Let L n be defined in (3.6) and γ ≥ , then we have kL n k H γ ≤ Cτ , n = 0 , . . . , Tτ − , where the constant C > depends only on T and k u k L ∞ ((0 ,T ); H γ +4 ) .Proof. For simplicity, we denote for n = 0 , , . . . , Tτ − w n ( s ) = v ( t n + s ) − v ( t n ) − s∂ t v ( t n ) , h n ( s ) = v ( t n + s ) + v ( t n ) + s∂ t v ( t n ) , s ≥ . Then from the definition, we have L n = 12 Z τ e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x w n ( s ) · e − ( t n + s ) ∂ x h n ( s ) i ds (3.28)+ 12 Z τ s e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x ∂ t v ( t n ) i ds. Noting that by Taylor’s expansion, w n ( s ) = Z s Z s ′ ∂ t v ( t n + t ) dtds ′ , and then by the formula in Lemma 3.5-(i), we see w n ( s ) = Z s Z s ′ e ( t n + t ) ∂ x ∂ x (cid:20) (cid:16) e − ( t n + t ) ∂ x ∂ x v ( t n + t ) (cid:17) + 13 (cid:16) e − ( t n + t ) ∂ x v ( t n + t ) (cid:17) (cid:21) dtds ′ . Plugging the above formula into (3.28), we get L n = L n + L n , where L n := 12 Z τ Z s Z s ′ e ( t n + s ) ∂ x ∂ x " e ( t − s ) ∂ x ∂ x (cid:20) (cid:16) e − ( t n + t ) ∂ x ∂ x v ( t n + t ) (cid:17) + 13 (cid:16) e − ( t n + t ) ∂ x v ( t n + t ) (cid:17) (cid:21) · e − ( t n + s ) ∂ x h n ( s ) dtds ′ ds, L n := 12 Z τ s e ( t n + s ) ∂ x ∂ x h e − ( t n + s ) ∂ x ∂ t v ( t n ) i ds. For L n , firstly we have kL n k H γ . Z τ Z s Z s ′ (cid:13)(cid:13)(cid:13) e ( t n + s ) ∂ x ∂ x h e ( t − s ) ∂ x ∂ x (cid:0) e − ( t n + t ) ∂ x ∂ x v ( t n + t ) (cid:1) · e − ( t n + s ) ∂ x h n ( s ) i(cid:13)(cid:13)(cid:13) H γ + (cid:13)(cid:13)(cid:13) e ( t n + s ) ∂ x ∂ x h e ( t − s ) ∂ x ∂ x (cid:16) e − ( t n + t ) ∂ x v ( t n + t ) (cid:17) · e − ( t n + s ) ∂ x h n ( s ) i(cid:13)(cid:13)(cid:13) H γ ! dtds ′ ds . Z τ Z τ Z τ (cid:13)(cid:13)(cid:13) e ( t − s ) ∂ x ∂ x (cid:0) e − ( t n + t ) ∂ x ∂ x v ( t n + t ) (cid:1) · e − ( t n + s ) ∂ x h n ( s ) (cid:13)(cid:13)(cid:13) H γ +1 + (cid:13)(cid:13)(cid:13) e ( t − s ) ∂ x ∂ x (cid:16) e − ( t n + t ) ∂ x v ( t n + t ) (cid:17) · e − ( t n + s ) ∂ x h n ( s ) (cid:13)(cid:13)(cid:13) H γ +1 ! dtds ′ ds. Then by using Lemma 3.1 (i), we obtain kL n k H γ . Z τ Z τ Z τ (cid:20)(cid:13)(cid:13)(cid:13) ∂ x (cid:0) e − ( t n + t ) ∂ x ∂ x v ( t n + t ) (cid:1) (cid:13)(cid:13)(cid:13) H γ +1 (cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 + (cid:13)(cid:13)(cid:13) ∂ x (cid:0) e − ( t