NUMERICAL ANALYSIS OF AN ENERGY-CONSERVATION SCHEME FOR TWO-DIMENSIONAL HAMILTONIAN WAVE EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 6, Number, Pages c 9 Institute for Scientific Computing and Information NUMERICAL ANALYSIS OF AN ENERGY-CONSERVATION SCHEME FOR TWO-DIMENSIONAL HAMILTONIAN WAVE EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS CHANGYING LIU, WEI SHI, AND XINYUAN WU Abstract. In this paper, an energ-conservation scheme is derived and analsed for solving Hamiltonian wave equations subject to Neumann boundar conditions in two dimensions. The energ-conservation scheme is based on the blend of spatial discretisation b a fourth-order finite difference method and time integration b the Average Vector Field AVF approach. The spatial discretisation via the fourth-order finite difference leads to a particular Hamiltonian sstem of second-order ordinar differential equations. The conservative law of the discrete energ is established, and the stabilit and convergence of the semi-discrete scheme are analsed. For the time discretisation, the corresponding AVF formula is derived and applied to the particular Hamlitonian ODEs to ield an efficient energ-conservation scheme. The numerical simulation is implemented for various cases including a linear wave equation and two nonlinear sine-gordon equations. The numerical results demonstrate the spatial accurac and the remarkable energ-conservation behaviour of the proposed energ-conservation scheme in this paper. Ke words. Two-dimensional Hamiltonian wave equation, finite difference method, Neumann boundar conditions, energ-conservation algorithm, average vector field formula.. Introduction The theme of this paper is the numerical analsis of an energ-conservation scheme for the following Hamiltonian wave equation in two-dimensional space: { utt a u u = fu,,, t Ω t, T ], u,, t = φ,, u t,, t = ϕ,,, Ω, where the function fu is the negative derivative of a potential energ V u, and Ω = l, r d, u. The initial functions φ and ϕ are wave modes or kinks and their velocit, respectivel see, e.g. [, 3, 38]. Here, we suppose that the sstem is supplemented with the homogenous Neumann boundar conditions: u u =, d u, =, l r, t [t, T ]. =r, l =d, u It is noted that the conservation of the energ is an essential propert of the Hamiltonian sstem -, which can be epressed as follows: 3 Et [ u t a u u ] V u dd = Et. Ω Therefore, it is ver important to design numerical schemes that can precisel preserve a discrete energ. This class of schemes is called conservative schemes. The often ield phsicall correct results and numerical stabilit [8]. In the literature, a considerable number of numerical schemes has been presented for solving the two-dimensional Hamiltonian wave equations. For eample, Received b the editors December 7, 7. Mathematics Subject Classification. 35C5, 65M6, 65M, 65M, 65M7. Corresponding author. 39

