MATHEMATICAL ENGINEERING TECHNICAL REPORTS. Numerical Integration of the Ostrovsky Equation Based on Its Geometric Structures

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1 MATHEMATICAL ENGINEERING TECHNICAL REPORTS Numerical Integration of the Ostrovsy Equation Based on Its Geometric Structures Yuto MIYATAKE, Taaharu YAGUCHI and Taayasu MATSUO METR April DEPARTMENT OF MATHEMATICAL INFORMATICS GRADUATE SCHOOL OF INFORMATION SCIENCE AND TECHNOLOGY THE UNIVERSITY OF TOKYO BUNKYO-KU, TOKYO -866, JAPAN WWW page:

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3 Numerical Integration of the Ostrovsy Equation Based on Its Geometric Structures Yuto MIYATAKE, Taaharu YAGUCHI and Taayasu MATSUO April 8th, Abstract We consider structure preserving numerical schemes for the Ostrovsy equation, which describes gravity waves under the influence of Coriolis force. This equation has two associated invariants: an energy function and the L norm. It is widely accepted that structure preserving methods such as invariants-preserving and multi-symplectic integrators generally yield qualitatively better numerical results. In this paper we propose five geometric integrators for this equation: energy-preserving and norm-preserving finite difference and Galerin schemes, and a multi-symplectic integrator based on a newly found multi-symplectic formulation. A numerical comparison of these schemes is provided, which indicates that the energy-preserving finite difference schemes are more advantageous than the other schemes. eyword Ostrovsy equation, Conservation, Multi-symplecticity, Discrete variational derivative method, Discrete partial derivative method Introduction In this paper we consider geometric numerical integration for the Ostrovsy equation [] under the periodic boundary condition of length L: u t + αuu βu γ u, ut, ut, + L, where α, β, γ are real parameters and the subscript t or, respectively denotes the differentiation with respect to time variable t or. This equation models gravity waves under the influence of Coriolis force. The parameter β measures the dispersion effects and γ measures the effect of the rotation. For eample, for β <, the equation models surface and internal waves in the ocean and surface waves in a shallow channel with uneven bottom [], and for β >, it models capillary waves on the surface of a liquid and magneto-acoustic Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Toyo, 7-- Hongo Bunyo-u, Toyo, -866, Japan. yuto miyatae@mist.i.u-toyo.ac.jp Department of Computational Science, Graduate School of System Informatics, Kobe University, - Roodai-cho, Nada-u, Kobe, 67-8, Japan yaguchi@pearl.obe-u.ac.jp Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Toyo, 7-- Hongo Bunyo-u, Toyo, -866, Japan. matsuo@mist.i.u-toyo.ac.jp This wor is a full version of our recent quic report [], with detailed discussions including etra schemes and numerical results.

4 waves in a plasma [, ]. In the absence of rotation γ, leads to the well-nown Korteweg de Vries KdV equation. Despite its physical importance for the case γ, only a few numerical methods have been proposed. Hunter [] used a finite difference scheme to investigate solutions under both positive and negative dispersion effects. In [6, ] Fourier-pseudospectral and Fourier-Galerin schemes are used to eamine the evolutions of soliton-lie solutions. Liu Pelinovsy Saovich studied the wave breaing phenomena for the case β theoretically and numerically []. More recently Yaguchi Matsuo Sugihara proposed four numerical schemes that inherit the following conservation properties of the Ostrovsy equation []. The Ostrovsy equation has three first integrals [7]: u d const., α 6 u + β u + γ u u d const., d const. In what follows, we call them the mass, energy, and L norm conservation laws, respectively. Note that in the mass conservation, we claimed that u is a zero-mean function. This is necessary so that u in the equation is also a periodic function. Generally, for mathematical convenience, the inverse operator is defined for any zero-mean and L-periodic function as u ut, s ds L ut, s ds d see, for eample, []. Notice this maps the function bac to a zero-mean and L-periodic function: u d, 6 as far as integration is allowed. For the PDEs with such invariants, it is widely accepted that structure preserving methods, for eample numerical methods that preserve the invariants, generally yield qualitatively better numerical results, and in the last two decades much effort has been devoted in this topic to finally find out several unified framewors. For eample, the Discrete Variational Derivative Method DVDM was proposed by Furihata [9] see also Furihata Matsuo [], the Average Vector Field Method AVFM was proposed by Celledoni et al. [] in finite difference contet, and the Discrete Partial Derivative Method DPDM was proposed by Matsuo [7] in finite element contet. From these standpoints, Yaguchi et al. have proposed four conservative numerical schemes: a finite difference scheme and a pseudospectral scheme that conserve the energy, and the same types of schemes that conserve the norm. The energy-preserving schemes are based on the following Hamiltonian structure: u t δg δu, Gu α 6 u + β u + γ u, 7 and the norm-preserving schemes are based on the following variational structure: u t α u + u + β + γ δh δu, u Hu. 8