n + t ) ∂ x v ( t n + t ) (cid:1) (cid:13)(cid:13)(cid:13) H γ +1 (cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 (cid:21) dtds ′ ds . Z τ Z τ Z τ (cid:20)(cid:13)(cid:13)(cid:13)(cid:0) e − ( t n + t ) ∂ x ∂ x v ( t n + t ) (cid:1) (cid:13)(cid:13)(cid:13) H γ +3 (cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 + (cid:13)(cid:13)(cid:13)(cid:0) e − ( t n + t ) ∂ x v ( t n + t ) (cid:1) (cid:13)(cid:13)(cid:13) H γ +3 (cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 (cid:21) dtds ′ ds. Using Lemma 3.1 (i) again, we get that kL n k H γ . Z τ Z τ Z τ h(cid:13)(cid:13) v ( t n + t ) (cid:13)(cid:13) H γ +4 (cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 + (cid:13)(cid:13) v ( t n + t ) (cid:13)(cid:13) H γ +3 (cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 i dtds ′ ds. Hence, in sum, we get kL n k H γ ≤ C Z τ Z τ Z τ (cid:16)(cid:13)(cid:13) v ( t n + t ) (cid:13)(cid:13) H γ +4 + (cid:13)(cid:13) v ( t n + t ) (cid:13)(cid:13) H γ +4 (cid:17)(cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 dtds ′ ds. (3.29)Now we control the term h n ( s ). From (2.2) and the Kato-Ponce inequality in Lemma 3.1, we have (cid:13)(cid:13) h n ( s ) (cid:13)(cid:13) H γ +1 . k v ( t n + s ) k H γ +1 + k v ( t n ) k H γ +1 + s k ∂ t v ( t n ) k H γ +1 . k v ( t n + s ) k H γ +1 + k v ( t n ) k H γ +1 + s (cid:13)(cid:13)(cid:13) (cid:16) e − t n ∂ x v ( t n ) (cid:17) (cid:13)(cid:13)(cid:13) H γ +2 . k v ( t n + s ) k H γ +1 + k v ( t n ) k H γ +1 + s (cid:13)(cid:13) v ( t n ) (cid:13)(cid:13) H γ +2 . k v k L ∞ ((0 ,T ); H γ +2 ) + τ k v k L ∞ ((0 ,T ); H γ +2 ) . (3.30)Inserting this estimate into (3.29), we get kL n k H γ ≤ Cτ , where C depends on k v k L ∞ ((0 ,T ); H γ +4 ) . OW-REGULARITY INTEGRATOR FOR KDV EQUATION 17
For L n , similarly as above, we have kL n k H γ . Z τ s (cid:13)(cid:13)(cid:13) ∂ x h e − ( t n + s ) ∂ x ∂ t v ( t n ) i ds (cid:13)(cid:13)(cid:13) H γ ds . Z τ s (cid:16)(cid:13)(cid:13) ∂ x ∂ t v ( t n ) (cid:13)(cid:13) H γ (cid:13)(cid:13) ∂ t v ( t n ) (cid:13)(cid:13) H γ + (cid:13)(cid:13) ∂ x ∂ t v ( t n ) (cid:13)(cid:13) L (cid:13)(cid:13) ∂ t v ( t n ) (cid:13)(cid:13) H γ + γ (cid:17) ds. Similarly as (3.30), we obtain that kL n k H γ . τ k v k L ∞ ((0 ,T ); H γ +2 ) . Combining the estimates on L n and L n , we finish the proof of the lemma. (cid:3) Proof of Theorem 2.1.