2 3 C. LIU, W. SHI, AND X. WU Bratsos [5] applied the method of lines to solve sstem. In [3], the authors constructed a high-order compact alternating direction implicit scheme. Based on the blend of spatial discretisation b different finite difference methods and time integration b predictor-corrector schemes, the authors see, e.g. [9,, 7] investigated nonlinear Hamiltonian wave equations numericall. Dehghan et al. see, e.g. [4, 5, 34, 35] proposed the meshless local Petrov-Galerkin MLPG method, the meshless local radial point interpolation LRPI method, the dual reciprocit boundar element DRBE method and the meshless local boundar integral e- quation LBIE method for solving the two-dimensional sine-gordon equations. In [9, 3, 3], Liu et al. first formulated the sstem as an abstract secondorder ODEs in a suitable infinite-dimensional function space and then proposed and analsed a class of arbitraril high-order time-stepping methods for solving Hamiltonian sstem. Mei et al. [33] presented an etension of the finite-energ condition for Runge Kutta Nström-tpe integrators solving nonlinear wave equations. The other numerical approaches also were investigated, such as the finite element method see, e.g. [, ], the perturbation method [4] and the spectral method see, e.g. [3, 8]. On the other hand, splitting methods see, e.g. [9,, 6, 7, 37] have been developed to investigate the numerical solutions of the evolutional PDEs in multi-dimensional case. Sheng et al. [38] provided highl efficient splitting cosine schemes for the two-dimensional sine-gordon equations b using a linearl implicit sequential splitting or Strangs splitting. In a recent stud of Hamiltonian wave equations, using the finite difference method and the average vector field AVF method, the authors consider a class of energ-conservation methods for one-dimensional Hamiltonian wave equations see [6]. However, there are ver few studies to pa attention to designing and analsing high-order energconserving schemes for the two-dimensional Hamiltonian wave equations subject to Neumann boundar conditions. Hence, in this work, we devote to constructing and analsing a novel and efficient energ-conservation scheme for the Hamiltonian wave equations with Neumann boundar conditions - in two dimensions. According to the homogenous Neumann boundar conditions and the structure of equation, we derive and analsis a kind of fourth-order finite difference operators to discrete the spatial derivatives of the sstem -. In such a wa, the conserved PDEs can be converted into a particular Hamiltonian sstem of ODEs epressed in the form Aw t Bwt = f wt, and the Hamiltonian of the obtained ODEs can be thought of as the approimate energ of the original continuous sstem. This motivates us to consider the AVF approach to the discretisation of the derived Hamiltonian ODEs sstem in time. Therefore, incorporating the fourthorder finite difference discretisation in space with the AVF time integrator ields a novel and efficient energ-conservation numerical scheme for the two-dimensional Hamiltonian wave equations -. The paper is organised as follows. In Section, a particluar Hamiltonian ODEs sstem is obtained b appling the fourth-order finite difference operators to discrete the spatial derivatives of the nonlinear sstem -. Then the conservation law, the stabilit and the convergence of the semi-discrete scheme are rigorousl analsed. Section 3 is devoted to describing in detail the idea of the AVF formula for the derived Hamiltonian sstem of ODEs. In Section 4, numerical eperiments on a linear Klein-Gordon equation and two sine-gordon equations are implemented and the numerical results show the convergence order of the spatial discretisation and the ecellent conservative propert of the proposed numerical scheme. Section 5 is concerned with conclusions.

3 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 3. Energ-conservation spatial discretisation via a fourth-order finite d- ifference method In this section, we first present an energ-conservation finite difference scheme for the two-dimensional Hamiltonian wave equations -, which is of order four for all the spatial grid points. Then the energ conservation law, the stabilit and the convergence of the semi-discrete scheme are rigorousl analsed... Notations and auiliar lemmas. Suppose M and M are two positive integers. let h = r l M and h = u d M be the stepsizes of spatial direction and discretise the region [ l, r ] [ d, u ] with the spatial grid Ω h = { } i, j i = l ih, j = d jh, i M, j M. For an grid function u = {u ij i, j Ω h }, v = {v ij i, j Ω h }, we use the following notations δ u i,j = u i,j u ij, δ h u ij = u i,j u ij u i,j u ij = u i,j u i,j, h u ij = u i,j u ij u i,j, and two finite difference operators 7 δ u 3h,j 6 δ u,j, i =, δ u 6h,j 7 6 δ u,j δ u,j, i =, 4 4 A u ij = 3 δ u ij 3 u ij, i M, δ u 6h M,j 7 6 δ u M,j δ u M,j, i = M, 7 δ u 3h M,j 6 δ u M,j, i = M, and 7 3h δ u i, 6 δ u i,, j =, δ u 6h i, 7 6 δ u i, δ u i,, j =, 4 5 A u ij = 3 δ u ij 3 u ij, j M, δ u 6h i,m 7 6 δ u i,m δ u i,m, j = M, 7 δ u 3h i,m 6 δ u i,m, j = M. Furthermore, b defining the inner product [ u, v =h h u, v, u M,v M, u,m v,m u M,M 4 v M,M h 4h, 6 i= j= ui, v i, u i,m v i,m u,j v,j u M,jv M,j M i= j= u ij v ij ],