5 They derived the conservative schemes based on the DVDM, but there was a difficulty that the original DVDM did not support the non-local operator. In order to wor around this, they etended the DVDM so that it can handle a discrete inverse operator δ that approimates and correctly replicates the zero-mean property 6. In this sense their trial was quite successful. There remained, however, a small drawbac that their operator δ failed to satisfy δ δ id., which is naturally epected corresponding to another basic property of : id. precise mathematical meaning of this identity will be described in Section.. This gave rise to unnatural average operators in the resulting schemes, which seemed to provoe undesirable oscillations in their numerical results the detail will be reviewed later. Taing these into consideration, we devote our effort to the following three points. Firstly, we propose new energy- or norm-preserving finite difference schemes where no such unnatural average operator appears. The ey here is to introduce the potential function ϕ satisfying ϕ u, and rewrite the Hamiltonian and variational structures with it. The technique is natural in the contet of dynamical systems. We then apply the DVDM to obtain conservative schemes. Our numerical eperiment strongly suggests that the qualitative behavior of numerical solutions by the new schemes is better than the eisting schemes. Secondly, we show that this equation has a multi-symplectic formulation, and provide a multi-symplectic scheme based on this formulation by applying the Preissman bo scheme. This formulation is motivated by the multi-symplectic formulation of the KdV equation by Ascher McLachlan []. Although this multi-symplectic scheme does not preserve the energy or the norm eactly, our numerical results show that the deviations are very small. Finally, we propose energy- or norm-preserving Galerin schemes. The eisting framewor the DPDM does not directly apply to the Ostrovsy equation due to the nonlocal term u. We solve this difficulty by introducing an L -projection technique. Remar. Note the difference between Hamiltonian structure 7 and variational structure 8. As is well nown, the KdV equation γ is a bi-hamiltonian PDE; i.e., both 7 and 8 give Hamiltonian structures. On the contrary, for γ, 8 should not give a Hamiltonian structure. In fact, if the Ostrovsy equation were a bi-hamiltonian PDE, it would have infinitely many first integrals. But it has been reported that this equation is not completely integrable [7], and the above three are believed to be the only invariants. This paper is organized as follows. In Section notation is introduced and the schemes by Yaguchi Matsuo Sugihara are summarized. In Section two finite difference schemes are presented. In Section a multi-symplectic formulation and a multi-symplectic scheme based on it are proposed. In Section two Galerin schemes are presented. In Section 6 some numerical results are provided. Concluding remars and comments are given in Section 7. Preliminaries In this section we prepare notation and review the approach by Yaguchi Matsuo Sugihara [].. Notation We first prepare notation used in Section, and. The interval [, L] is discretized by uniform grids with the space mesh size L/N, where N is the number of nodes. Numerical

6 solutions are denoted by U n un, or Φ n ϕn,, where is the time mesh size. We often write the solutions as a vector U n U n,..., U n. For simplicity, we also denote U n+ U n+ condition, we consider {U n } restriction by the discrete periodic boundary condition U n + U n /. In order to treat the periodic boundary, an infinitely long vector, and then its N-dimensional U n mod N for all Z. We denote the latter space by X d : {U {U } Z U R, U U mod N, for all Z}, and introduce its zero-mean subspace X d : {U U, U X d }. These are natural discretizations of continuous spaces X and X, which will be introduced in Section.. The standard central difference operators that approimate,,, are denoted by δ, δ, δ, δ δ U n respectively: U n + U n δ δ δ, δ δ,, δ U n U n n + U + U n, and the forward and bacward difference operators, which are also the approimations of, are denoted by δ + U n U n + U n, δ U n U n U n. We denote the approimations of by δ and δ, whose eplicit definitions will be provided later. We use the forward difference operator δ t + U n+ U n / which is the discretization of t. In Section we utilize the inner product defined by f, g fg d. As for the difference operators, the following summation-by-parts formulas are useful. Lemma. [9]. The difference operators δ and δ that for any two sequences U, V X d, U δ V + δ U V U δ V + Eq. 9 corresponds to the integration-by-parts formula: u v d + uv d for any two L-periodic functions ut, and vt,. u v d +. Approach by Yaguchi Matsuo Sugihara N are sew-symmetric in the sense δ U V. 9 uv d, In this subsection the previous approach by Yaguchi et al. [] is summarized. They assumed that the initial condition is given such that it satisfies U, which corresponds to, and defined the operator δ by a summation operator n δ U U n + U n j + U n U n j + U n L l + U n j, j j l

7 which corresponds to. The second term of the right hand side of enforces δ U n, which corresponds to the zero-mean property 6. We review the energy-preserving scheme first. A discrete version of the energy G αu /6 + βu / + γ u /, and accordingly a discrete variational derivative that approimates δg/δu αu / βu γ u are defined by G n α n U 6 + β δg δu n+, U n α 6 Then the scheme is defined as follows. δ + U n U n+ + δ U n + γ δ U n, + U n+ U n + U n βδ U n+ γ δ U n+. Scheme Yaguchi et al. s energy-preserving finite difference scheme []. Given an initial approimate solution U X d, we compute U n n,,... by U n+ U n δ δg δu n+, U n,..., N. Eq. corresponds to the Hamiltonian structure 7. Numerical solutions by Scheme conserve both the total mass and the energy. Theorem. Conservation of the total mass and the energy []. Under the periodic boundary condition, the numerical solutions by Scheme conserve the total mass and the energy: U n U, G n G n,,.... Net, the norm-preserving finite difference scheme is summarized. A discrete version of the norm H u /, and accordingly a discrete variational derivative that approimates δh/δu u are defined by H n n U, δh δu n+, U n U n+. Then the scheme is defined as follows. Scheme Yaguchi et al. s norm-preserving finite difference scheme []. Given an initial approimate solution U X d, we compute U n n,,... by U n+ U n α U n+ δ + δ U n+ + βδ + γ δ δh δu n+, U n,..., N.

