A sixth-order dual preserving algorithm for the Camassa-Holm equation

Transcription

1 A sith-order dal preserving algorithm for the Camassa-Holm eqation Pao-Hsing Chi Long Lee Tony W. H. She November 6, 29 Abstract The paper presents a sith-order nmerical algorithm for stdying the completely integrable Camass- Holm (CH) eqation. The proposed sith-order accrate method preserves both the dispersion relation and the Hamiltonians of the CH eqation. The CH eqation in this stdy is written as an evoltion eqation, involving only the first-order spatial derivatives, copled with the Helmholtz eqation. We propose a two-step method that first solves the evoltion eqation by a sith-order symplectic Rnge- Ktta method and then solves the Helmholtz eqation sing a three-point sith-order compact scheme. The first-order derivative terms in the first step are approimated by a sith-order dispersion-relationpreserving scheme that preserves the physically inherent dispersive natre. The compact Helmholtz solver, on the other hand, allows s to se relatively few nodal points in a stencil, while achieving a higher-order accracy. The sith-order symplectic Rnge-Ktta time integrator is preferable for an eqation that possess a Hamiltonian strctre. We illstrate the ability of the proposed scheme by eamining eamples involving peakon or peakon-like soltions. We compare the compted soltions with eact soltions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accracy, elapsed compting time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method. keywords: Camassa-Holm eqation; Hamiltonians; symplectic Rnge Ktta; sith-order; Helmholtz eqation; dispersion-relation-preserving. 1 Introdction The Camassa-Holm (CH) eqation [2], t + 2κ t + 3 = 2 +, (1.1) reslts from an asymptotic epansion of the Eler eqations governing the motion of an inviscid flid whose free srface can ehibit gravity-driven wave motion [3, 17]. The small parameters sed to carry ot the Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Da-an District, Taipei City 16, Taiwan (R.O.C.). f @nt.ed.tw Department of Mathematics, Ross Hall 212, University of Wyoming, Dept 336, 1 E. University Ave., Laramie, WY , USA. llee@wyo.ed Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Da-an District, Taipei City 16, Taiwan (R.O.C.). twhshe@nt.ed.tw 1

2 epansion are the aspect ratio, whereby the depth of the flid is assmed to be mch smaller than the typical wavelength of the motion, and the amplitde ratio, or ratio between a typical amplitde of wave motion and the average depth of the flid. Ths, the eqation is a member of the class of weakly nonlinear (de to the smallness assmption on the amplitde parameter) and weakly dispersive (de to the long wave assmption parameter) models for water wave propagation. However, at variance with its celebrated close relatives in this class, sch as the Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) eqations, these small parameters are assmed to be linked only by a relative ordering, rather than by a power-law relation. This allows s to retain terms on the right-hand-side that wold be of higher order with respect to both the KdV and BBM epansions, and, in principle, to consider dynamical regimes in which nonlinearity is somewhat dominant with respect to wave dispersion. This eqation possesses the remarkable property of complete integrability, as evidenced by its La-pair representation, and permits an infinite nmber of nonlocal conserved properties [2,2]. When κ = in eqation (1.1), the eqation becomes a non-dispersive eqation that admits peakon soltions. There is etensive literatre on nmerical analysis and implementation for the KdV type of eqations; however, nmerical algorithms for the CH eqation have only received attention recently. While no attempt will be made here to provide a detailed reference list, the following are eamples of recent algorithms developed for the CH eqation. In [5 9], a completely integrable particle method is introdced that solves the eqation in infinite domains, semi-infinite domains with the zero bondary condition on one side, periodic domains, and homogeneos finite domains. The particle method is based on the Hamiltonian strctre of the eqation, an algorithm corresponding to a completely integrable particle lattice. Each particle in this method travels along a characteristic crve of the shallow-water wave model, determined by solving a system of nonlinear integro-differential eqations. This system of nonlinear integro-differential eqations can be viewed as particle interaction throgh a long-range potential (here position and momentm dependent). Besides the particle method, a method based on mltipeakons is developed in [15] for eqation (1.1). The convergence proof of this method is given in [16]. A psedospectral method is developed in [18] for the travelling wave soltion of eqation (1.1). Similar methods, a semi-discretization Forier-Galerkin method and a Forier-collocation method, are developed in [19]. A finite volme method, within the adaptive pwinding contet, is developed for the peakon soltion of eqation (1.1) [1]. A local discontinos Galerkin finite element method is developed in [23]. Eqation (1.1) involves two third-order derivative terms, and t. For most nmerical schemes, ecept the particle method and the related methods, certain care is reqired to discretize those terms in order to achieve a higher-order accracy while preserving the physically inherent dispersive natre of the eqation. In the stdy, however, we avoid discretizing both of the third-order derivative terms by applying the Helmholtz operator to m(,t) (1 2 )(,t), (1.2) and rewrite the eqation (1.1) into an eqivalent formlation m t = 2(m + κ) m. (1.3) We call eqations (1.2) and (1.3) the m-formlation of the CH eqation. Note that eqation (1.3) involves only the first-order derivative terms. As a reslt of the new formlation, we develop a sith-order twostep iterative nmerical algorithm that first solves the evoltion eqation (1.3) by a sith-order symplectic Rnge-Ktta method and then solves the Helmholtz eqation (1.2) with a three-point sith-order compact scheme. The first-order derivative terms in the first step are approimated by a sith-order dispersionrelation-preserving scheme that preserves the physically inherent dispersive natre. The compact Helmholtz 2