4 3 C. LIU, W. SHI, AND X. WU the notations of the discrete l -norm u = u, u and other discrete norms δ u = h h i= u = h h i= [ δ u i M, δ u i j= [ M u i, j=,j ] δ u i,m, ] u i,j u i,m, can be ielded. Likewise, the notations δ u i,j, δ u ij, u ij, u ij, δ u and u can be defined as well. Lemma.. For an grid function u = {u ij i, j Ω h }, we define the norm b 7 u = 4 δ u δ u 3 3 and then, the following inequalit holds 8 u u, u u, where the H semi-norm is given b u = δ u δ u. Proof. It follows from the definition of the operators u ij, u ij and the norms u, u that and u = h h 4 i= [ δ u i, δ u i δ u i,m δ u i,m ] h h = δ u, u = h h 4 i= j= [ δ u i M, M, δ u i j= [ δ u,j δ u,j ] δ u M,j δ u M,j h h = δ u. j= [ δ u,j M i= j= M δ u i,j δ u i,j,j i= δ u i,j ] δ u i,m δ u i,j δ u i,j ] δ u M,j We then obtain u = 4 δ u δ u u u δ u δ u. 3 3 The lemma is confirmed. Lemma.. The operators A and A are smmetric and positive semi-definite with respect to the inner product, defined b 6.

5 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 33 Proof. For an grid functions u, v defined on Ω h, it follows from using the summation b parts and that and A u, v = 4 M 3 h h i= [ M δ u i, δ v i, δ u i,mδ v i,m ] M j= = u, A v, A u, u = 4 M 3 h h i= M 3 h h i= j= M 3 h h i= u i,j v i,j u i,m v i,m ] [ δ u i M, δ u i j= [ M u i, j= = 4 3 δ u 3 u δ u. Likewise, it can be verified that δ u i,j δ v i,j [ u i, v i,,j ] δ u i,m ] u i,j u i,m A u, v = u, A v and A u, u 4 3 δ u 3 u δ u. Hence, the conclusions of the lemma are true. Using the five lemmas stated in the paper [5], we will conclude that the difference operators A and A are important to the construction of the fourth-order discretisation for the second-order derivative in all the spatial grid points. Actuall, the following lemma is the promotion of the five lemmas stated in the paper [5]. It is eas to verif the conclusion of the lemma b using the Talor epansion with integral residual items. Lemma.3. Suppose that ω, C 6,6 [ l, r ] [ d, u ] satisfies the following conditions 9 k ω l, = k ω r, =, k ω, d = k ω, u =, k =, 3, 5. We then have ω i, = A ω i, R i, i M, [ d, u ], and ω, j = A ω, j R j, j M, [ l, r ], where the residuals R i and R j satisf the following approimations R i = Oh 4 and R j = Oh 4.

6 34 C. LIU, W. SHI, AND X. WU Proof. Following the five lemmas given in the paper [5], we obtain 7 δ ω 3h, 6 δ ω, R, i =, δ ω 6h, 7 6 δ ω, δ ω, R, i =, 4 ω i, = 3 δ ω i, 3 ω i, R i, i M, δ ω 6h M, 7 6 δ ω M, δ ω M, R M, i = M, 7 δ ω 3h M, 6 δ ω M, R M, i = M, i.e., ω i, = A ω i, R i, i M, [ d, u ], and there eists a positive constant C such that Similarl, we also have R i C h 4. ω, j = A ω, j R j, j M, [ l, r ], and there eists a positive constant C such that R j C h 4. The conclusions of the lemma have been proved... Fourth-order spatial semidiscretisation, stabilit and convergence. It can be obtained straightforwardl from the Hamiltonian wave equation that a u = u tt a u fu, a u 3 = u tt a u f uu, a u 4 = u tt a u f uu f uu, a u 5 = u tt a u f uu 3 3f uu u f uu 3. Therefore, the boundar conditions deliver 3 u 3 l =, u 3 r =, u 5 l =, u 5 r =, d u, t < t T. In a similar wa, we also have 4 u 3 d =, u 3 u =, u 5 d =, u 5 u =, l r, t < t T. Under the assumption of u,, t C 6,6, Ω [t, T ], and from the equations 3, 4 and the Lemma.3, we dispose the spatial derivatives of the Hamiltonian sstems - and arrive at the following fourth-order semi-discrete scheme U ijt a 5 A U ij t A U ij t = f U ij t R ij t, i M, j M, t t T, where U ij t represent the eact solutions u,, t on the spatial grid Ω h [t, T ], and the residuals R ij t satisf R ij t = Oh 4 h 4, i M, j M, t t T.