Now, combining the local error estimate and the stability results,we give the proof of Theorem 2.1. As described in the subsection 3.2, it is sufficient to estimate k v ( t n ) − v n k H γ . From (3.5), Lemma 3.10 and Lemma 3.9, there exit constants C > τ > < τ ≤ τ , we have k v ( t n +1 ) − v n +1 k H γ ≤ Cτ + (1 + Cτ ) k v ( t n ) − v n k H γ , n = 0 , , . . . , Tτ − , where C, τ depend on T and k v k L ∞ ((0 ,T ); H γ +4 ) . By iteration and Gronwall’s inequality, we get k v ( t n +1 ) − v n +1 k H γ ≤ τ n X j =0 (1 + Cτ ) j ≤ Cτ , n = 0 , , . . . , Tτ − , which proves Theorem 2.1. (cid:3) Numerical results
In this section, we carry out numerical experiments of the presented LRI scheme (2.5) for justifyingthe convergence theorem. Also, we provide the numerical investigations of convergence of the Strangsplitting scheme [12, 13] (or see the Appendix A) as comparisons.To get an initial data with the desired regularity, we construct u ( x ) by the following strategy[26]. Choose N > T with grid points x j = j πN for j = 0 , . . . , N . Take a uniformly distributed random vectors rand( N, ∈ [0 , N and denote U N = rand( N, . Then we define u ( x ) := | ∂ x,N | − θ U N k| ∂ x,N | − θ U N k L ∞ , x ∈ T , (4.1)where the pseudo-differential operator | ∂ x,N | − θ for θ ≥ l = − N/ , . . . , N/ − (cid:0) | ∂ x,N | − θ (cid:1) l = ( | l | − θ if l = 0 , l = 0 . Thus, we get u ∈ H θ ( T ) for any θ ≥
0. We implement the spatial discretizations of the numericalmethods within discussions by the Fourier pseudo-spectral method [29] with a large number of gridpoints N = 2 in the torus domain T . We shall present the error u ( x, t n ) − u n in the H γ -norm( γ = 0 or 2) at the final time t n = T = 2, where the exact solution is obtained numerically by theLRI scheme (2.5) with τ = 10 − . Figure 1 shows the convergence results of the LRI scheme (2.5) byusing different time step τ under the initial data of different regularities. In Figure 2, we show thecorresponding convergence curves of the Strang splitting scheme (A.1) from [12, 13]. The details ofthe implementations of the Strang splitting scheme is given in the Appendix A.Based on the numerical results from Figures 1 & 2, we have the following observations:1) The presented LRI scheme (2.5) has the second order accuracy in time under H γ -norm withinitial data in H γ +4 for any γ ≥ H γ +4 (see the red dash-dot lines in Figure 1), the LRI scheme shows some convergence orderreduction. This indicates that our theoretical estimate in Theorem 2.1 is optimal and the regularityassumption is sharp. -6 -5 -4 -3 -2 || u - u n || L / || u || L H -dataH -dataO( ) -5 -4 -3 -2 || u - u n || H / || u || H H -dataH -dataO( ) Figure 1.
Convergence of the LRI scheme: relative error k u − u n k L / k u k L (left)and k u − u n k H / k u k H (right) at t n = T = 2 under initial data of different regu-larities. -6 -5 -4 -3 -2 || u - u n || H / || u || H H -dataH -dataO( ) Figure 2.
Convergence of the Strang splitting scheme: relative error k u − u n k H / k u k H at t n = T = 2 under initial data of different regularities.2) The Strang splitting scheme (A.1) converges at the second order rate in H γ with initial datain H γ +5 (see the blue solid line in Figure 2), which confirms the theoretical result proved in [13].With less regular initial data, e.g. H γ +4 initial data, the scheme still converges but with an unstableorder (see the red dash-dot line in Figure 2). The implicity of Strang splitting scheme makes thecomputations very time-consuming.3) The error from LRI (2.5) and the Strang splitting scheme (A.1) are rather similar (cf. theright one in Figure 1 and Figure 2), while the LRI (2.5) is much more efficient.5. Conclusion
In this work, we have studied numerically the KdV equation on a torus under rough initial data.By some rigorous tools from harmonic analysis, we established the sharp convergence theorem ofan exponential-type integrator as outlined in [11]. The theoretical result shows that the presentedintegrator can reach the second order accuracy in H γ space with initial data from H γ +4 for any OW-REGULARITY INTEGRATOR FOR KDV EQUATION 19 γ ≥
0. Compared with classical numerical methods, the presented integrator requires less regularityof the solution for optimal convergence rate and is more efficient for solving the KdV equation underrough initial data case.
