메뉴 건너뛰기




Volumn 46, Issue 3, 2008, Pages 1554-1579

A convergent finite difference scheme for the Camassa-Holm equation with general H1 initial data

Author keywords

Camassa Holm equation; Convergence; Finite difference scheme; Hyperbolic elliptic system; Weak solution

Indexed keywords

CAMASSA-HOLM EQUATION; CONVERGENCE; FINITE DIFFERENCE SCHEME; HYPERBOLIC-ELLIPTIC SYSTEM; WEAK SOLUTION;

EID: 49749109221     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060673242     Document Type: Article
Times cited : (49)

References (39)
  • 1
    • 33644862175 scopus 로고    scopus 로고
    • Numerical simulation of Camassa-Holm peakons by adaptive upwinding
    • R. ARTEBRANT AND H. J. SCHROLL, Numerical simulation of Camassa-Holm peakons by adaptive upwinding, Appl. Numer. Math., 56 (2006), pp. 695-711.
    • (2006) Appl. Numer. Math , vol.56 , pp. 695-711
    • ARTEBRANT, R.1    SCHROLL, H.J.2
  • 2
    • 0034714658 scopus 로고    scopus 로고
    • Multipeakons and the classical moment problem
    • R. SEALS, D. H. SATTINGER, AND J. SZMIGIELSKI, Multipeakons and the classical moment problem, Adv. Math., 154 (2000), pp. 229-257.
    • (2000) Adv. Math , vol.154 , pp. 229-257
    • SEALS, R.1    SATTINGER, D.H.2    SZMIGIELSKI, J.3
  • 3
    • 51149109959 scopus 로고    scopus 로고
    • Global dissipative solutions of the Camassa-Holm equation
    • A. BRESSAN AND A. CONSTANTIN, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), pp. 1-27.
    • (2007) Anal. Appl. (Singap.) , vol.5 , pp. 1-27
    • BRESSAN, A.1    CONSTANTIN, A.2
  • 4
    • 33846878957 scopus 로고    scopus 로고
    • Global conservative solutions of the Camassa-Holm equation
    • A. BRESSAN AND A. CONSTANTIN, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), pp. 215-239.
    • (2007) Arch. Ration. Mech. Anal , vol.183 , pp. 215-239
    • BRESSAN, A.1    CONSTANTIN, A.2
  • 5
    • 33845317002 scopus 로고    scopus 로고
    • An optimal transportation metric for solutions of the CamassaHolm equation
    • A. BRESSAN AND M. FONTE, An optimal transportation metric for solutions of the CamassaHolm equation, Methods Appl. Anal., 12 (2005), pp. 191-219.
    • (2005) Methods Appl. Anal , vol.12 , pp. 191-219
    • BRESSAN, A.1    FONTE, M.2
  • 6
    • 0041760859 scopus 로고    scopus 로고
    • Characteristics and the initial value problem of a completely integrable shallow water equation
    • R. CAMASSA, Characteristics and the initial value problem of a completely integrable shallow water equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), pp. 115-139.
    • (2003) Discrete Contin. Dyn. Syst. Ser. B , vol.3 , pp. 115-139
    • CAMASSA, R.1
  • 7
    • 12044254491 scopus 로고
    • An integrable shallow water equation with peaked solitons
    • R. CAMASSA AND D. D. HOLM, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), pp. 1661-1664.
    • (1993) Phys. Rev. Lett , vol.71 , pp. 1661-1664
    • CAMASSA, R.1    HOLM, D.D.2
  • 8
    • 56349110516 scopus 로고
    • A new integrable shallow water equation
    • R. CAMASSA, D. D. HOLM, AND J. HYMAN, A new integrable shallow water equation, Adv. Appl. Mech, 31 (1994), pp. 1-33.
    • (1994) Adv. Appl. Mech , vol.31 , pp. 1-33
    • CAMASSA, R.1    HOLM, D.D.2    HYMAN, J.3
  • 9
    • 30444460126 scopus 로고    scopus 로고
    • On a completely integrable numerical scheme for a nonlinear shallow-water wave equation
    • R. CAMASSA, J. HUANG, AND L. LEE, On a completely integrable numerical scheme for a nonlinear shallow-water wave equation, J. Nonlinear Math. Phys., 12 (2005), pp. 146-162.
    • (2005) J. Nonlinear Math. Phys , vol.12 , pp. 146-162
    • CAMASSA, R.1    HUANG, J.2    LEE, L.3
  • 10
    • 85066350078 scopus 로고    scopus 로고
    • Integral and integrable algorithms for a nonlinear shallow-water wave equation
    • to appear
    • R. CAMASSA, J. HUANG, AND L. LEE, Integral and integrable algorithms for a nonlinear shallow-water wave equation, J. Comput. Phys., to appear.