7 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 35 Omitting the residuals R ij t in 5 and using u ij t U i, j, t result in the following semi-discrete scheme 6 u ijt a A u ij t A u ij t = f u ij t, i M, j M, t t T. Let ut = {u ij t i, j Ω h }. Then the equations in 6 can be rewritten in a compact form as follows: 7 u t a A ut A ut = f ut, t t T. With the semi-discrete scheme 6, we have the following energ conservation law. Theorem.. Suppose that ut = {u ij t i, j Ω h } is the solution of the semi-discrete scheme 6 or 7. We then have the following discrete energ conservation law: 8 Et u t a ut Ṽ ut = Et, where the potential function Ṽ ut is given b 9 Ṽ ut =h h V u ij t h h [V u, t V u M,t 4 i= j= V u,m t V u M,M t ] h h h h j= [ V u,j t V u M,jt ]. i= [V u i, t V u i,m t ] Proof. Taking the inner product of 7 with u t and following careful calculations, we have [ ] u t, u t a A ut, u t A ut, u t f ut, u t =, where u t, u t = d dt h h h h { h h [ u 4, t u M,t u t ],M u t M,M [ u i, t ] u t i,m i= j= h h i= [ u,j t u M,j ] j= u ij t = d dt u t, u t = d dt u t,

8 36 C. LIU, W. SHI, AND X. WU [ a A ut, u t = a d 4 dt 3 h h = a and M 3 h h i= 4 M 3 h h j= M 3 h h j= i= ] A ut, u t [ δ u i [ u i, t M [ δ u,j, t M δ u i j= j= M t δ u i,j i= [ u,j t M i=,j t u i,j t u i,m t ] t δ u i,m t ] δ u M,j t ] u i,j t u M,jt ] [ d 4 δ ut dt 3 δ ut ut ut ] = a 3 f ut, u t = d { h h [V u, t V u M,t dt 4 V u,m t V u M,M t ] d dt ut, 3 h h h h i= j= h h i= [V u i, t V u i,m t ] [V u,j t V u M,jt ] j= V u ij t d ut. dtṽ Inserting -3 into ields 4 d dt u t a ut Ṽ ut =. Therefore, the energ conservation law is valid. Following the discrete energ conservation law 8 shown b Theorem., we can obtain the following result on stabilit analsis. Theorem.. Assume that the potential function V is positive and ut = {u ij t i, j Ω h } is the solution of the semi-discrete scheme 6 or 7. Then the semi-discrete scheme is stable under the discrete H -norm, that is, there eists a constant C 3 such that 5 ut H φ C 3 Et, where ut H = ut δ ut δ ut.

9 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 37 Proof. According to the discrete energ conservation law 8 and Lemma., we have 6 u t Et and δ ut δ ut a Et. For an t t, the Cauch-Schwarz inequalit ields t u ijt u ijt u ijζdζ t 7 Therefore, it follows from 7 that u ijt t t 8 ut ut t t t u ijζdζ. t t t u ζ dζ. Combining the inequalities 6 and 8 conforms the conclusion of the theorem. Remark.. Since the potential function V is onl used through its gradient, an V c can fit for the equations with a suitable constant c. Under this assumption that V is positive, together with the discrete energ conservation law, we can prove the stabilit of the semi-discrete scheme. In what follows, we assume that the function f satisfies a Lipschitz condition with respect to u, namel, there is a positive constant L, s.t. 9 fu fv L u v, u, v. We now quote the well-known Gronwall inequalit summarised below, which will pla an important role in the proof of the convergence for the semi-discrete scheme 6 or 7. Lemma.4. Suppose that ft, gt are two nonnegative functions defined on [t, T ], and satisf the following differential inequalit f t C 4 ft gt. Then [ t ] ft e C 4t ft e C4s gsds, t [t, T ], t where C 4 is a nonnegative constant. Theorem.3. It is assumed that u,, t C 6,6, Ω [t, T ] is the solution of - and ut = {u ij t i, j Ω h } is the solution of the semi-discrete scheme 6 or 7. Then, we have 3 et H = Oh 4 h 4, t [t, T ], where e ij t = U ij t u ij t, i M, j M. Proof. Subtracting 6 from 5, and noticing the initial conditions, we obtain the error sstem e t a A et A et = f Ut f ut Rt, t t T, et =, e t =,