3 solver, on the other hand, allows s to se relatively few nodal points in a stencil, while achieving a higherorder accracy. The sith-order symplectic Rnge-Ktta time integrator preserves the Hamiltonians of the eqation. The principle of the two-step iterative algorithm is to solve the first-order eqation and then to solve the Helmholtz eqation, repeating the process ntil the convergence criterions are satisfied. A second-order accrate scheme based on the same principle for solving the CH eqation and a class of partial differential eqations involving the Helmholtz eqation is developed in [1, 11]. 2 Two-step iterative algorithm The evoltion eqation (1.3) can be solved by a standard method of lines (MOL). Let m n = m(t n,) and m n+1 = m(t n + t,) be the semi-discretized m vales at time level n and n + 1, respectively. Letting F(m,) = ( 2κ m 2 m), eqation (1.3) becomes m t = F (m,). (2.1) A sith-order accrate symplectic Rnge-Ktta scheme, developed in [21], is employed in the MOL for solving eqation (2.1): [ 5 m (1) = m n + t 36 F(1) + ( c 3 )F(2) + ( c ] 3 )F(3), (2.2) [ m (2) = m n + t ( c 12 )F(1) + ( 2 9 )F(2) + ( c ] 12 )F(3), (2.3) [ m (3) = m n + t ( 5 36 c 3 )F(1) + ( c 3 )F(2) + 5 ] 36 F(3), (2.4) [ 5 m n+1 = m n + t 18 F(1) F(2) + 5 ] 18 F(3). (2.5) where c = and F(i) = F(m (i), (i) ), i = 1,2,3. In this symplectic Rnge-Ktta method, in order to obtain m n+1 from eqation (2.5), we need to solve eqations (2.2) (2.4) simltaneosly for obtaining m (1), m (2), and m (3). After obtaining m n+1, we solve the Helmholtz eqation to obtain n+1. However, since the evoltion eqation (2.1) is copled with the Helmholtz eqation (1.2), in order to obtain m n+1 and n+1 from m n and n, it is necessary to introdce an iterative scheme. Therefore, instead of forming a linear system and solving eqations (2.2) (2.4) simltaneosly, we incorporate the three eqations into an iterative scheme. The iterative scheme solves eqation (2.1) and the Helmholtz eqation alternately ntil the convergence criterions are satisfied. The iterative steps are described as follows: Step 1: m n and n are known. We perform the fied-point iteration on eqations (2.2) (2.4) to obtain m (i) : (I) Given an initial gess for m (i) and (i), denoted m [],(i) and [],(i), respectively, i = 1,2,3 from m n and n. Solve eqations (2.2) (2.4) to obtain m [1],(i), i = 1,2,3. (II) Using m [1],(i) and the three-point sith-order compact scheme, we solve the Helmholtz eqation (1.2) to obtain [1],(i), i = 1,2,3. 3