Appendix A. Strang splitting scheme
As firstly used in [31], the Strang splitting method applies to the KdV equation (1.1) by splittingit into a linear part: Φ tA : ∂ t u ( t, x ) + ∂ x u ( t, x ) = 0 , t > , x ∈ T , and an inviscid Burgers equation:Φ tB : ∂ t u ( t, x ) = 12 ∂ x ( u ( t, x )) , t > , x ∈ T , where Φ tA ( · ) and Φ tB ( · ) denote the propagators. Then the Strang splitting scheme reads: denote u n = u n ( x ) ≈ u ( t n , x ) and for n ≥ u n +1 = Φ τ/ A ◦ Φ τB ◦ Φ τ/ A ( u n ) . (A.1)The propagator Φ tA ( u ) = e − t∂ x u is given exactly. Here to implement the Strang splittng schemeas has been analyzed in [12, 13], we seek for the exact solution at the Burgers step (at least up tomachine precision). The solution of Φ tB ( u ) can be given by the characteristics method as follows.For x ∈ T , let x = x ( t ) satisfying˙ x ( t ) = − u ( t, x ( t )) , t > , x (0) = x . Along the characteristics we have ddt u ( t, x ( t )) = 0, and so ˙ x ( t ) = − u (0 , x ) which gives x ( t ) − x = − tu (0 , x ) , t ≥ . Hence, with u (0 , x ) = u ( x ) known, if we want to compute u ( t, x j ) at the grid point x j ∈ T , we set x ( t ) = x j and so u ( t, x j ) = u ( x ). Then we solve the nonlinear equation x j = x − tu ( x ) forthe initial position x , which can be done by for example the Newton’s iteration. Afterwards, weinterpolate u at x , which can be obtained accurately by the non-uniform fast Fourier transform(NUFFT) [8]. In our implementation, we apply the NUFFT to the accuracy δ = 10 − and thesame for the Newton’s iteration: δ = x j − x ( n )0 + tu (cid:16) x ( n )0 (cid:17) . The full scheme is implicit. Acknowledgements
Y. Wu is partially supported by NSFC 11771325 and 11571118. X. Zhao is partially supportedby the Natural Science Foundation of Hubei Province No. 2019CFA007, the NSFC 11901440 andthe starting research grant of Wuhan University. Part of the work was done while the authors werevisiting the Shanghai Center for Mathematical Sciences.
References [1]
J. Bao, Y. Wu , Global well-posedness for the periodic generalized Korteweg-de Vries equation, Indiana Univ.Math. J. 66 (2017) pp. 1797-1825.[2]
J. Bourgain, D. Li , On an endpoint Kato-Ponce inequality, Differential Integral Equations 27 (2014) pp. 1037-1072.[3]
Y. Bruned, K. Schratz , Resonance based schemes for dispersive equations via decorated trees,arXiv:2005.01649v1 [math.NA].[4]
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao , Sharp global well-posedness for KdV andmodified Kdv on R and T , J. Amer. Math. Soc. 16 (2003) pp. 705-749.[5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao , Multilinear estimates for periodic KdV equa-tions, and applications, J. Funct. Anal. 211 (2004) pp. 173-218.[6]
C. Court`es, F. Lagouti`ere, F. Rousset , Error estimates of finite difference schemes for the Korteweg-de Vriesequation, IMA J. Numer. Anal. 40 (2020) pp. 628-685.[7]
B. Guo, J. Shen , On spectral approximations using modified Legendre rational functions: Application to theKorteweg-de Vries equation on the half line, Indiana Univ. Math. J. 50 (2001) pp. 181-204.[8]
L. Greengard, J.Y. Lee , Accelerating the nonuniform fast Fourier transform, SIAM Rev. 46 (2004) pp. 443-454.[9]
M. Gubinelli , Rough solutions for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Anal. 11 (2012)pp. 709-733. [10]
M. Hochbruck, A. Ostermann , Exponential integrators, Acta Numer. 19 (2010) pp. 209-286.[11]