10 38 C. LIU, W. SHI, AND X. WU where Rt = { R ij t i M, j M }. Taking the inner product with e t ields e t, e t a [ A et, e t A et, e t ] = f Ut f ut, e t Rt, e t. A careful calculation gives d e t a et L et e t Rt e t dt C 5 e t a et Rt, i.e., d e t a et C 5 e t a et Rt, dt where C 5 is a constant. Appling the Gronwall inequalit to the above inequalit with the eact initial conditions, we obtain e t a et t t ep C 5 t s Rs ds = Oh 8 h 8. The estimations of e t and et can be ielded e t = Oh 8 h 8 and et = Oh 8 h 8. Similarl to the proof process of Theorem., we obtain et et t t On noticing that et =, we have t et = Oh 8 h 8. Using Lemma. to the estimation of et leads to t e ζ dζ. et = δ et δ et = Oh 8 h 8. Therefore, the error estimation under the H semi-norm is obtained et H = et δ et δ et = Oh 8 h 8, t [t, T ]. The conclusion of the theorem is confirmed..3. Corresponding Hamiltonian ODEs. Let w j t = u,j t, u,j t,..., u M,jt and f wj t = f u,j t, f u,j t,..., f u M,jt, the semidiscrete scheme 6 or 7 is identical to the following nonlinear ODEs w t a IM D M 3 h D M I M h wt = f wt, t < t T, wt = φ =, φ,..., φ M w t = ψ = ψ,..., ψ, M

11 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 39 where I d are d d identit matri with d = M, M, is the Kronecker product, wt = w t,..., w M t, D d = , and f wt = f w t,..., f w M t, φj = ψ j = ψ, j,..., ψ M, j. d d φ, j,..., φ M, j, Multipling the sstem 3 with the matri A = h h A M A M, we obtain { Aw t Bwt = f wt, t < t T, 3 wt = φ, w t = ψ, which is equivalent to 3, where A d = diag,,...,, for d = M and M, and B = a A IM D M D M I M is a positive semi-definite matri, and f wt = h h Af wt is the negative gradient of the energ potential function Ṽ wt defined b 9. The sstem 3 is a Hamiltonian sstem with the Hamiltonian 33 Hw t, wt = w t Aw t wt Bwt Ṽ wt. Theorem.4. Assume that ut = {u ij t i, j Ω h } is the solution of the semi-discrete scheme 6, or equivalentl, the Hamiltonian sstem 3. Then we have H w t, wt = Et, t t T. Proof. Following the formula 33 and through careful calculations, we obtain H w t, wt = w t Aw t wt Bwt Ṽ wt The conclusion of the theorem is proved. = u t a ut Ṽ ut = Et. 3. Enverg-preserving time integrators: AVF formula In this section, we concentrate on constructing energ-conservation time integrators for the following nonlinear Hamiltonian sstem of ODEs { Aw t Bwt = f wt, t < t T, 34 wt = φ, w t = ψ. Let qt = wt, pt = Aq t. Then the sstem 34 can be epressed as the following first-order Hamiltonian sstem of ODEs { p t = q Hp, q, 35 q t = p Hp, q,