8 Eq. corresponds to the variational structure 8. Although Yaguchi et al. actually described the scheme in a different form U n+ U n + α U n+ δ U n+ + δ U n+ βδ U n+ γ δ n+ U, it is easy to show it is equivalent to. Numerical solutions by Scheme conserve both the total mass and the norm. Theorem. Conservation of the total mass and the norm []. Under the periodic boundary condition, the numerical solutions by Scheme conserve the total mass and the norm: U n U, H n H, n,,.... Remar. Actually, the scheme corresponds to the discretization of by the midpoint rule. u t + α u u + u β u γ u, In this way, Yaguchi et al. succeeded in designing conservative schemes. There remained, however, a small disadvantage that although in an analogy with the continuous case the operator δ is epected to satisfy δ δ id., it turned out not, and an unnatural operator appears as follows. Lemma.. For any U X d, it holds δ δ U δ δ U U + U + U +. The unnecessary average operator broadens the stencil in the resulting schemes, and the behavior of the numerical solution can get slightly worse. Actually, Yaguchi et al. [] found in their paper that the schemes can get slightly unstable when the space mesh size is increased. New conservative finite difference schemes In view of the above bacground, we here propose two new conservative finite difference schemes free of such an unepected average operator. This is done by reformulating the Hamiltonian and variational structures so that no inverse operator is required eplicitly. In this section, we first consider the reformulation and then derive schemes based on the reformulated structures.. Reformulation of the Hamiltonian and variational structures with the potential ϕ We denote by X the set of L-periodic and sufficiently smooth functions, and by X its subset of all zero-mean functions. In order to eliminate the inverse operator, we here introduce 6

9 a potential function ϕ u, where is defined by. From this definition, it immediately follows that ϕ u, and that the equality 6 implies ϕ d in terms of ϕ. This means that ϕ also belongs to X. It is easy to chec that id. in X; i.e., they map u X bac to u itself. The equation can then be rewritten with ϕ to ϕ t + αϕ ϕ βϕ γϕ. Note that this does not include the inverse operator. By introducing an energy function, this can be cast into ϕ t δg δϕ, Gϕ α 6 ϕ + β ϕ + γ ϕ. 6 If we further multiply both sides by note that δg/δϕ αϕ ϕ + βϕ + γϕ X, we finally reach a Hamiltonian form: ϕ t δg δϕ. The sew-symmetry of will be shown in Lemma. below. Although the Hamiltonian form includes, actually the previous epression 6 suffices to show the energy preservation see Theorem. below. In the subsequent subsection, we will fully utilize this to construct an energy-preserving scheme avoiding inverse operators. Similarly, Eq. can be rewritten in another variational way: ϕ t By multiplying, we find α u + u β γ δh δϕ ϕ t α u + u β γ δh δϕ., Hϕ ϕ. 7 As above, the epression 7 can imply the desired norm preservation property see Theorem.. But in this case the epression 7 still eplicitly includes the inverse operator. Fortunately it can be avoided if we substitute δh/δϕ ϕ into 7, and epand the right hand side to obtain ϕ t α u + uϕ + βϕ + γϕ. 8 We will do eactly the same thing later in discrete setting to eliminate discrete inverse operators. The sew-symmetry of the operators in the Hamiltonian and variational forms are summarized in the following lemma. Lemma.. If ut, X, then the operator α u + u β γ is sewsymmetric in the sense that for any v, w X, it holds v α u + u β γ w d α u + u β γ v w d. 7

10 Proof. The sew-symmetry of u + u is well-nown see, for eample, [8]. Since is obviously sew-symmetric, it remains to show the sew-symmetry of. From the sew-symmetry of, v w d v w d vw d v w d. We can prove the energy and norm conservation laws based on the epression 6 and 7 as follows. Theorem.. Let ϕ X be a solution to. Then it preserves the following quantities: d dt G d, d dt Proof. The conservation of the energy G is immediate: d dt G d δg δϕ ϕ t d H d. ϕ t ϕ t d. The second equality follows from the epression 6. We net prove the norm H conservation property: d dt H d. δh δϕ ϕ t d δh δϕ δh δϕ ϕ t d α u + u β γ δh δϕ d Note that δh/δϕ ϕ. The first equality is just the chain rule. The second is from the sew-symmetry of, and the third is from the variational structure 7. The last follows from the sew-symmetry of α u + u β γ. Based on these Hamiltonian and variational structures, we derive energy-preserving and norm-preserving finite difference schemes in the following two subsections.. The new energy-preserving scheme In this subsection a finite difference scheme that conserves both the total mass and the energy is presented. We define a discrete version of the energy G αϕ /6+βϕ /+γϕ /, and accordingly a discrete variational derivative that approimates δg/δϕ αϕ ϕ + βϕ + γϕ by G n α 6 δ Φ n + β δ δg δφ n+, Φ n α 6 δ Φ n δ + γ Φn Φ n+, + δ + βδ Φ n+ + γφ n+. 8 Φ n+ δ Φ n + δ Φ n