    • J. Comput. Phys
    • CAMASSA, R.1    HUANG, J.2    LEE, L.3
  • 11
    • 33746133952 scopus 로고    scopus 로고
    • Global weak solutions to a generalized hyperelastic-rod wave equation
    • G. M. COCLITE, H. HOLDEN, AND K. H. KARLSEN, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), pp. 1044-1069.
    • (2005) SIAM J. Math. Anal , vol.37 , pp. 1044-1069
    • COCLITE, G.M.1    HOLDEN, H.2    KARLSEN, K.H.3
  • 13
  • 14
    • 85129402022 scopus 로고    scopus 로고
    • Global existence and blow-up for a shallow water equation
    • A. CONSTANTIN AND J. ESCHER, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26 (1998), pp. 303-328.
    • (1998) Ann. Sc. Norm. Super. Pisa Cl. Sci , vol.26 , Issue.4 , pp. 303-328
    • CONSTANTIN, A.1    ESCHER, J.2
  • 15
    • 0000468550 scopus 로고    scopus 로고
    • Global weak solutions for a shallow water equation
    • A. CONSTANTIN AND J. ESCHER, Global weak solutions for a shallow water equation, Indiana, Univ. Math. J., 47 (1998), pp. 1527-1545.
    • (1998) Indiana, Univ. Math. J , vol.47 , pp. 1527-1545
    • CONSTANTIN, A.1    ESCHER, J.2
  • 16
    • 0000985293 scopus 로고    scopus 로고
    • Wave breaking for nonlinear nonlocal shallow water equations
    • A. CONSTANTIN AND J. ESCHER, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), pp. 229-243.
    • (1998) Acta Math , vol.181 , pp. 229-243
    • CONSTANTIN, A.1    ESCHER, J.2
  • 17
    • 0042279206 scopus 로고    scopus 로고
    • On the geometric approach to the motion of inertial mechanical systems
    • A. CONSTANTIN AND B. KOLEV, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), pp. R51-R79.
    • (2002) J. Phys. A , vol.35
    • CONSTANTIN, A.1    KOLEV, B.2
  • 18
    • 0242350978 scopus 로고    scopus 로고
    • Geodesic flow on the diffeomorphism group of the circle
    • A. CONSTANTIN AND B. KOLEV, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), pp. 787-804.
    • (2003) Comment. Math. Helv , vol.78 , pp. 787-804
    • CONSTANTIN, A.1    KOLEV, B.2
  • 19
    • 0033415933 scopus 로고    scopus 로고
    • A shallow water equation on the circle
    • A. CONSTANTIN AND H. P. MCKEAN, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), pp. 949-982.
    • (1999) Comm. Pure Appl. Math , vol.52 , pp. 949-982
    • CONSTANTIN, A.1    MCKEAN, H.P.2
  • 20
    • 0034349440 scopus 로고    scopus 로고
    • Global weak solutions for a shallow water equation
    • A. CONSTANTIN AND L. MOLINET, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), pp. 45-61.
    • (2000) Comm. Math. Phys , vol.211 , pp. 45-61
    • CONSTANTIN, A.1    MOLINET, L.2
  • 21
    • 0031679566 scopus 로고    scopus 로고
    • Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod
    • H. H. DAI, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), pp. 193-207.
    • (1998) Acta Mech , vol.127 , pp. 193-207
    • DAI, H.H.1
  • 22
    • 0000830886 scopus 로고    scopus 로고
    • A few remarks on the Camassa-Holm equation
    • R. DANCHIN, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), pp. 953-988.
    • (2001) Differential Integral Equations , vol.14 , pp. 953-988
    • DANCHIN, R.1
  • 23
    • 0043159153 scopus 로고    scopus 로고
    • A note on well-posedness for Camassa-Holm equation
    • R. DANCHIN, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), pp. 429-444.
    • (2003) J. Differential Equations , vol.192 , pp. 429-444
    • DANCHIN, R.1
  • 24
    • 33750982275 scopus 로고    scopus 로고
    • Dynamics of Viscous Compressible Fluids
    • Oxford University Press, Oxford
    • E. FEIREISL, Dynamics of Viscous Compressible Fluids, Oxford Lecture Ser. Math. Appl. 26, Oxford University Press, Oxford, 2004.
    • (2004) Oxford Lecture Ser. Math. Appl , vol.26
    • FEIREISL, E.1
  • 25
    • 0035311815 scopus 로고    scopus 로고
    • Integrable vs. nonintegrable geodesic soliton behavior
    • O. B. FRINGES. AND D. D. HOLM, Integrable vs. nonintegrable geodesic soliton behavior, Phys. D, 150 (2001), pp. 237-263.