12 33 C. LIU, W. SHI, AND X. WU with the initial values pt = A ψ, qt = φ and the Hamiltonian 36 Hp, q = p A p q Bq Ṽ q, which is identical to the following Hamiltonian 37 H w t, wt = w t Aw t wt Bwt Ṽ wt. The energ conservation is one of the essential properties of the Hamiltonian sstem, whose Hamiltonians can be regarded as the approimate energ of the original PDEs. Therefore, it is of great importance to construct energ-conservation time integrators see, e.g. [4, 7,, 8, 9, 7, 3, 36, 39, 4]. For instance, we consider the following Hamiltonian sstem 38 ż = J Hz, where J is a skew-smmetric matri with the form I 39 J = I and H is the Hamiltonian. McLachlan et al. derived the AVF formula in [3], i.e., 4 z n = z n τ J H sz n sz n ds. The Hamiltonian of the sstem 38 can be preserved eactl b the AVF formula 4. Appling the AVF formula 4 to Hamiltonian sstem 35 with a straightforward calculation, we obtain q n = q n τa p n τ A g sq n sq n ds, 4 p n = p n τ g sq n sq n ds, which can preserve the energ or Hamiltonian of the sstem 34 eactl, namel, Hw n, w n = Hw n, w n, where gw = Bw fw and t n = t n τ with τ is the time stepsize. Let q n = w n and p n = Aw n. Then, the formula 4 is identical to w n =w n τw n τ A g sw n sw n ds, 4 w n =w n τa g sw n sw n ds. Since gw = Bw fw, we can simplif the integral appeared in the formula 4 or 4 as I g = g sw n sw n ds = B w n w n where fw = Afw and f w = f sw n sw n ds, f w,..., f wm with f wj = f u,j, f u,j,..., f um,j. Moreover, from the fact that the function fu satisfies

13 fu = ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 33 dv u du, we have f su n ij su n V u n ij V u n ij ij ds = u n. ij u n ij Here, it should be noted that, if u n V u n ij V u n ij u n ij u n ij is, which can be understood as ij V u n ij V u n ij u n ij u n ij Therefore, the nonlinear integral f sw n sw n ds = u n ij, i M, j M, then = dv un ij du = fu n ij. f sw n sw n ds can be epressed as: 43 = A V wn V w n w n w n V w n V w n = A w n, V wn V w n w n w n,..., V wn M V wm n w n w n M wm n with 44 V w n j V w n j w n wj n V u n,j = V un,j u n, V un,j V un,j,j u n,j u n,..., V un M V,j un M,j,j u n,j u n M,j un M,j According to the above analsis, we are now in a position to present the energconservation time integrator for the oscillator Hamiltonian ODEs sstem 34. Theorem 3.. The following modified AVF formula w n = w n τw n τ 4 A Bw n w n τ V w n V w n w 45 n w n, w n = w n τ A Bw n w n τ V wn V w n w n w n. for the Hamiltonian sstem of ODEs 34 can preserve the energ 37 eactl: Hw n, w n = Hw n, w n. Here, V wn V w n w n w n is denoted b 43 and 44. Remark 3.. Actuall, the modified AVF formula 45 is a kind of discrete gradient, which is called the mean value discrete gradient and is a second-order approimation to the gradient Ṽ of the energ potential function 9 at the midpoint of w n and w n see, e.g. [, 4]. Moreover, the smmetr of a method is also an essential propert in long time integration. The definition of smmetr is given below see [4, 4]. Definition 3.. The adjoint method Φ τ of a method Φ τ is defined as the inverse map of the original method with reversed time step τ; i.e., Φ τ := Φ τ. A method with Φ τ = Φτ is called smmetric.