11 We can easily chec the following ey equality: G n+ G n δg Φ δφ n+, Φ n n+ Φ n. 9 Using the above discrete variational derivative, we define the energy-preserving scheme as follows. Scheme Energy-preserving finite difference scheme. Given an initial approimate solution Φ X d, we compute Φ n n,,... by δ Φ n+ Φ n δg δφ n+, Φ n,..., N. Eq. corresponds to the Hamiltonian structure 6. Notice that in this scheme we claimed that the initial solution Φ should satisfy Φ X d, which is natural in view of the continuous case. The initial value Φ can be generated either by integrating u, analytically or by the summing of U via. Scheme defines a map X d X d in the following sense. Lemma.. Consider one step of Scheme starting from Φ n X d. If Φ n+ solves, then Φ n+ X d. Proof. By applying to both sides of, we find the assumption Φ n X d implies Φ n+ X d. Φn+ Numerical solutions by Scheme preserve both the total mass and the energy.. Then Theorem. Scheme : Conservation of the total mass and the energy. Under the periodic boundary condition, the numerical solutions by Scheme conserve the total mass and the energy: δ Φ n δ Φ, G n G, n,,.... Proof. The conservation of the total mass follows from the property of the difference operator and the discrete periodic boundary condition. We prove the conservation of the energy: δ. G n+ G n δg Φ δφ n+, Φ n n+ Φ δ n+ Φ n Φ n n+ Φ Φ n The first equality is from 9, the second is from, and the third follows from the sew-symmetry of δ. 9

12 . The new norm-preserving scheme In this subsection a finite difference scheme that conserves both the total mass and the norm is presented. The story becomes slightly more complicated than the previous case. We define a discrete version of norm H ϕ /, and accordingly a discrete variational derivative that approimates δh/δϕ ϕ by H n δ Φ u Note that these satisfy the ey equality: H n+ H n, δh δφ n+, Φ n δ Φ n+. δh Φ δφ n+, Φ n n+ Φ n. With this discrete variational derivative, we can formally define a prototype of the normpreserving scheme, analogously to the continuous variational structure 7. Scheme Norm-preserving finite difference scheme: a prototype. Given an initial approimate solution Φ X d, we compute Φ n n,,... by δ Φ n+ Φ n where U n+ δ α Φ n+ : X d X d is a discrete inverse operator approimating such that δ δ n+ U δ + δ U n+ βδ γδ, and δ δ δ δ δh δφ n+, Φ n,..., N, id. in X d. Note that here we used a new discrete inverse operator δ, which is different from δ used in Yaguchi et al. s scheme. Such an operator δ in fact eists. Lemma.. There eists an operator δ in X d. Proof. The Moore Penrose pseudo-inverse of δ epression of δ : X d X d such that δ δ δ δ id. is an eample of δ. In fact, the matri, say D, is a circulant matri which can be diagonalized via the discrete Fourier transform. The eigenvalues are i sinπj/n/ j,..., N, which becomes for the eigenvector,...,. This corresponds to in other words, the vector is orthogonal to X d. The Moore Penrose pseudo-inverse of D, which we denote by D +, is also a circulant matri with the same eigenvectors, and the corresponding eigenvalues are the reciprocals and the zero see, for eample, []. This means that both D + D and D D + have the zero eigenvalue for,...,, while all other eigenvalues are equal to. This implies the claim. Given such an operator δ, Scheme is in fact conservative. Lemma.6 Scheme : Conservation of the total mass and the norm. Under the periodic boundary condition, the numerical solutions by Scheme conserve the total mass and the norm: δ Φ n δ Φ, H n H, n,,....

13 For the proof, we need the following lemma cf. Lemma.. α Lemma.7. If U X d, then an operator U n+ δ + δ U n+ βδ γδ is sew-symmetric in the sense that for any sequences V, W X d, α δ n+ V U δ + δ U n+ βδ γδ δ W α n+ U δ + δ U n+ βδ γδ δ V Proof. The sew-symmetry of U n+ δ + δ see [8]. Since δ From the sew-symmetry of δ, U n+ δ W. is straightforward for eample, is also sew-symmetric, it remains to show the sew-symmetry of δ δ V δ δ W V δ W δ δ δ V W V δ W.. Proof. Proof of Lemma.6. The conservation of the total mass is the same as before. The conservation of the norm can be shown as follows.. H n+ H n δh Φ δφ n+, Φ n δh δ δ n+ Φ n δφ n+, Φ n δh δφ n+, Φ n α δ Φ n+ Φ n n+ U δ + δ U n+ βδ γδ δ δh δφ n+, Φ n The first equality is from. In the second equality, we have used the assumption δ δ id. and the summation-by-parts formula 9 also note that δh/δφ n+, Φ n X d. Then the scheme and Lemma.7 were used. Scheme is, however, not satisfactory as is in the following senses: i it still contains the inverse operator δ, and ii it seems not straightforward to show for Scheme a counterpart of Lemma.. In order to wor around these difficulties, we consider slightly modifying the scheme: by combining with, we see that Scheme is equivalent to δ Φ n+ Φ n