4 (III) Repeat (I) with [k],(i) and m [k],(i) and (II) for the net iteration ntil the (k + 1) th iteration, for which the residals, in the maimm norm, of eqations (2.2) (2.4) and the Helmholtz eqation (1.2) satisfy or convergence criterions: m [k+1],(i) m n ( ma F m [k+1],(i), [k+1],(i)) ε, j, j=1,n t m ma [k+1],(i) j, j=1,n ( [k+1],(i) [k+1],(i) ) ε, (2.6) where N is the nmber of grid points and i = 1,2,3. The vale for the threshold error ε is typically chosen to be 1 12 throghot or comptations. Or nmerical eperiments indicate that typically the nmber of iterations needed for convergence is less than 2 (i.e. k + 1 2). Step 2: Use eqation (2.5) to pdate m n+1. Step 3: Solve eqation (1.2) to obtain n+1. Retrn to Step 1. 3 Dispersion-relation-preserving scheme The spacial accracy of the proposed scheme depends on how accrately we can approimate the first-order derivative terms. In particlar, if the eqation of interest is a dispersive eqation, sch as the CH eqation, a dispersion-relation-preserving scheme is necessary to ensre the accracy of nmerical soltions. In this section, we develop a dispersion-relation-preserving scheme for the first-order derivative terms. Sppose that the first derivative term at the grid point i is approimated by the following algebraic eqation: m i = 1 h (c 1m i 5 + c 2 m i 4 + c 3 m i 3 + c 4 m i 2 + c 5 m i 1 +c 6 m i + c 7 m i+1 + c 8 m i+2 + c 9 m i+3 ). (3.1) For simplicity, we consider the case involving only the positive convective coefficient in the above eqation, since the derivation will be the same for the negative convective coefficient. Derivation of epressions for c 1 c 9 is followed by applying the Taylor series epansions for m i±1, m i±2, m i±3, m i 4 and m i 5 with respect to m i and then eliminating the seven leading error terms derived in the modified eqation. Elimination of these error terms enables s to derive the following set of algebraic eqations: c 1 + c 2 + c 3 + c 4 + c 5 + c 6 + c 7 + c 8 + c 9 =, (3.2) 5c 1 4c 2 3c 3 2c 4 c 5 + c 7 + 2c 8 + 3c 9 = 1, (3.3) 25 2 c 1 + 8c c 3 + 2c c c 7 + 2c c 9 =, (3.4) c c c c c c c c 9 =, (3.5) c c c c c c c c 9 =, (3.6) c c c c c c c c 9 =, (3.7) c c c c c c c c 9 =. (3.8) 4

5 To niqely determine all nine introdced coefficients shown in (3.1), we need two more eqations. Following the sggestion in [22], we derive the eqations by preserving the dispersion relation that governs the relation between the anglar freqency and the wavenmber of the first-order dispersive term. To obtain the two etra eqations based on the principle of preservation of the dispersion relation, we note that the Forier transform pair for m is m(k) = 1 2π m() = Z + Z + m() e i k d, (3.9) m(α) e i k dk. (3.1) If we perform the Forier transform on each term shown in Eq. (3.1), we obtain that the wavenmber k is approimated by the following epression k i h (c 1 e i 5kh + c 2 e i 4kh + c 3 e i 3kh + c 4 e i 2kh i kh + c 5 e +c 6 + c 7 e i kh + c 8 e i 2kh + c 9 e i 3kh ), (3.11) where i = 1. Spposing that the effective wavenmber k is eactly eqal to the right-hand side of Eq. (3.11) [22], we have k k. In order to acqire a better dispersive accracy, k shold be made as close to k as possible. This implies that E defined in the sense of the 2-norm of the error between k and k will be the local minimm for sch a k. The error E is defined as follows Z π 2 E(k) = k h k h Z 2 π 2 d(kh) = γ γ 2 dγ, (3.12) π 2 π 2 where h is denoted as the grid size and γ = kh. For E to be a local minimm, we assme the following two etreme conditions E =, c 4 (3.13) E =. c 5 (3.14) Under the above prescribed etreme conditions, the two algebraic eqations needed for the coefficients to be niqely determined are 4 3 c 1 + 4c 3 + 2πc 4 + 4c c c 9 + π =, (3.15) 4 3 c 2 + 4c 4 + 2πc 5 + 4c c =. (3.16) We remark that for a trly dispersion-relation-preserving scheme, i.e. the error E is trly a local minimm on the parameter space, one will need to impose E/ c i = for i = to obtain 9 eqations for the coefficients. Or approach, instead, (i) ensres the higher-order accracy by letting the coefficients satisfy the Taylor series epansions and (ii) partially enforces the reqirements for a dispersion-relation-preserving scheme. Or nmerical eperiments show that the pwinding scheme for the first-order derivative obtained by taking the derivatives abot c 4 and c 5 for E (eqations (3.13) and (3.14)) prodces the least nmerical 5

6 errors. It is also worth noting that the integration interval shown in eqation (3.12) needs to be sfficiently wide to cover a complete period of sine (or cosine) waves. Eqations (3.15) and (3.16) together with eqations (3.2) to (3.8) yield the coefficients : c 1 = 1 5 ( 1575π2 834π ), 12432π π2 (3.17) c 2 = 3 1 ( 7875π2 4248π ), 12432π π2 (3.18) c 3 = π (55125π2 ), 12432π π2 (3.19) c 4 = π2 ( 6216π ), π π2 (3.2) 12 c 5 = 5(21π 64), (3.21) c 6 = π (17325π2 ), 12432π π2 (3.22) c 7 = π (55125π2 ), 12432π π2 (3.23) c 8 = 9 5 ( 2625π2 1544π ), 12432π π2 (3.24) c 9 = 1 15π 44 ( ). 6 15π 272 (3.25) It is easy to show that the proposed pwinding scheme for the first-order derivative is sith-order spatially accrate : m = m eact ( 15π π π 2 )h6 7 m 7 + ( 7875π2 3936π π π 2 )h7 8 m 8 + O(h8 ) +. (3.26) 4 Three-point sith-order accrate compact Helmholtz solver We introdce a compact scheme for solving the Helmholtz eqation in this section. It is well known that in order to obtain a higher-order nmerical method for the Helmholtz eqation, one can always introdce more points in a stencil. The improved accracy, however, comes at the cost of an epensive matri calclation, de to the wider stencil. With the aim of developing a nmerical scheme that is higher-order accrate while sing relative few stencil points in the finite difference discretization, we introdce a compact scheme involving only three points in a stencil, bt is sith-order accrate. Consider the following prototype eqation 2 k = f (). (4.1) 2 6