14 33 C. LIU, W. SHI, AND X. WU According to the definition of smmetr, it can be verified that the modified AVF formula 45 is unaltered b echanging n n and τ τ. Therefore, the modified AVF formula 45 can be verified to be smmetric. In addition, the modified AVF formula 45 can be interpreted as the second-order mean value discrete gradient see [, 4]. Therefore, it is eas to clarif that the global error accurac of the modified AVF formula 45 is second-order. In the following theorem, we show the smmetr and the global error accurac of the modified AVF formula 45 for solving the resulting semi-discrete scheme 34. Theorem 3.. The modified AVF formula 45 for the semi-discrete Hamiltonian ODEs sstem 34 is smmetric with respect to the time variable and convergent of order two. Proof. The smmetr of the modified AVF formula 45 can be confirmed b echanging n n and τ τ, and we skip the detailed proof process. Since the modified AVF formula 45 is the second-order mean value discrete gradient see [], the modified AVF formula 45 is convergent of order two. From the above analsis, it is clear that we have derived an energ-conservation full discrete scheme for solving the two-dimensional Hamiltonian wave equations - b discreting the spatial derivatives of the PDEs via the fourth-order finite difference method and appling the AVF approach to the resulted Hamiltonian sstem of ODEs. Since the convergence order of the modified AVF method is Oτ, the global error accurac of the full discrete scheme can get to Oτ h 4 h Numerical eperiments As stated in the introduction of this paper, the energ conservation law is an essential propert for the two-dimensional Hamiltonian wave equations -. The conserved quantities of the constructed numerical scheme can be used to evaluate the stabilit of the numerical schemes. In what follows, we will appl the derived energ-conservation scheme to three practical problems to verif the numerical accurac and the numerical behaviour of energ preservation. For comparison, the time integrators we selected are: HBVM5,: the energ-preserving Hamiltonian Boundar Value Method of order two given in [6, 7]. SV: the smplectic Störmer-Verlet formula of order two given in []; As is known, iterative solutions are required for implicit schemes. We use fiedpoint iteration for the modified AVF formula 45 and the HBVM5, method. We set the error tolerance as 5 for each time-step iteration procedure and set the maimum number of iterations to be. 4.. Test problem: linear wave equation. In order to observe the accurac of the spatial discretisation, we consider the following linear wave equation in two dimensions 46 u tt u u = π u, over the region, [, ] [, ] with the initial conditions 47 u,, = sinπ sinπ, u t,, = sinπ sinπ. The eact solution of the problem is 48 u,, t = e t sinπ sinπ.

15 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 333 Obviousl, the problem satisfies the homogenous Neumann boundar conditions. We compute the problem using the energ-conservation scheme proposed in this paper with fied time stepsize τ =. and several spatial steps h, h at time T =. The numerical results in Table demonstrate that the spatial discretisation converges with order four under the l -norm, H -norm and l -norm. Table. The convergence order in space of numerical solutions at T = with different spatial steps h, h and sufficientl small time stepsize τ =. correspond to different norms. h, h l -norm H -norm l -norm u n U n order u n U n H order u n U n order, * * *, , , We then choose the stepsizes h = h =.5 and τ =. for solving the problem and illustrate the conservative propert of the new scheme in Fig.. The logarithms of the discrete energ error Et n Et and the relative discrete energ error Et n Et /Et are plotted in Fig. a and b, respectivel. It can be observed that the energ obtained b the new scheme is well conserved. 4 4 error of energ AVF SV HBVM5, error of relative energ AVF SV HBVM5, t a Et n Et t b Et n Et /Et Figure. The logarithms of the discrete energ error a and the relative discrete energ error b. 4.. Simulation of two-dimensional sine-gordon equations. In this subsection, the proposed new energ-conservation scheme is applied to simulate the two-dimensional sine-gordon equation: 49 u tt u u = sinu, < t T, over the region Ω = [ a, a] [ b, b]. The problem is equipped with the homogeneous Neumann boundar conditions 5 u ±a,, t = u, ±b, t =, and the initial conditions 5 u,, = f,, u t,, = g,.