14 α U n+ δ + δ U n+ δ Φ n+ + βδ Φ n+ + γδ δ Φ n+. 6 Note that the assumption δ δ id. applies only to those in X d, and until Φ n+ X d is proved, the operator δ δ in the right hand side cannot be erased. But let us put this issue aside for the moment, and consider the following modified scheme, which corresponds to 8. Scheme Norm-preserving finite difference scheme. Given an initial approimate solution Φ X d, we compute Φ n n,,... by δ Φ n+ Φ n α U n+ δ + δ U n+ U n+ + βδ Φ n+ + γφ n+,..., N. 7 This scheme is completely free of the inverse operator δ. Furthermore, the solution of Scheme turns out to happily remain in X d as follows. Lemma.8. Assume Φ n X d, and consider one step of Scheme. Then Φ n+ X d. Proof. From 7, we obtain δ Φ n+ Φ n α U n+ δ + δ U n+ U n+ + βδ Φ n+ + γφ n+. Since the left hand side, and the first and second terms of the right hand side vanish from the sew-symmetry of δ, U n+ δ + δ U n+ and δ, we get Φn+. This implies the claim. Consequently, the solution of Scheme also satisfy 6, and thus it is also a solution of Scheme. This implies the conservation properties of Scheme ; we summarize this in the net theorem. Theorem.9. Under the periodic boundary condition, the numerical solution by Scheme enjoys the conservation properties. Multi-symplectic integrator In this section a new multi-symplectic formulation and the associated local conservation laws are shown. The multi-symplectic discretization is also proposed by means of the Preissman bo scheme.. Multi-symplectic partial differential equations and their integrators We start by briefly reviewing the concept of multi-symplecticity in a general contet [,, 6, ]. A partial differential equation F u, u t, u, u t,... is said to be multi-symplectic if it can be written as a system of first order equations: Mz t + Kz z Sz, 8

15 with z R d a vector of state variables, typically consisting of the original variable u as one of its components. The constant d d -matrices M and K are sew-symmetric, and S is a smooth function depending on z. A ey observation for the multi-symplectic formulation 8 is that it has a multi-symplectic conservation law: where ω and κ are differential two forms: t ω + κ, 9 ω dz Mdz, κ dz Kdz. Another ey property is the following conservation laws. conservation laws: The system 8 has local t Ez + F z, t Iz + Gz, where Ez, F z, Iz, and Gz are the density functions defined by Ez Sz z K z, F z z t K z, Gz Sz z t M z, Iz z M z. Thus integrating the densities Ez and Iz over the spatial domain under appropriate assumptions on F z and Gz leads to the global invariants: Ez Ez d, Iz Iz d. A scheme is called to be multi-symplectic if it satisfies some discrete version of the multi-symplectic conservation law 9. As multi-symplectic schemes, the Preissman bo scheme and the Euler bo scheme are widely nown. We adopt the Preissman bo scheme in this paper. The Preissman bo scheme also nown as the centered bo scheme was first introduced by Preissman in 96, and widely used in hydraulics. It reads Mδ t + Zn + Kδ + + Z n+ z SZ n+, + where the sub-/super-indices + and n + mean the abbreviations: Z n+ Z n+ + Zn+ + Z n, Z n + Z n + Z n + + Zn+ Zn + + Zn + Z n+ + In what follows we use the same abbreviation also for other quantities i.e., all the quantities with +/ inde mean the averages; not the quantities on staggered grid. The scheme was proved to be multi-symplectic by Bridges Reich []; the result is summarized in the net theorem. For the detail, including the definition of discrete symbols, readers may refer [].., In the contet of multi-symplectic integration, the discretization of z is usually denoted by Z,n and the numerical solution by U,n. But in this paper we use the same notation Z n or U n as in the finite difference schemes.

16 Theorem. []. Let us define discrete differential two forms by ω,n U n, V n MU n, V n, κ,n U n, V n KU n, V n, where, is the Euclidean scalar product on R d, and suppose that U n two solutions of the discrete variational equation and V n are any Mδ t + dzn + Kδ + + dz n+ D zz SZ n+ dz n+ + + associated with. The Preissman bo scheme is multi-symplectic in the sense that the discretization satisfies ω +,n+ ω +,n + ω +,n+ ω,n+. Remar. It is sometimes possible to derive schemes preserving discrete versions of the local conservation laws, when Sz is quadratic [8]. Unfortunately, however, it is not the case for the Ostrovsy equation, as shown below.. A multi-symplectic formulation and an integrator for the Ostrovsy equation In this subsection, a multi-symplectic formulation for the Ostrovsy equation is presented. With z ϕ, u, v, w, it is given by / γϕ / z t + z w + V u β v, u where Sz uw+v u+v /β γϕ / and V u αu /6. This formulation is motivated by the multi-symplectic formulation for the KdV equation []. In fact, for γ, reduces to the one in []. From, the density functions E and I defined above are eplicitly given by Ez Sz z K z α 6 + β u γ ϕ + uw [ϕ w + u v v u w ϕ], Iz z M z ϕ u u ϕ. Under periodic or vanishing boundary condition, we have the following two global conserved quantities: α Ez 6 u + β u + γ ϕ d, Iz u d. Based on this epression, we derive the following multi-symplectic scheme. Scheme 6 The multi-symplectic scheme. Given an initial approimate solution Φ X d, we compute Φ n n,,... by U n+ + U n + where U n+ + + αu n+ U n+ + U n+ + β U n+ + U n+ + + U n+ U n+ Φ n+ + Φ n+ /. γφ n+ +,..., N,