7 We first denote the vales of 2 / 2, 4 / 4 and 6 / 6 at a nodal point i as 2 2 i = s i, (4.2) 4 4 i = v i, (4.3) 6 6 i = w i. (4.4) Development of the compact scheme at point i starts with relating v, s and w with as follows: δ h 6 w i + γ h 4 v i + β h 2 s i = α 1 i+1 + α i + α 1 i 1. (4.5) Based on physics, it is legitimate to set α 1 = α 1 since the Helmholtz eqation is elliptic in natre. Having set α 1 = α 1, the derivation is followed by epanding i±1 with respect to i. Sbstittion of these Taylorseries epansion eqations into Eq. (4.5) leads to δ h 6 w i + γ h 4 v i + β h 2 s i = (α + 2α 1 ) i + h2 2! (2α 1) 2 i 2 + h4 4! (2α 1) 4 i 4 + h6 6! (2α 1) 6 i 6 + h8 8! (2α 1) 8 i +. (4.6) 8 Throgh a term-by-term comparison of the derivatives shown in Eq. (4.6), five simltaneos algebraic eqations can be derived. Hence, the introdced free parameters can be determined as α 1 = α 1 = 1, α = 2, β = 1, γ = 1 12 and δ = Note that w i = k 3 i + k 2 f i + k 2 f i + 4 f i, v 2 4 i = k 2 i + k f i + 2 f i, 2 and s i = k i + f i. Eqation (4.5) can then be epressed as It follows that α 1 i+1 + ( α β h 2 k γ h 4 k 2 δ h 6 k 3) i + α 1 i 1 ( = [h 2 β f i + h 4 γ k f i + 2 f i 2 ( i h 2 k h4 k ) ( + h 6 δ k 2 f i + k 2 f i f i 4 ) 36 h6 k 3 = h 2 f i + 1 ( ) 12 h4 k f i + 2 f i h6 )]. (4.7) i + i 1 ( ) k 2 f i + k 2 f i f i 4. (4.8) Using the proposed scheme, the corresponding modified eqation for (4.1) can be derived as follows, after performing some algebraic maniplation: 2 2 k = f + h h H.O.T.. (4.9) 1 Eqation (4.9) shows that the 3-point stencil scheme is indeed sith-order accrate. We implement a mltigrid method sing the V-cycle and flly-weighted projection/prolongation with the red-black Gass-Seidel smoother to solve the system of algebraic eqations arising from discretization of the proposed scheme. 5 Nmerical reslts In this section, we provide several test problems to validate the proposed scheme and elcidate its comptational properties. 7

8 5.1 Travelling wave soltion in periodic domains The first eample is the traveling wave soltion in periodic domains considered in [8, 9]. The periodic travelling wave soltion is given by (,t) = U( ct), provided that the minima of are located at = and the wave elevation is positive. In this case one finds the soltion of the travelling wave eqation is given by U = ± U 3 + (c 2κ)U 2 +C(A)U, (5.1) c U where c and A are denoted as the wave speed and the wave amplitde, respectively, and the integration constant C is a fnction of A. Integration of eqation (5.1) leads to the epression, = 2 b1 (b 2 b 3 ) (b 1 b 2 )Π(ϕ,β 2,T ). (5.2) The wavelength L of this periodic soltion can be written as L = 4 b1 (b 2 b 3 ) (b 1 b 2 )Π(ϕ,β 2,T ). (5.3) Details abot the variable, ϕ, β, b i (i = 1 3), c, and T are discssed in [8, 9]. 2 Present, N = 128 at t = Eact, at t = initial data 2 L 2 -error norm = 4.762E-6 Present, N = 128 Eact (a) (b) Figre 5.1: The predicted traveling wave soltion at (a) t = , (b) t = (over one period). Both are compared with the eact soltion, the dotted lines. The domain is L The nmber of cells sed for this calclation is N = 128 and t = 1 4. The parameters sed in the test problem are c = 2, κ = 1/2, and the integration constant C = 1, which altogether yield the wavelength (period) of L according to eqation (5.3). The total time for the wave to travel throgh the domain and back to the initial position is t = The time step sed in this calclation is t = 1 4 while the grid size is =.492 (or the nmber of cells N = 128). Figre 5.1(a) shows the nmerical and the eact soltions at t =.788. The initial data is the dashed line. A good 8

