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Development of a Numerical Phase Optimized Upwinding Combined Compact Difference Scheme for Solving the Camassa–Holm Equation with Different Initial Solitary Waves C

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Development of a Numerical Phase Optimized Upwinding Combined Compact Difference Scheme for Solving the Camassa–Holm Equation with Different Initial Solitary Waves C Development of a Numerical Phase Optimized Upwinding Combined Compact Difference Scheme for Solving the Camassa–Holm Equation with Different Initial Solitary Waves C. H. Yu,1,2 Tony W. H. Sheu,2,3,4 C. H. Chang,4 S. J. Liao5 1Department of Ocean Science and Engineering, Zhejiang University, Yuhangtang Road, Hangzhou, Zhejiang, People’s Republic of China 2Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan, Republic of China 3Institute of Applied Mathematical Sciences, Department of Mathematics, National Taiwan University, Taipei, Taiwan, Republic of China 4Center of Advanced Study in Theoretical Sciences (CASTS), Department of Mathematics, National Taiwan University, Taipei, Taiwan, Republic of China 5School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Received 31 January 2013; accepted 6 January 2015 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/num.21965 In this article, the solution of Camassa–Holm (CH) equation is solved by the proposed two-step method. In the first step, the sixth-order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first-order derivative term. For the purpose of retaining both of the long-term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure-like variable is approximated by the sixth-order accurate three-point centered compact difference scheme. Through the fundamental and numeri- cal verification studies, the integrity of the proposed high-order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first-order advection term κux in the CH equation will be addressed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: 000–000, 2015 Keywords: Camassa–Holm equation; Hamiltonian; long-term accurate; upwinding combined compact difference scheme Correspondence to: Tony W. H. Sheu, Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan, Republic of China (e-mail: [email protected]) Contract grant sponsor: National Science Council; contract grant number: NSC 99-2221-E-002-225-MY3 Contract grant sponsor: Fundamental Research Funds for the Central Universities; contract grant number: 2014QNA4030 © 2015 Wiley Periodicals, Inc. 2 YUETAL. I. INTRODUCTION Unlike the integrable KdV and Boussinesq equations applied to study soliton wave interaction, the integrable Camassa–Holm (CH) equation given below can be used to model not only the soliton interaction but also the breaking of wave ut + 2κux − uxxt + 3uux = 2ux uxx + uuxxx. (1) The solution of the nonlinear dispersive equation given above will be sought subject to the +| | 1+l | − |+| |+ initial condition which satisfies w(x,0) > 0 and R (1 x ) ( w(x,0) κ wx (x,0) |wxx(x,0)|)dx < ∞ for integers l. The momentum variable at t =0isw(x,0) = u(x,0) − uxx(x,0)+κ [1]. Equation (1) derived under the shallow water assumption has a positive constant 1/2 κ = (gh0) in the inviscid limit, where g stands for the gravity and h0 denotes the undisturbed water depth. Provided that κ = 0, Eq. (1) turns out to be the special case of the b-family of the equations ut − uxxt + (b + 1)uux = bux uxx + uuxxx [2]. This class of equations includes the integrable CH equation, where b = 2, and the Degasperis–Procesi equation, where b = 3 [3]. The CH equation has been the subject of intensive studies for many years for understanding its possible generation of breaking or peaked solitary wave, and other solitary waves whose natures are different at the peaks. Provided that there is a discontinuity in the first derivative at the solution peak, it is referred to as the peakon solution. In other words, peakon (or peaked solitary wave), by definition, is a soliton accommodating the finite-valued discontinuous first derivative term. As Eq. (1) permits peakon and cuspon solutions, calculation of these locally high gradient solutions near wave crest numerically is a challenging task. In comparison with the first-order spatial derivative term and the temporal derivative term, both of the third-order spatial derivative term and the mixed space-time derivative term shown in Eq. (1) were much less studied numerically. For this reason, the third-order CH equation has been transformed to its equivalent system of equations containing only the first-order spatial and tem- poral derivative terms. In this spirit, the u – m formulation was used in [4] to retain its symplectic solution property using the sixth-order accurate iterative symplectic Runge–Kutta scheme. All the first-order derivative terms were approximated by the proposed