17 Conservative Galerin schemes In this section, we propose conservative Galerin finite element schemes for the Ostrovsy equation. As a framewor to derive conservative Galerin schemes, the DPDM [7] is nown, which also fully utilizes Hamiltonian or variational structures. But unfortunately one can not apply the DPDM directly to the Ostrovsy equation due to the non-local operator. Even worse, the Hamiltonian/variational structures with the potential ϕ u are not so useful in this case, since that would require epensive smooth elements. We circumvent this situation by incorporating the idea of L -projection. We here only show the resulting schemes, and leave the discussion on the general framewor to our future wor [9]. Below, we first consider two wea formulations for the energy and the norm preservations, respectively. Then conservative Galerin schemes are shown based on these formulations.. Wea formulations.. A wea formulation for the energy preservation Recall that the energy function G is defined by Gϕ, ϕ, ϕ αϕ /6 + βϕ / + γϕ /, and the partial derivatives are G ϕ γϕ, G ϕ α ϕ, G ϕ βϕ. Then we consider the following wea formulation: Find ϕt,, pt, H S such that for any v, v H S, G G ϕ t, v ϕ, v +, v p, v, ϕ G p, v, v. ϕ The symbol S means the torus of length L, and H S is the standard first order Sobolev space on the torus we often drop S where no confusion occurs. From the wea forms,, the desired conservation property can be deduced as shown in the net theorem. Theorem. Energy conservation property of the wea forms,. Suppose ϕt,, pt, H S are the solutions of the wea forms,. Also assume that ϕ t t,, ϕ t t, H S, then it holds Proof. d dt d dt G d, with Gϕ, ϕ, ϕ α 6 ϕ + β ϕ + γ ϕ. G Gϕ, ϕ, ϕ d ϕ, ϕ t G ϕ, ϕ t ϕ t, ϕ. + G +, ϕ t ϕ G, ϕ t ϕ G +, ϕ t ϕ p, ϕ t The first equality is just the chain rule. The second and third equalities follow from with v ϕ t and with v ϕ t, respectively. The last is from the sew-symmetry of.

18 .. A wea formulation for the norm preservation We net consider the following wea formulation: Find ϕt,, pt, H S such that for any v, v H S, α ϕ t, v ϕ, v p, v + γϕ, v, p, v βϕ, v. 6 From the wea forms, 6, the desired conservation property can be deduced as shown in the net theorem. Theorem. Norm conservation property of the wea forms, 6. Suppose ϕt,, pt, H S are the solutions of the wea forms, 6. Also assume that ϕ t,, ϕ t, H S, then it holds Proof. d dt H d, with H ϕ. d H d ϕ t, ϕ dt α ϕ, ϕ p, ϕ + γϕ, ϕ α ϕ, ϕ + βϕ, ϕ + γϕ, ϕ. The first equality is just the chain rule. The second and third equalities follow from with v ϕ and 6 with v ϕ, respectively. The last is from the periodic boundary condition and the sew-symmetry of.. An energy-preserving Galerin scheme In this subsection, an energy-preserving Galerin scheme based on the wea forms and is constructed and its properties are shown. Before presenting schemes, we define the L -projection operator P W onto the finite-dimensional approimation space W H L. We also denote P W u by D W u for convenience. Roughly speaing, D W p p is the operator that approimates p. As for the above operators, the following formulas are straightforward. Lemma.. For any u L and v W, it holds and for any u H and v W, it holds u, v P W u, v, 7 D W p u, v D W p u, v p. 8 Utilizing the above operator, we define the following Galerin scheme. Let S H S be an appropriately chosen trial space, and W H S a test space. For eample, we can simply choose the standard periodic piecewise linear function space. In what follows, we basically assume this. 6