9 error in L 2 norm nmber of crrent 2 nd -order two-step iterative particle cells method method method N = E E E-2 N = E-4 9.8E E-3 N = E E E-4 N = E E-5 1.9E-4 Table 5.1: The comparison of errors in L 2 norm among three methods for the problem considered in Figre 5.1 (b). The time step sed in the calclation is c t = 1 4. rate of convergence nmber of crrent 2 nd -order two-step iterative particle cells method method method N = 32 N = N = N = Table 5.2: The comparison of rates of convergence among three methods for the problem considered in Figre 5.1 (b). The time step sed in the calclation is c t = 1 4. CPU times (seconds) nmber of crrent 2 nd -order two-step iterative particle cells method method method N = E E-2 1.E-3 N = E E E-2 N = E E-2 N = E-1 Table 5.3: The comparison of CPU times among three methods for the problem considered in Figre 5.1 (b). The time step sed in the calclation is c t =

10 agreement with the analytic soltion is clearly demonstrated. To show that the proposed scheme is phase accrate, we also plot the predicted soltion at t = As Figre 5.1(b) is shown, the waveform over one period of time and the waveform of the initial data are visally identical. Tables 5.1 Table 5.3 show the comparisons of errors in L 2 norm, rates of convergence, and CPU times among three different methods: the proposed method, a second-order two-step iterative method [11], and a second-order particle method [9]. The soltions are compted at a fied ratio c t =.25, where c = 2 is the wave speed. As epected, the tables show that the proposed method is sith-order accrate and has smallest errors in L 2 norm among the three methods for fine grids, bt the proposed method is less efficient than the other two methods. 5.2 Smooth travelling wave soltion In [4], an eact travelling wave soltion of eqation (1.1) is given by (,t) = U( ct) U(s), where c = 8κ/3 and U(s) = 8 (1 3 κ 3 ) 3 + 6sin2z (1 + 2cos2z)(2 3cos2z, (5.4) 3cos4z + 2sin2z + sin4z) with z = arctan(e s/2 )/3. The initial condition () = U() yields the initial data for the ailiary variable m, ( c 2 ) m () = κ (c U()) 2 1. (5.5) The CH eqation (1.1) is a complete integrable eqation. There are many Hamiltonians associated with the eqation. Among these conserved qantities, the mass M, and the Hamiltonians H 1 and H 2 are given as follows: Z M = d, H 1 = 1 Z ( 2 + ( ) 2) d, H 2 = 1 Z ( 3 + ( ) 2 + 2κ 2 ) d. (5.6) 2 2 The conserved qantity W associated with eqation (1.1) is derived in [5]. The brief derivation for W is described as follows. First, eqation (1.1) is written as W t + (W ) where W is defined as W = m + κ, where m =. If we define then we can write eqation (5.7) as W = Z =, (5.7) W d, (5.8) W t + W =. (5.9) This is an advection eqation, where the conserved qantity W is advected by and therefore is a constant in time. Figre 5.2 (a) is a plot for the compted travelling wave soltion. The compted soltion is compared with the eact soltion at time t = 5 with κ = 1. The figre shows clearly that the compted soltion and the eact soltion are visally identical, which means that the proposed scheme evolves the solitary wave while perfectly maintaining its shape. Besides maintaining the waveform, the sith-order symplectic time integrator employed in the proposed scheme has the ability to preserve the Hamiltonians. Figre 5.2 (b) shows that all the conserved qantities, M, H 1, H 2, and W, are well preserved by the proposed algorithm. The comptational domain for this eample is [ 18,18], while the nmber of cells sed is N = 248 ( =.17578). The time step sed is t =.5. 1