    • (2001) Phys. D , vol.150 , pp. 237-263
    • FRINGES, O.B.1    HOLM, D.D.2
  • 26
    • 49049150360 scopus 로고
    • Symplectic structures, their Bäcklund transformations and hereditary symmetries
    • B. FUCHSSTEINER, AND A. S. FOKAS, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), pp. 47-66.
    • (1981) Phys. D , vol.4 , pp. 47-66
    • FUCHSSTEINER, B.1    FOKAS, A.S.2
  • 27
    • 34447276776 scopus 로고    scopus 로고
    • Convergence of a finite difference scheme for the Camassa-Holm equation
    • H. HOLDEN AND X. RAYNAUD, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J. Numer. Anal., 44 (2006), pp. 1655-1680.
    • (2006) SIAM J. Numer. Anal , vol.44 , pp. 1655-1680
    • HOLDEN, H.1    RAYNAUD, X.2
  • 28
    • 33644554516 scopus 로고    scopus 로고
    • A convergent numerical scheme for the Camassa-Holm equation based on multipeakons
    • H. HOLDEN AND X. RAYNAUD, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), pp. 505-523.
    • (2006) Discrete Contin. Dyn. Syst , vol.14 , pp. 505-523
    • HOLDEN, H.1    RAYNAUD, X.2
  • 29
    • 55349120802 scopus 로고    scopus 로고
    • Global conservative solutions of the Camassa-Holm equation - A Lagrangian point of view
    • to appear
    • H. HOLDEN AND X. RAYNAUD, Global conservative solutions of the Camassa-Holm equation - A Lagrangian point of view, Comm. Partial Differential Equations, to appear.
    • Comm. Partial Differential Equations
    • HOLDEN, H.1    RAYNAUD, X.2
  • 30
    • 0001971504 scopus 로고    scopus 로고
    • The Euler-Poincaré equations and semidirect products with applications to continuum theories
    • D. D. HOLM, J. E. MARSDEN, AND T. S. RATIU, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), pp. 1-81.
    • (1998) Adv. Math , vol.137 , pp. 1-81
    • HOLM, D.D.1    MARSDEN, J.E.2    RATIU, T.S.3
  • 31
    • 4043101891 scopus 로고    scopus 로고
    • Wave structure and nonlinear balances in a family of evolutionary PDEs
    • 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), pp. 323-380.
    • (2003) SIAM J. Appl. Dyn. Syst , vol.2 , pp. 323-380
    • HOLM, D.D.1    STALEY, M.F.2
  • 32
    • 0037171407 scopus 로고    scopus 로고
    • Camassa-Holm, Korteweg-de Vries and related models for water waves
    • R. S. JOHNSON, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), pp. 63-82.
    • (2002) J. Fluid Mech , vol.455 , pp. 63-82
    • JOHNSON, R.S.1
  • 35
    • 0034688728 scopus 로고    scopus 로고
    • Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation
    • Y. A. LI AND P. J. OLVER, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), pp. 27-63.
    • (2000) J. Differential Equations , vol.162 , pp. 27-63
    • LI, Y.A.1    OLVER, P.J.2
  • 36
    • 0031989845 scopus 로고    scopus 로고
    • A shallow water equation as a geodesic flow on the Bott-Virasoro group
    • G. MLSIOLEK, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), pp. 203-208.
    • (1998) J. Geom. Phys , vol.24 , pp. 203-208
    • MLSIOLEK, G.1
  • 37
    • 0035479874 scopus 로고    scopus 로고
    • On the Cauchy problem for the Camassa-Holm equation
    • G. RODRIGUEZ-BLANCO, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), pp. 309-327.
    • (2001) Nonlinear Anal , vol.46 , pp. 309-327
    • RODRIGUEZ-BLANCO, G.1
  • 38
    • 0034394008 scopus 로고    scopus 로고
    • On the weak solutions to a shallow water equation
    • Z. XIN AND P. ZHANG, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), pp. 1411-1433.
    • (2000) Comm. Pure Appl. Math , vol.53 , pp. 1411-1433
    • XIN, Z.1    ZHANG, P.2
  • 39
    • 0036913863 scopus 로고    scopus 로고
    • On the uniqueness and large time behavior of the weak solutions to a shallow water equation
    • Z. XIN AND P. ZHANG, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2002), pp. 1815-1844.
    • (2002) Comm. Partial Differential Equations , vol.27 , pp. 1815-1844
    • XIN, Z.1    ZHANG, P.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.