Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa-Holm equation with different initial solitary waves
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...
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| Published in | Numerical methods for partial differential equations Vol. 31; no. 5; pp. 1645 - 1664 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
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Blackwell Publishing Ltd
01.09.2015
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| Abstract | 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 numerical 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 31: 1645–1664, 2015 |
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| AbstractList | 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 numerical 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 [kappa]ux in the CH equation will be addressed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1645-1664, 2015 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 numerical 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 31: 1645–1664, 2015 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 numerical 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 Kappa u sub(x) in the CH equation will be addressed. Numer Methods Partial Differential Eq 31: 1645-1664, 2015 |
| Author | Liao, S. J. Sheu, Tony W. H. Chang, C. H. Yu, C. H. |
| Author_xml | – sequence: 1 givenname: C. H. surname: Yu fullname: Yu, C. H. organization: Department of Ocean Science and Engineering, Zhejiang University, Yuhangtang Road, Hangzhou, Zhejiang, People's Republic of China – sequence: 2 givenname: Tony W. H. surname: Sheu fullname: Sheu, Tony W. H. email: twhsheu@ntu.edu.tw organization: Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, Taiwan, Republic of China – sequence: 3 givenname: C. H. surname: Chang fullname: Chang, C. H. organization: Center of Advanced Study in Theoretical Sciences (CASTS), Department of Mathematics, National Taiwan University, Taiwan, Taipei, Republic of China – sequence: 4 givenname: S. J. surname: Liao fullname: Liao, S. J. organization: School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China |
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| References | A. B. De Monvel, A. Kostenko, D. Shepelsky, and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J Math Anal 41 (2009), 1559-1588. P. C. Chu and C. Fan, A three-point combined compact difference scheme, J Comput Phys 140 (1998), 370-399. R. I. McLachlan, Symplectic integration of Hamiltonian wave equations, Numer Math 66 (1994), 465-492. A. Constantin and W. A. Strauss, Stability of Camassa-Holm solitons, J Nonlinear Sci 12 (2002), 415-422. P. H. Chiu and T. W. H. Sheu, On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation, J Comput Phys 228 (2009), 3640-3655. T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J Physics A 39 (2006), 5287-5320. S. Hakkaev, Stability of peakons for an integrable shallow water equation, Phys Letter A 354 (2006), 137-144. W. Oevel and M. Sofroniou, Symplectic Runge-Kutta schemes II: classification of symmetric method, University of Paderborn, Germany, Preprint, 1997. A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Commun Math Phys 211 (2000), 45-61. R. Camassa and L. Lee, Complete integrable particle methods and the recurrence of initial states for a nonlinear shallow-water wave equation, J Comput Phys 227 (2008), 7206-7221. R. Camassa, Characteristics and initial value problem of a completely integrable shallow water equation, Discrete Continuous Dyn Syst Ser B 3 (2003), 115-139. H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm Partial Diff Equat 32 (2006), 1511-154. Tony W. H. Sheu, P. H. Chiu, and C. H. Yu, On the development of a high-order compact scheme for exhibiting the switching and dissipative solution natures in the Camassa-Holm equation, J Comput Phys 230 (2011), 5399-5416. T. Matsuo and H. Yamaguchi, An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations, J Comput Phys 228 (2009), 4346-4358. D. Cohen, B. Owren, and X. Raynaud, Multi-symplectic integration of the Camassa-Holm equation, J Comput Phys 227 (2008), 5492-5512. C. K. W. Tam and J. C. Webb, Dispersion-relation-preserving finite difference schemes for computational acoustics, J Comput Phys 107 (1993), 262-281. Y. Ohta, K -I. Maruno and B. -F. Feng, An integrable semi-discretization of the Camassa-Holm