11 1.8 Initial data Nmerical soltion Eact soltion 7 6 M H 1 H 2 W (a) (b) time Figre 5.2: (a) The smooth travelling wave. Comparison between the compted soltion and the eact soltion at t = 5. They are visally identical. (b) Verification of the proposed algorithm. It shows that the qantities M, H 1, H 2, and W for the travelling wave problem in (a) are well preserved by the proposed algorithm. Note that the vale of W is divided by Interacting solitons In [9], the CH eqation (1.1) in periodic domains is scaled into the wave-dispersion regimes. Under this set of scales, eqation (1.1) is compatible with the KdV eqation stdied in the paper by Zabsky and Krskal [24]. Ths, the CH eqation shold in principle spport soltions that behave similarly to those of the KdV eqation and ehibit phenomena similar to those described in Zabsky and Krskal s paper [24], i.e. soliton formation with interactions, and the recrrence of smooth initial states. Consider the following scaled CH eqation in a periodic domain: t + 2κ t + 3 = 2 +, (,) = 1 3δ cos(πδ), (5.1) (,t) = (2/δ,t), where κ = 1/(2δ) and δ 2 is the small parameter in front of the dispersive term in the KdV eqation. Note that when δ is small, the reqired periodic domain for the stdy is large. Hence a large nmber of grid points may be necessary for obtaining flly resolved comptations, which makes the initial vale problem (5.1) a challenging problem for a long-time behavior stdy. Using the parameter δ =.22 in (5.1), Figre 5.3 (a) shows that eight solitons appear in the domain at the same time. The peaks and phases of the solitons are identical to those compted by the particle method. The nmber of cells sed in this calclation is N = 8192 in the periodic domain L = 2/δ The total nmber of time steps sed is 3785 for the final time t = Note that in order to se the reslt of the particle method as a referenced soltion, we se a large nmber of particles, N = 21, in or calclation. Similar to the eample in the previos section, we also compte the conserved qantities, M, H 1, H 2, and W. We remark that in terms of preserving these conserved qantities, this eample is a harder problem compared with the smooth travelling wave eample in Section 5.2. The reason is that this eample involves the formation of eight sharp solitons. Figre 5.3 (b) 11

12 shows that the proposed algorithm preserves all the conserved qantities dring the formation of solitons. Note that in Figre 5.3 (b), the vale of H 1 is divided by 1, the vale of H 2 is divided by 1, and the vale of W is divided by two-step method particle method initial data M H 1 H 2 W (a) (b) time Figre 5.3: (a) shows that eight solitons appear in the domain at the same time. The peaks and phases of the solitons are identical to those compted by the particle method. (b) shows that the proposed algorithm preserves all the conserved qantities dring the formation of solitons. 5.4 The α-formlation In the limit of κ =, the non-dispersive CH eqation t t + 3 = 2 + (5.11) admits peakon soltions. For peakon soltions, the second derivative of ( ) at the peaks is a Dirac delta fnction. Hence the ailiary variable m = is also a Dirac delta fnction. Therefore solving problems involving peakson soltions poses a challenge for the proposed m-formlation two-step algorithm. In fact, for problems of this type, nmerical simlations pose a challenge as well to the Elerian based schemes, inclding global spectral or psedo-spectral schemes. A global spectral or psedo-spectral scheme is definitely not a good choice for this type of problem, de to the hge nmber of terms reqired in the corresponding epansion (since the second derivative term of the sharp peaked wave is approaching to a Dirac delta fnction whose Forier epansion coefficients do not decay). In this stdy, to avoid this nmerical difficlty, following the sggestion in [13], we write the nondispersive CH eqation (5.11) into an eqivalent system of eqations: t + + P =, (5.12) P + P = 1 2 (2 + α), (5.13) α t + (α) = ( 3 2P), (5.14) where α = The two-step iterative method developed for solving eqations (1.3) and (1.2), the m- formlation, can be sed to solve the above system of eqations. That is, in the first step, instead of 12

13 solving one eqation (1.3), we solve eqations (5.12) and (5.14) for and α. In the second step, we solve a Helmholtz eqation (5.13) for the ailiary variable P. The dispersion-relation-preserving scheme, the threepoint compact Helmholtz solver, and the symplectic sith-order time integrator all remind nchanged. We call eqations (5.12)-(5.14) the α-formlation. Note that the α-formlation involves estimating only the firstorder spatial derivative of. This is advantageos for problems involving peakon soltions, since the firstorder spatial derivatives near the peaks can be accrately approimated by a finite difference approimation Sharp peaked travelling waves An eample in [5 7] shows that for the non-dispersive CH eqation (5.11), the soltion (,t), corresponding to the initial condition m () = asech 2 (), forms a rather sharply peaked wave and moves to the right, followed by others emerging from the location of the initial hmp m. The peaks are sharpened over time and eventally the first-order spatial derivatives of the peaks become discontinos (corners) when time approaches infinity. For the particle method developed in [5 7], one can observe that the particles rapidly clster in the region of the peaks of the solitary waves. Sch pile-p phenomenon sggests that particles get very close to each other in this region. When the distance between particles is so close that the machine precision can no longer distingish between locations of the coalescing particles, particle collisions occr nmerically. This effect is prely nmerical, since particle collisions cannot take place in finite time [7]. As a conseqence of sch a nmerical artifact, the particle method breaks down shortly after the nmerical collision occrs. Nevertheless, becase of the particle clstering, the particle method can captre the peak location accrately before the method breaks down. To resolve the nmerical artifact, one can se a higherprecision arithmetic to etend the time for the first occrrence of sch nmerical particle collisions [7]. Besides higher-precision arithmetics, a redistribtion algorithm is introdced in [7] to resolve the nmerical particle collision, so that a lower-precision arithmetic can be sed for solving the problem. We se reslts compted from the particle method as the referenced soltions. We se a = 1/2 in the initial condition, m () = asech 2 (), and solve the Helmholtz eqation (1.2) to obtain. Figre 5.4 (a) shows the first soliton, the second soliton, and the formation of the third soltion compted by the proposed algorithm with the α-formlation. Figre 5.4 (b) is the magnification of the third soliton. The soltions compted by the proposed method is compared with those compted by the particle method. The reslts are indistingishable between the two methods. The final time of the calclation is t = 4. The nmber of cells sed for the proposed method is N = in the domain [ 5,15] ( =.35176), while the resoltion for the particle method is =.75. The time step sed is t =.125 for the proposed method and t =.325 for the particle method. Table 5.4 compares the magnitdes of the first and the second solitons compted by the proposed and the particle methods at t = 4, with those predicted by the theory as t [5]. Both methods have done a good job of predicting the magnitdes. The proposed method The particle method Theoretical prediction (t=4) (t=4) (t ) The first soliton /3 The second soliton /3 Table 5.4: Comparison of the predicted magnitdes of the first and the second solitons among the proposed method, the particle method, and the theory. 13