19 Recall that the energy function Gϕ, ϕ, ϕ is defined as G αϕ /6 + βϕ / + γϕ /, and the corresponding partial derivatives are G/ ϕ γϕ, G/ ϕ α ϕ, G/ ϕ βϕ. In view of this, we define discrete partial derivatives by ϕ n+, ϕ n γ ϕn+ + ϕ n, D W ϕ n+, D W ϕ n α D W ϕ n+ + D W ϕ n+ D W ϕ n + D W ϕ n, D W ϕ n+, D W ϕ n β D W ϕn+ + D W ϕn It is easy to chec that they satisfy the following discrete chain rule Gϕ n+, D W ϕ n+, D W ϕ n+ Gϕ n, D W ϕ n, D W ϕ n d ϕ n+, ϕ n, ϕn+ ϕ n + D W ϕ n+, D W ϕ n, D W ϕ n+ D W ϕ n + D W ϕ n+, D W ϕ n, D W ϕ n+ D W ϕ n. 9 Scheme 7 Energy-preserving Galerin scheme. Suppose ϕ n S is given. Find ϕ n+ S and p n+ S such that, for any v W, v W, ϕ n+ ϕ n, v ϕ n+, ϕ n, v + P W D W ϕ n+, D W ϕ n, v p n+, v p n+, v D W ϕ n+, D W ϕ n, v. Obviously and correspond to and, respectively. Numerical solutions by Scheme 7 conserve both the total mass and the energy. Theorem. Scheme 7 : Conservation of the total mass and the energy. Assume that the trial and test spaces S, W are given such that S W. Then Scheme 7 is conservative in the sense that Proof. Since ϕ n+ d ϕ n d, Gϕ n+, D W ϕ n+, D W ϕ n+ d ϕ n d [ϕ n] L,. Gϕ n, D W ϕ n, D W ϕ n d. the conservation of the total mass is obvious. We net prove the conservation of the energy: Gϕ n+, D W ϕ n+, D W ϕ n+ Gϕ n, D W ϕ n, D W ϕ n d 7

20 ϕ n+, ϕ n, ϕn+ ϕ n + D W ϕ n+, D W ϕ n, D W ϕ n+ D W ϕ n + D W ϕ n+, D W ϕ n, D W ϕ n+ D W ϕ n ϕ n+, ϕ n, ϕn+ ϕ n + P W D W ϕ n+, D W ϕ n, D W ϕ n+ D W ϕ n + D W ϕ n+, D W ϕ n, D W ϕ n+ D W ϕ n ϕ n+, ϕ n, ϕn+ ϕ n + P W D W ϕ n+, D W ϕ n, ϕn+ ϕ n p n+, D W ϕ n+ D W ϕ n ϕ n+, ϕ n, ϕn+ ϕ n + P W D W ϕ n+, D W ϕ n, ϕn+ ϕ n. p n+, ϕn+ ϕ n ϕ n+ ϕ n, ϕn+ ϕ n The first equality is just the discrete chain rule 9. In the second equality, we have used 7 and 8 with p. In the third equality, 8 with p, and with v D W ϕ n+ D W ϕ n / W were used. The fourth equality follows from 8 with p, and the fifth from with v ϕ n+ ϕ n / S W.. A norm-preserving Galerin scheme In this subsection a norm-preserving Galerin scheme based on the wea forms and 6 is constructed, and its properties are shown. Scheme 8 Norm-preserving Galerin scheme. Suppose ϕ n S is given. Find ϕ n+ S and p n+ S such that, for any v W, v W, ϕ n+ ϕ n, v α D W ϕ n+, v p n+, v + γϕ n+, v, p n+, v βd W ϕ n+, v, where ϕ n+ ϕ n+ + ϕ n /. Eq. and correspond to and 6, respectively. Then the scheme enjoys the following the total mass and the norm conservation properties. 8

21 Theorem. Scheme 8 : Conservation of the total mass and the norm. Assume that the trial and test spaces S, W are set such that S W. Then Scheme 8 is conservative in the sense that ϕ n+ d where H n D W ϕ n /. ϕ n d, H n+ d H n d, Proof. Similar to Scheme 7, the conservation of the total mass is obvious. The conservation of the norm goes as follows. H n+ H n d D W ϕ n+ D W ϕ n ϕ n+ ϕ n, D W ϕ n+ α D W ϕ n+, D W ϕ n+ α D W ϕ n+, D W ϕ n+ α D W ϕ n+, D W ϕ n+ + γϕ n+, ϕ n+., D W ϕ n+ + D W ϕ n p n+, D W ϕ n+ p n+, D W ϕ n+ + βd W ϕ n+, D W ϕ n+ + γϕ n+, D W ϕ n+ γϕ n+, ϕ n+ 6 Numerical eamples Scheme Scheme Scheme Scheme Scheme 6 Scheme 7 Scheme 8 Table : Numerical schemes used in this paper. The Energy-Preserving Finite Difference Scheme by Yaguchi et al. The Norm-Preserving Finite Difference Scheme by Yaguchi et al. The Energy-Preserving Finite Difference Scheme The Norm-Preserving Finite Difference Scheme The Multi-Symplectic Scheme The Energy-Preserving Galerin Scheme The Norm-Preserving Galerin Scheme We compare the proposed schemes numerically. The parameters are set to α, β., γ. The length of the spatial period was set to L π. The initial condition was set to u, sin, and accordingly the potential to ϕ, cos. Hunter reported that oscillations were observed in this setting [], and Yaguchi et al. confirmed this with the above parameters []. We too mesh sizes. and L/N, with N for Scheme,,,, 6, and N for Scheme 7, 8 the difference is to avoid singularities in the coefficient matrices. Although in principle we can use non-uniform grids 9