equation and its determinant solution, J Phys A: Math Theor 41 (2008), 355205. R. Camassa, J. Huang, and L. Lee, Integral and integrable algorithm for a nonlinear shallow-water wave equation, J Comput Phys 216 (2006), 547-572. Y. Xu and C. W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J Numer Anal 46 (2008), 1998-2021. R. Camassa, D. Holm, and J. Hyman, A new integrable shallow water equation, Adv Appl Mech 31 (1994), 1-33. B. -F. Feng, K. Maruno and Y. Ohta, A self-adaptive mesh method for the Camassa-Holm equation, J Comput Appl Math 235 (2010), 229-243. P. H. Chiu, L. Lee, and T. W. H. Sheu, A sixth-order dual preserving algorithm for the Camassa-Holm equation, J Comput Appl Math 233 (2010), 2267-2278. H. Kalisch and X. Raynaud, Convergence of a spectral projection of the Camassa-Holm equation, Numer Method Part D E 22 (2006), 1197-1215. S. J. Liao, Two kinds of peaked solitary waves of the KdV, BBM and Boussinesq equations, Sci China, Phys, Mechan Astron 55 (2012), 2469-2475. H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin Dyn Syst 14 (2006), 503-523. P. H. Chiu, L. Lee, and T. W. H. Sheu, A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation, J Comput Phys 228 (2009), 8034-8052. A. M. Wazwaz, A modified KdV-equaiton that admits a variety of travelling wave solutions: kinks, solitons, peakons and cuspons, Phys Scr 86 (2012), 045501. T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys Letter A 284 (2001), 184-193. M. Stanislavova and A. Stefanov, On global finite energy solutions of the Camassa-Holm equation, J Fourier Anal Appl 11 (2005), 511-531. H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J Numer Anal 44 (2006), 1655-1680. G. Ashcroft and X. Zhang, Optimized prefactored compact schemes, J Comput Phys 190 (2003), 459-477. D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J Appl Dyn Syst 2 (2003), 323-380. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys Rev Lett 71 (1993), 1661-1664. H. Kalisch and J. Lenells, Numerical study of traveling-wave solutions for the Camassa-Holm equation, Chaos, Solitons Fractals 25 (2005), 287-298. R. Camassa and L. Lee, A complete integral particle method for a nonlinear shallow-water wave equation in periodic domains, DCDIS, Series A 14 (2007), 1-5. R. Camassa, J. Huang, and L. Lee, On a completely integral numerical solution for a nonlinear shallow-water wave equation, J Nonlinear Math Phys 12 (2005), 146-162. R. Artebrant and H. J. Schroll, Numerical simulation of Camassa-Holm peakons by adaptive upwinding, Appl Numer Math 56 (2006), 695-711. J. Lenells, Stability of periodic peakons, Int Math Res Notices 10 (2004), 485-499. 2009; 41 1993; 107 2001; 284 2006; 56 2006; 32 2002; 12 2006; 39 2006; 14 1994; 66 1997 2008; 227 2003; 190 2000; 211 2006; 216 2012; 55 2006; 354 2007; 14 1999 2011; 230 2005; 25 2004; 10 1993; 71 2006; 44 2006; 22 2010; 235 2010; 233 2003; 2 2003; 3 2008; 46 2008; 41 2009; 228 2005; 11 1998; 140 2005; 12 2012; 86 1994; 31 |
| References_xml | – reference: A. B. De Monvel, A. Kostenko, D. Shepelsky, and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J Math Anal 41 (2009), 1559-1588. – reference: Y. Ohta, K -I. Maruno and B. -F. Feng, An integrable semi-discretization of the Camassa-Holm equation and its determinant solution, J Phys A: Math Theor 41 (2008), 355205. – reference: R. Camassa, J. Huang, and L. Lee, Integral and integrable algorithm for a nonlinear shallow-water wave equation, J Comput Phys 216 (2006), 547-572. – reference: B. -F. Feng, K. Maruno and Y. Ohta, A self-adaptive mesh method for the Camassa-Holm equation, J Comput Appl Math 235 (2010), 229-243. – reference: H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm Partial Diff Equat 32 (2006), 1511-154. – reference: J. Lenells, Stability of periodic peakons, Int Math Res Notices 10 (2004), 485-499. – reference: P. H. Chiu, L. Lee, and T. W. H. Sheu, A sixth-order dual preserving algorithm for the Camassa-Holm equation, J Comput Appl Math 233 (2010), 2267-2278. – reference: R. Camassa and L. Lee, Complete integrable particle methods and the recurrence of initial states for a nonlinear shallow-water wave equation, J Comput Phys 227 (2008), 7206-7221. – reference: R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys Rev Lett 71 (1993), 1661-1664. – reference: G. Ashcroft and X. Zhang, Optimized prefactored compact schemes, J Comput Phys 190 (2003), 459-477. – reference: D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J Appl Dyn Syst 2 (2003), 323-380. – reference: H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J Numer Anal 44 (2006), 1655-1680. – reference: A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Commun Math Phys 211 (2000), 45-61. – reference: R. Camassa, Characteristics and initial value problem of a completely integrable shallow water equation, Discrete Continuous Dyn Syst Ser B 3 (2003), 115-139. – reference: P. H. Chiu, L. Lee, and T. W. H. Sheu, A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation, J Comput Phys 228 (2009), 8034-8052. – reference: T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J Physics A 39 (2006), 5287-5320. – reference: D. Cohen, B. Owren, and X. Raynaud, Multi-symplectic integration of the Camassa-Holm equation, J Comput Phys 227 (2008), 5492-5512. – reference: S. Hakkaev, Stability of peakons for an integrable shallow water equation, Phys Letter A 354 (2006), 137-144. – reference: H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin Dyn Syst 14 (2006), 503-523. – reference: R. Camassa, D. Holm, and J. Hyman, A new integrable shallow water equation, Adv Appl Mech 31 (1994), 1-33. – reference: R. I. McLachlan, Symplectic integration of Hamiltonian wave equations, Numer Math 66 (1994), 465-492. – reference: Tony W. H. Sheu, P. H. Chiu, and C. H. Yu, On the development of a high-order compact scheme for exhibiting the switching and dissipative solution natures in the Camassa-Holm equation, J Comput Phys 230 (2011), 5399-5416. – reference: H. Kalisch and J. Lenells, Numerical study of traveling-wave solutions for the Camassa-Holm equation, Chaos, Solitons Fractals 25 (2005), 287-298. – reference: S. J. Liao, Two kinds of peaked solitary waves of the KdV, BBM and Boussinesq equations, Sci China, Phys, Mechan Astron 55 (2012), 2469-2475. – reference: W. Oevel and M. Sofroniou, Symplectic Runge-Kutta schemes II: classification of symmetric method, University of Paderborn, Germany, Preprint, 1997. – reference: P. H. Chiu and T. W. H. Sheu, On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation, J Comput Phys 228 (2009), 3640-3655. – reference: A. Constantin and W. A. Strauss, Stability of Camassa-Holm solitons, J Nonlinear Sci 12 (2002), 415-422. – reference: P. C. Chu and C. Fan, A three-point combined compact difference scheme, J Comput Phys 140 (1998), 370-399. – reference: M. Stanislavova and A. Stefanov, On global finite energy solutions of the Camassa-Holm equation, J Fourier Anal Appl 11 (2005), 511-531. – reference: A. M. Wazwaz, A modified KdV-equaiton that admits a variety of travelling wave solutions: kinks, solitons, peakons and cuspons, Phys Scr 86 (2012), 045501. – reference: H. Kalisch and X. Raynaud, Convergence of a spectral projection of the Camassa-Holm equation, Numer Method Part D E 22 (2006), 1197-1215. – reference: T. Matsuo and H. Yamaguchi, An energy-conserving Galerkin scheme for a class of nonlinear dispersive equations, J Comput Phys 228 (2009), 4346-4358. – reference: T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys Letter A 284 (2001), 184-193. – reference: Y. Xu and C. W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J Numer Anal 46 (2008), 1998-2021. – reference: R. Artebrant and H. J. Schroll, Numerical simulation of Camassa-Holm peakons by adaptive upwinding, Appl Numer Math 56 (2006), 695-711. – reference: R. Camassa, J. Huang, and L. Lee, On a completely integral numerical solution for a nonlinear shallow-water wave equation, J Nonlinear Math Phys 12 (2005), 146-162. – reference: C. K. W. Tam and J. C. Webb, Dispersion-relation-preserving finite difference schemes for computational acoustics, J Comput Phys 107 (1993), 262-281. – reference: R. Camassa and L. 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| Snippet | 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... 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... |
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| SubjectTerms | Approximation Camassa-Holm equation Derivatives Fluid dynamics Hamiltonian Integrity long-term accurate Mathematical analysis Mathematical models Partial differential equations upwinding combined compact difference scheme Wave propagation |
| Title | Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa-Holm equation with different initial solitary waves |
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