14 .4.3 SRK6 particle method initial condition SRK6 particle method (a) (b) Figre 5.4: (a) shows the first soliton, the second soliton, and the formation of the third soltion. (b) is the magnification of the third soliton. The soltions compted by the proposed method is compared with those compted by the particle method. The reslts are indistingishable between the two methods Soliton-antisoliton collision The two-soliton dynamics of the non-dispersive CH eqation (5.11) are stdied in detail in [2, 3]. An eact soltion is given for the perfectly antisymmetric soliton-antisoliton collision case. This is a nmerically challenging problem, since the term tends toward a sm of delta fnctions when the collision occrs. This sggests that the right-hand-side of eqation (1.3) becomes the derivative of a delta fnction when the collision occrs. Similar to the previos eample, we avoid the nmerical difficlty by sing the α- formlation in or nmerical simlations. Consider the soliton-antisoliton initial condition () = e +5 e 5. (5.15) The collision time t c and the wave speed c can be obtained by solving eqation (4.26) in [3] 1 = 2log[sech( ct c )], 2c 2 = tanh( ct c ). (5.16) Solving the above eqations, we have c and t c Following the notations in [3], we write the soltion of eqation (5.11) for the soliton-antisoliton collision as c [ (,t) = e q(t) e +q(t) ], (5.17) tanh(c(t t c )) where q(t) = log [ sech 2 (t t c ) ]. (5.18) Figre 5.5 (a) (d) show simlations of the soliton-antisoliton collision: (a) is the initial condition, (b) is the beginning of the collision, (c) is the approimate time of the collision, and (d) is post collision. The 14

15 1 t = 1 t = SRK6 eact soltion (a) (b) t = SRK6 eact soltion t = 11 SRK6 eact soltion (c) (d) Figre 5.5: The soliton-antisoltoin collision: (a) is the initial condition, (b) is the beginning of the collision, (c) is the approimate time of the collision, and (d) is post collision. The theoretical wave speed is c , and the theoretical collision time is t c The compted soltions are compared with the eact soltions in the Figres. The simlation figres show that the proposed scheme not only accrately captres the wave speed and the collision time, bt they are indistingishable from the eact soltions. 15

16 Figres compare soltions compted by the proposed scheme with the eact soltions fond by eqations (5.17) and (5.18). They show that the proposed scheme not only accrately captres the wave speed and the collision time, bt they are indistingishable from the eact soltions. The nmber of cells sed in the simlations is N = in the domain [ 25,25], or =.351. The time step is t =.1. 6 Conclsion and ftre work A two-step iterative algorithm for a completely integrable CH eqation is developed in this stdy. The algorithm is sith-order accrate and preserves the dispersion relation and the Hamiltonians of the eqation. In the first step, we introdce a sith-order accrate dispersion-relation-preserving scheme to approimate the first-order spatial derivatives, and in the second step we develop a three-point sith-order accrate Helmholtz solver. A sith-order symplectic Rnge-Ktta time integrator that well preserves the Hamiltonians of the completely integrable CH eqation is employed as the time-stepping scheme in the first step. Strength of the proposed algorithm is validated throgh several eamples to demonstrate the method s efficiency and accracy over time. We assess the comptational qality of the proposed algorithm by the compted errors, the CPU times, and the rates of convergence. Note that for peakon or peakon-like soltions, sch as the eamples in Sections and 5.4.2, it makes sense to implement some type of adaptive-mesh-refinement (AMR) scheme to resolve the sharp peaks, in particlar for the Elerian type of schemes like the proposed algorithm. However, since this stdy is a one-dimensional problem, instead of an AMR scheme, we se very fine grids to resolve the peakon type of soltions, while maintaining reasonable comptational times. While we have developed an efficient higher-order method to stdy soltion properties of the CH eqation, there are still qestions left nanswered by this stdy. One of them regards the splitting errors introdced by the two-step iterative process. The two-step iterative method developed here is similar to operator splitting methods. It is not clear what splitting error is introdced by solving two eqations alternately. We are developing a posterior error estimator to evalate the splitting error introdced by the iterative scheme. 7 Acknowledgment This work was spported by the National Science Concil of the Repblic of China nder Grants NSC E-2-4 and partially spported by NSF throgh the Grant DMS T. W. H. She wold like to thank the Department of Mathematics at the University of Wyoming in Laramie for its provision of ecellent research facilities dring his visiting professorship. In addition, we were stimlated considerably by the discssion with R. Camassa in the corse of condcting this stdy. References [1] R. Artebrant and H. J. Schroll, Nmerical simlation of Camassa-Holm peakons by adaptive pwinding, Appl. Nmer. Math., 56 (26) [2] R. Camassa and D. Holm, An integrable shallow water eqation with peaked solitons, Phys. Rev. Letters, 71 (1993) [3] R. Camassa, D. Holm, and J. M. Hyman, A new integrable shallow water eqation, Advan. Appl. Mech., 31 (1994)