modified wavenumber opti- mized fourth-order accurate spatial scheme. This u – m formulation failed to numerically resolve the peaked wave crest. The u – P equations, which contain the equation ut + uux =−Px and the inhomogeneous Helmholtz equation for P, were therefore applied to conserve not only the symplecticity but also preserve the numerical modified wavenumber. Moreover, application of the u − P − α formulation helps to capture peakon and cuspon sharp profiles. In this light, the = 2 + 2 energy density α u ux is taken into account in the simulation of CH equation by solving the + =− = − 3 − 2 transport equation αt (uα)x Qx for α, where Q 2Pu u 2κu [6, 5]. In 1944, Camassa et al. showed that smooth solitary wave solutions are permitted for all κ > 0 but solutions with the derivative discontinuity are not allowed to appear at the peak of solitary wave [7]. In the stability analysis of the CH solitons, Constantin and Strauss [8] proved however that CH solution is stable under small disturbances. Similar to the finding in [9], Liao derived the peaked solitary wave for the case of κ > 0 through the Homotopic Analysis Method (HAM). However, such a peaked solitary wave has never been observed experimentally as κ > 0. This is the impetus for us to clarify numerically whether or not peaked solution is permitted in CH equation under the condition of κ >0. The rest of this article is organized as follows. The nonlinear CH equation and some of its intriguing solution natures will be presented in Section II. In Section III, the CH equation is trans- formed to a system of equations containing one equation for u with the reduced differential order Numerical Methods for Partial Differential Equations DOI 10.1002/num DEVELOPMENT OF A NUMERICAL PHASE OPTIMIZED UPWINDING 3 and the other inhomogeneous Helmholtz equation for P. In Sections IV.A—IV.C, the numerical methods developed for solving the equations shown in the u – P formulation are described. Section IV.D is devoted to the dispersion analysis of the proposed combined compact difference scheme. In Section V, the CH solutions will be sought subject to different initial conditions for the CH equation with/without inclusion of the linear first-order advection term. Our objective is to show their traveling/nontraveling one-solitary wave natures. Finally, some concluding remarks will be drawn in Section VI. II. CH EQUATION AND ITS DISTINGUISHED FEATURES The nonlinear dispersive CH equation has many mathematically intriguing properties and phys- ically rich phenomena. For example, CH equation has bi-Hamiltonian structure and has two associated infinite dimensional Hamiltonian formulations. This integrable equation possesses therefore infinitely many conservation laws. This mathematically fruitful wave equation accom- modates many remarkable geometric structures. First, CH equation can be completely integrated through the Lax pair equations constructed by Camassa and Holm [10]. The CH equation is also classified to be multisymplectic. According to the work of Bridges and Reich [11], two variables ∂ω + ∂κ = (ω, κ)exists which can conserve symplecticity because of ∂t ∂x 0. Given an initial condition 1 u0(x, t = 0) belonging to the Sobolev space H , Eq. (1) investigated at κ = 0, for example, has been shown to have the well-known conservation laws given in Section V [12–14]. III. ITERATIVE SOLUTION ALGORITHM The dimensionless CH equation has two space-time mixed derivative and third-order dispersive terms. We are therefore motivated to rewrite Eq. (1) to the equivalent system of equations having only the first-order derivative terms. One can reduce the differential order of the CH equation by rewriting (1) to the equation mt + umx + 2ux m =−2κux , where the momentum variable is defined by m = u − uxx − κ [15, 16]. The CH equation can be also transformed to the u – P system of equations. In the current study, the u – P formulation is adopted as this formulation has been known to be more suitable to predict solitary wave propagation involving either a peakon or a cuspon. Given a properly prescribed boundary condition, the solution u for (1) will be first sought subject to an initial condition u(x, t = 0) = f ∈ H 1 from the following nonlinear equation ut + uux =−Px .(2) The variable P is known as the dimensionless pressure (or surface elevation) and is governed by 2 P − Pxx = u + uux + 2κu. (3) Thanks to Eqs. (2) and (3), Eq. (1) is classified to be elliptic-hyperbolic provided that the solution remains smooth. IV. SYMPLECTIC SEMI-DISCRETIZATION METHOD In the literature, Camassa and Lee [17–20] solved the Eq. (1), subject to different specified bound- ary conditions, by introducing a proper number of “particles” along the characteristic curve of CH Numerical Methods for Partial Differential Equations DOI 10.1002/num 4 YUETAL.
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