M. Hofmanov´a, K. Schratz , An exponential-type integrator for the KdV equation, Numer. Math. 136 (2017)pp. 1117-1137.[12]
H. Holden, C. Lubich, N.H. Risebro , Operator splitting for partial differential equations with Burgers nonlin-earity, Math. Comp. 82 (2012) pp. 173-185.[13]
H. Holden, K.H. Karlsen, N.H. Risebro, T. Tao , Operator splitting methods for the Korteweg-de Vriesequation, Math. Comp. 80 (2011) pp. 821-846.[14]
H. Holden, K.H. Karlsen, N.H. Risebro , Operator splitting methods for generalized Korteweg-de Vries equa-tions, J. Comput. Phys. 153 (1999) pp. 203-222.[15]
T. Kato, G. Ponce , Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl.Math. 41 (1988) pp. 891-907.[16]
C. Klein , Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schr¨odinger equation,ETNA 29 (2008) pp. 116-135.[17]
M. Kn¨oller, A. Ostermann, K. Schratz , A Fourier integrator for the cubic nonlinear Schr¨odinger equationwith rough initial data, SIAM J. Numer. Anal. 57 (2019) pp. 1967-1986.[18]
T. Kappeler, P. Topalov,
Global wellposedness of KdV in H − ( T , R ), Duke Math. J. 135 (2006) pp. 327-360.[19] D. Li , On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam. 35 (2019) pp. 23-100.[20]
H. Liu, J. Yan , A local discontinuous Galerkin method for the Kortewegde Vries equation with boundary effect,J. Comput. Phys. 215 (2006) pp. 197-218.[21]
Ch. Lubich , On splitting methods for Schr¨odinger-Poisson and cubic nonlinear Schr¨odinger equations, Math.Comp. 77 (2008) pp. 2141-2153.[22]
H. Ma, W. Sun , Optimal error estimates of the Legendre–Petrov–Galerkin method for the Korteweg–de Vriesequation, SIAM J. Numer. Anal. 39 (2001) pp. 1380-1394.[23]
Y. Maday, A. Quarteroni , Error analysis for spectral approximation of the Korteweg-de Vries equation,RAIRO-Mod´elisation math´ematique et analyse num´erique 22 (1988) pp. 821-846.[24]
R.I. McLachlan, G.R.W. Quispel , Splitting methods, Acta Numer. 11 (2002) pp. 341-434.[25]
A. Ostermann, F. Rousset, K. Schratz , Error estimates of a Fourier integrator for the cubic Schr¨odingerequation at low regularity, to appear on Found. Comput. Math. (2020).[26]
A. Ostermann, K. Schratz , Low regularity exponential-type integrators for semilinear Schr¨odinger equations,Found. Comput. Math. 18 (2018) pp. 731-755.[27]
A. Ostermann, C. Su , A Lawson-type exponential integrator for the Korteweg-de Vries equation, to appear onIMA J. Numer. Anal. (2020) https://doi.org/10.1093/imanum/drz030.[28]
J. Shen , A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: applicationto the KdV equation, SIAM J. Numer. Anal. 41 (2003) pp. 1595-1619.[29]
J. Shen, T. Tang, L. Wang , Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011.[30]
K. Schratz, Y. Wang, X. Zhao , Low-regularity integrators for nonlinear Dirac equations, to appear on Math.Comp. (2020).[31]
F. Tappert . Numerical solutions of the Korteweg-de Vries equation and its generalizations by the split-stepFourier method. In: (A.C. Newell, editor) Nonlinear Wave Motion, Amer. Math. Soc. 1974, pp. 215-216.[32]
J. Yan, C.W. Shu , A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal. 40(2002) pp. 769-791.
Y. Wu: Center for Applied Mathematics, Tianjin University, 300072, Tianjin, China
E-mail address : [email protected] X. Zhao: School of Mathematics and Statistics & Computational Sciences Hubei Key Laboratory, WuhanUniversity, Wuhan, 430072, China
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