17 [4] R. Camassa and A.I. Zenchck, On the initial vale problem for a completely integrable shallow water wave eqation, Phys. Lett. A, 281:26 33, 21. [5] R. Camassa, Characteristics and initial vale problem of a completely integrable shallow water eqation, DCDS-B, 3 (23) [6] R. Camassa, J. Hang, and L. Lee, On a completely integral nmerical scheme for a nonlinear shallowwater wave eqation, J. Nonlin. Math. Phys., 12 (25) [7] R. Camassa, J. Hang, and L. Lee, Integral and integrable algorithm for a nonlinear shallow-water wave eqation, J. Comp. Phys., 216 (26) [8] R. Camassa and L. Lee, A complete integral particle method for a nonlinear shallow-water wave eqation in periodic domains, DCDIS-A, 14(S2) (27) 1-5. [9] R. Camassa and L. Lee, Complete integrable particle methods and the recrrence of initial states for a nonlinear shallow-water wave eqation, J. Comp. Phys., 227 (28) [1] R. Camassa, P. H. Chi, L. Lee, and T. W. H. She, Nmerical investigation of Helmholtz reglarizations in a class of partial differential eqations, in sbmission. [11] P. H. Chi, L. Lee, and T. W. H. She, A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave eqation, J. Compt. Phys. 228 (29) [12] G. M. Coclite, K. H. Karlsen and N. H. Risebro, A convergent finite difference scheme for the Camassa-Holm eqation with general H 1 initial data, Dept. Math. /CMA University of Oslo, 16 (26) [13] D. Cohen, B. Owren, and X. Raynad, Mlti-symplectic integration of the Camassa-Holm eqation, J. Comp. Phys., 227 (28) [14] J. De Frtos and J. M. Sanz-Serna, An easily implementable forth-order method for the time integration of wave problems, J. Compt. Phys., 13 (1992) [15] H. Holden and X. Raynad, A convergent nmerical scheme for the Camassa-Holm eqation based on mltipeakons, DCDS-B, 14 (26) [16] H. Holden and X. Raynad, Convergence of a finite difference scheme for the Camassa-Holm eqation, SIAM J. Nmer. Anal., 44 (26) [17] R.S. Johnson Camassa-Holm, Korteweg-de Vries and related models for water waves, JFM, 455 (22) [18] H. Kalisch and J. Lenells, Nmerical stdy of traveling-wave soltions for the Camassa-Holm eqation, Chaos, Solitons & Fractals, 25 (25) [19] H. Kalisch and X. Raynad, Convergence of a spectral projection of the Camassa-Holm eqation, Nmerical Methods for Partial Differential Eqations, 22 (26) [2] J. Lenells, Conservation laws of the Camassa-Holm eqation, J. Phys. A., 38 (25)

18 [21] W. Oevel, W. Sofronio, Symplectic Rnge-Ktta schemes II: classification of symplectic methods, Univ. of Paderborn, Germany, Preprint, [22] C. K. W. Tam and J. C. Webb, Dispersion-relation-preserving finite difference schemes for comptational acostics, J. Comp. Phys., 17 (1992) [23] Y. X and C. W. Sh, A local discontinos Galerkin method for the Camassa-Holm eqation, SIAM J. Nmer. Anal., 46 (28) [24] N. J. Zabsky and M. D. Krskal, Interaction of solitons in a collisionless plasma and the recrrence of initial states, Phys. Rev. Lett., 15 (1965)