16 334 C. LIU, W. SHI, AND X. WU In general, the eact solution of the D sine-gordon equation cannot be obtained eactl. Therefore, the conservative propert of the energ becomes significant to eamine the superiorit of a numerical approach. Moreover, it has been known that different initial conditions will lead to different numerical phenomena. In what follows, we will use the new scheme to simulate two kinds of particular circular ring solitons. The initial conditions and parameters are chosen similarl to those in [5, 38]. Problem. Circular ring soliton We choose the initial conditions as: 5 f, = 4 arctan ep 3, g, =, over the two-dimensional domain, [, ] [, ]. We simulate the problem b using the spatial steps h = h =. and time step τ =.. The simulation results and the corresponding contours at times t =,, 4, 6, 8, in terms of sinu/ are plotted in Fig. and Fig. 3, respectivel. The conservative behaviour of the new scheme is shown in Fig. 4 from which it can be observed that the energ is well conserved. In comparison with the classical SV method and the HBVM5, method, the modified AVF method 45 has superior propert of energ conservation. a T= b T= c T=4 d T=6 e T=8 f T= Figure. Circular ring solitons: the function of sinu/ for the numerical solutions at times t =,, 4, 6, 8 and.

17 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS 335 sinu/, T= sinu/, T= sinu/, T= sinu/, T=6 sinu/, T=8 sinu/, T= Figure 3. Circular ring solitons: contours of sinu/ for the numerical solutions at times t =,, 4, 6, 8 and. 4 error of energ AVF SV HBVM5, error of relative energ AVF SV HBVM5, t a Et n Et t b Et n Et /Et Figure 4. The logarithms of the discrete energ error a and the relative discrete energ error b. Problem. The collision of two circular solitons Furthermore, we take the following initial conditions: 53 f, = 4 arctan ep g, = 4.3sech ep ,, 7 7,,, 7 7,

18 336 C. LIU, W. SHI, AND X. WU and epand the solution across the sides = and = 7 using the smmetr properties of the problem. We tr to seek the solutions over the domain, [ 3, ] [, 7] with the spatial steps h =.4, h =.8 and time step τ =.. The simulating results and the corresponding contour maps in terms of sinu/ at times t =,, 4, 6, 8, are depicted in Fig.5 and Fig.6, respectivel. The numerical results show the collision between two epanding circular ring solitons. The conservative behaviour of the discrete energ is demonstrated in Fig. 7. It again show that the modified AVF method 45 is more advantageous to the classical SV method and the HBVM5, method. a T= b T= c T=4 d T=6 e T=8 f T= Figure 5. Circular of two ring solitons: the function of sinu/ for the numerical solutions at times t =,, 4, 6, 8 and. 5. Conclusions In this paper, incorporating the fourth-order finite difference method and the AVF approach, we derived and analsed a novel energ-conservation scheme for the two-dimensional Hamiltonian wave equations with the homogenous Neumann boundar conditions. In this work, we first analsed some properties of the fourthorder finite difference operators and proposed the spatial semi-discrete scheme. The energ conservation law, the stabilit and the convergence of the finite difference scheme were analsed in detail. Moreover, the semi-discrete scheme can be epressed as the particular Hamiltonian sstem of ODEs 34, and its Hamiltonian 37 is equivalent to the discrete energ 8. For the time integration, the AVF approach is applied to preserve the Hamiltonian of the sstem 34 obtained from the spatial discretisation. The perfect combination of the fourth-order finite difference method and the AVF approach ields a novel and efficient numerical scheme

19 ENERGY-CONSERVATION SCHEME FOR HAMILTONIAN WAVE EQUATIONS sinu/, T= 7 sinu/, T= 7 sinu/, T= sinu/, T=6 7 sinu/, T=8 7 sinu/, T= Figure 6. Circular of two ring solitons: contours of sinu/ for the numerical solutions at times t =,, 4, 6, 8,. 4 error of energ AVF SV HBVM5, error of relative energ AVF SV HBVM5, t a Et n Et t b Et n Et /Et Figure 7. The logarithms of the discrete energ error a and the relative discrete energ error b. for the two-dimensional Hamiltonian wave equations. Three illustrative numerical eperiments were implemented and the numerical results show the convergence order of the spatial semidiscretisation and the remarkable conservative behaviour of the novel energ-conservation algorithm proposed in this paper. References [] M. J. Ablowitz, B. M. Herbst, C. Schober, On the numerical solution of the sine-gordon equation, J. Comput. Phs

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