22 in the Galerin schemes, we did not do this since in the following numerical eamples nonuniform grids does not necessarily seem advantageous. We employed the P elements, for which it is easy to chec the assumptions in Theorem. and.. In order to solve the proposed implicit schemes, we used fsolve in MATLAB with the tolerances T olf un and T olx set to 6. Figs. and show the evolutions of the energies and the norms. As shown in the theorems in the previous sections, Scheme, and 7 conserve the energies, and Scheme, and 8 the norms. Scheme 6 conserve neither of them, but the deviations are small. In fact, as for the norm, the deviation by Scheme 6 is smaller than those by Scheme, and 7 which do not conserve the norm. A similar result is also observed for the energy. Thus Scheme 6 seems to be the best of all the proposed schemes. We also notice that if we compare Scheme, and 7, Scheme and 7 seem better than Scheme, because the deviations of the energy are smaller. As for the energy-preserving schemes, in a similar way, Scheme and 8 seem better than Scheme. Net, let us evaluate each scheme in view of qualitative behaviors. The numerical solutions are shown in Figs. 9 respectively. All figures show the oscillation that Hunter reported and Yaguchi et al. observed. But as Yaguchi et al. observed, there eist small differences in the qualities of the solutions. The results by the conservative finite difference schemes see Figs. and 6 are smoother, especially in t >, than other schemes. From this standpoint, conservative finite difference schemes, especially the energy-preserving finite difference scheme Scheme, are the most advantageous in all schemes. These results seem to support the effectiveness of the schemes epressed in the potential ϕ Energy Scheme Scheme Scheme Scheme Scheme 6 Scheme 7 Scheme 8 time Figure : Evolution of the energy for each scheme.

23 Norm Scheme Scheme Scheme Scheme Scheme 6 Scheme 7 Scheme 8 time Figure : Evolution of the norm for each scheme. u t Figure : The numerical solution obtained by Scheme the energy-preserving finite difference scheme by Yaguchi et al. with N and..

24 u t Figure : The numerical solution obtained by Scheme the norm-preserving finite difference scheme by Yaguchi et al. with N and.. u t Figure : The numerical solution obtained by Scheme the energy-preserving finite difference scheme with N and..

25 u t Figure 6: The numerical solution obtained by Scheme the norm-preserving finite difference scheme with N and.. u t Figure 7: The numerical solution obtained by Scheme 6 the multi-symplectic scheme with N and..

26 u t Figure 8: The numerical solution obtained by Scheme 7 the energy-preserving Galerin scheme with N and.. u t Figure 9: The numerical solution obtained by Scheme 8 the norm-preserving Galerin scheme with N and..

27 7 Concluding remars We have proposed five new structure preserving numerical schemes for the Ostrovsy equation: the energy-preserving finite difference scheme, the norm-preserving finite difference scheme, the multi-symplectic integrator, the energy-preserving Galerin scheme and the norm-preserving Galerin scheme. We also lie to emphasize that in this paper we have also found a multi-symplectic formulation of the Ostrovsy equation. Numerical eamples confirmed the effectiveness of the proposed schemes, in particular, the finite difference schemes based on the potential epression. The reason why this approach improved the results, and whether or not it wors also for other PDEs, such as the KdV, are interesting future research topics. For the Galerin schemes, in the present paper we have introduced the idea of L -projection. This technique seems useful in more general contet; the discussion and the etension of the DPDM are left to our future wor[9], which will be reported soon elsewhere. References [] U. M. Ascher and R. I. McLachlan. Multisymplectic bo schemes and the Korteweg de Vries equation. Appl. Numer. Math., 8: 69,. [] E. S. Benilov. On the surface waves in a shallow channel with an uneven bottom. Stud. Appl. Math., 87:, 99. [] T. J. Bridges. Multi-symplectic structures and wave propagation. Math. Proc. Cambridge Philos. Soc., :7 9, 997. [] T. J. Bridges and S. Reich. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A, 8:8 9,. [] E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. R. J. O Neale, B. Owren, and G. R. W. Quispel. Preserving energy resp. dissipation in numerical PDEs, using the Average Vector Field method. NTNU preprint series: Numerics No.7/9, [6] G. Y. Chen and J. P. Boyd. Analytical and numerical studies of wealy nonlocal solitary waves of the rotation-modified Korteweg de Vries equation. Physica D, :,. [7] R. Choudhury, R. I. Ivanov, and Y. Liu. Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsy equation. Chaos Solitons Fractals, :, 7. [8] D. Cohen, B. Owren, and X. Raynaud. Multi-symplectic integration of the Camassa Holm equation. J. Comput. Phys., 7:9, 8. α δg [9] D. Furihata. Finite difference schemes for u t δu that inherit energy conservation or dissipation property. J. Comput. Phys., 6:8, 999. [] D. Furihata and T. Matsuo. Discrete Variational Derivative Method: A structurepreserving numerical method for partial differential equations. Chapman & Hall/CRC, Boca Raton,. [] V. N. Galin and Y. A. Stepanyants. On the eistence of stationary solitary waves in a rotating fluid. J. Appl. Math. Mech., 6:99 9, 99.

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