메뉴 건너뛰기




Volumn 39, Issue 19, 2006, Pages 5287-5320

Numerical methods for Hamiltonian PDEs

Author keywords

[No Author keywords available]

Indexed keywords


EID: 33646271105     PISSN: 03054470     EISSN: 13616447     Source Type: Journal    
DOI: 10.1088/0305-4470/39/19/S02     Document Type: Article
Times cited : (206)

References (153)
  • 3
    • 33744482805 scopus 로고
    • Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow (Part I)
    • 10.1016/0021-9991(66)90015-5 0021-9991
    • Arakawa A 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow (Part I) J. Comput. Phys. 1 119-43
    • (1966) J. Comput. Phys. , vol.1 , Issue.1 , pp. 119-143
    • Arakawa, A.1
  • 5
    • 0033293798 scopus 로고    scopus 로고
    • The midpoint scheme and variants for Hamiltonian systems: Advantages and pitfalls
    • 10.1137/S1064827597316059 1064-8275
    • Ascher U and Reich S 1999 The midpoint scheme and variants for Hamiltonian systems: advantages and pitfalls SIAM J. Sci. Comput. 21 1045-65
    • (1999) SIAM J. Sci. Comput. , vol.21 , Issue.3 , pp. 1045-1065
    • Ascher, U.1    Reich, S.2
  • 6
    • 1042304391 scopus 로고    scopus 로고
    • Multisymplectic box schemes and the Korteweg-de Vries equation
    • 10.1016/j.apnum.2003.09.002 0168-9274
    • Ascher U M and McLachlan R I 2004 Multisymplectic box schemes and the Korteweg-de Vries equation Appl. Numer. Math. 48 255-69
    • (2004) Appl. Numer. Math. , vol.48 , Issue.3-4 , pp. 255-269
    • Ascher, U.M.1    McLachlan, R.I.2
  • 8
    • 21344491735 scopus 로고
    • On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms
    • 10.1007/BF02188219 0022-4715
    • Benetin G and Giorgilli A 1994 On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms J. Stat. Phys. 74 1117-43
    • (1994) J. Stat. Phys. , vol.74 , Issue.5-6 , pp. 1117-1143
    • Benetin, G.1    Giorgilli, A.2
  • 9
    • 0001046947 scopus 로고
    • Extension of the classical Cartan form
    • 10.1103/PhysRevD.29.599 0556-2821 D
    • Betounes D E 1984 Extension of the classical Cartan form Phys. Rev. D 29 599-606
    • (1984) Phys. Rev. , vol.29 , Issue.4 , pp. 599-606
    • Betounes, D.E.1
  • 10
    • 0001564321 scopus 로고
    • Dangers of multiple time step methods
    • 10.1006/jcph.1993.1220 0021-9991
    • Biesiadecki J J and Skeel R D 1993 Dangers of multiple time step methods J. Comput. Phys. 109 318-28
    • (1993) J. Comput. Phys. , vol.109 , Issue.2 , pp. 318-328
    • Biesiadecki, J.J.1    Skeel, R.D.2
  • 12
    • 77954848699 scopus 로고    scopus 로고
    • Generalized finite differences in computational electromagnetics
    • Bossavit A 2001 Generalized finite differences in computational electromagnetics Prog. Electromagn. Res., PIER 32 45-64
    • (2001) Prog. Electromagn. Res., PIER , vol.32 , pp. 45-64
    • Bossavit, A.1
  • 13
    • 0030695841 scopus 로고    scopus 로고
    • A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities
    • 10.1098/rspa.1997.0075 1364-5021 A
    • Bridges T J 1997 A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities Proc. R. Soc. A 453 1365-95
    • (1997) Proc. R. Soc. , vol.453 , Issue.1962 , pp. 1365-1395
    • Bridges, T.J.1
  • 14
    • 0042137401 scopus 로고    scopus 로고
    • Multi-symplectic structures and wave propagation
    • 10.1017/S0305004196001429 0305-0041
    • Bridges T J 1997 Multi-symplectic structures and wave propagation Math. Proc. Camb. Phil. Soc. 121 147-90
    • (1997) Math. Proc. Camb. Phil. Soc. , vol.121 , Issue.1 , pp. 147-190
    • Bridges, T.J.1
  • 15
    • 33749326404 scopus 로고    scopus 로고
    • Canonical multi-symplectic structure on the total exterior algebra bundle
    • 1364-5021 A
    • Bridges T J 2006 Canonical multi-symplectic structure on the total exterior algebra bundle Proc. R. Soc. A DOI:10.1098/rspa.2005.1629
    • (2006) Proc. R. Soc.
    • Bridges, T.J.1
  • 16
    • 33746421439 scopus 로고    scopus 로고
    • Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry
    • 10.1098/rspa.1999.0411 1364-5021 A
    • Bridges T J and Derks G 1999 Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry Proc. R. Soc. A 455 2427-69
    • (1999) Proc. R. Soc. , vol.455 , Issue.1987 , pp. 2427-2469
    • Bridges, T.J.1    Derks, G.2
  • 17
    • 0035582791 scopus 로고    scopus 로고
    • The symplectic Evans matrix, and the instability of solitary waves and fronts
    • 10.1007/s002050100123 0003-9527
    • Bridges T J and Derks G 2001 The symplectic Evans matrix, and the instability of solitary waves and fronts Arch. Ration. Mech. Anal. 156 1-87
    • (2001) Arch. Ration. Mech. Anal. , vol.156 , Issue.1 , pp. 1-87
    • Bridges, T.J.1    Derks, G.2
  • 18
    • 0036436270 scopus 로고    scopus 로고
    • Linear instability of solitary wave solutions of the Kawahara equation and its generalizations
    • 10.1137/S0036141099361494 0036-1410
    • Bridges T J and Derks G 2002 Linear instability of solitary wave solutions of the Kawahara equation and its generalizations SIAM J. Math. Anal. 33 1356-78
    • (2002) SIAM J. Math. Anal. , vol.33 , Issue.6 , pp. 1356-1378
    • Bridges, T.J.1    Derks, G.2
  • 19
    • 0037112236 scopus 로고    scopus 로고
    • Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework
    • 10.1016/S0167-2789(02)00655-3 0167-2789 D
    • Bridges T J, Derks G and Gottwald G 2002 Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework Physica D 172 190-216
    • (2002) Physica , vol.172 , Issue.1-4 , pp. 190-216
    • Bridges, T.J.1    Derks, G.2    Gottwald, G.3
  • 20
    • 33646272034 scopus 로고    scopus 로고
    • Covariant multi-symplectic discretizations on the total exterior algebra bundle
    • Bridges T J and Hydon P E 2005 Covariant multi-symplectic discretizations on the total exterior algebra bundle Technical Report (University of Surrey)
    • (2005) Technical Report
    • Bridges, T.J.1    Hydon, P.E.2
  • 21
    • 22244442343 scopus 로고    scopus 로고
    • Vorticity and symplecticity in Lagrangian fluid dynamics
    • 0305-4470 015
    • Bridges T J, Hydon P E and Reich S 2005 Vorticity and symplecticity in Lagrangian fluid dynamics J. Phys. A: Math. Gen. 38 1403-18
    • (2005) J. Phys. A: Math. Gen. , vol.38 , Issue.6 , pp. 1403-1418
    • Bridges, T.J.1    Hydon, P.E.2    Reich, S.3
  • 23
    • 0037832748 scopus 로고    scopus 로고
    • Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity
    • 10.1016/S0375-9601(01)00294-8 0375-9601 A
    • Bridges T J and Reich S 2001 Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity Phys. Lett. A 284 184-93
    • (2001) Phys. Lett. , vol.284 , Issue.4-5 , pp. 184-193
    • Bridges, T.J.1    Reich, S.2
  • 24
    • 0039152065 scopus 로고    scopus 로고
    • Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow-water equations
    • 10.1016/S0167-2789(01)00188-9 0167-2789 D
    • Bridges T J and Reich S 2001 Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow-water equations Physica D 152 491-504
    • (2001) Physica , vol.152-153 , Issue.1-2 , pp. 491-504
    • Bridges, T.J.1    Reich, S.2
  • 25
    • 0041030044 scopus 로고    scopus 로고
    • On the geometry of multisymplectic manifolds
    • 0263-6115 A
    • Cantrijn F, Ibort A and de Len M 1999 On the geometry of multisymplectic manifolds J. Aust. Math. Soc. A 66 303-30
    • (1999) J. Aust. Math. Soc. , vol.66 , pp. 303-330
    • Cantrijn, F.1    Ibort, A.2    De Len, M.3
  • 26
    • 0035311279 scopus 로고    scopus 로고
    • Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient
    • 10.1016/S0168-9274(00)00033-7 0168-9274
    • Castillo J E, Hyman J M, Shashkov M and Steinberg S 2001 Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient Appl. Numer. Math. 37 171-87
    • (2001) Appl. Numer. Math. , vol.37 , Issue.1-2 , pp. 171-187
    • Castillo, J.E.1    Hyman, J.M.2    Shashkov, M.3    Steinberg, S.4
  • 27
    • 0042315367 scopus 로고    scopus 로고
    • Total variation in discrete multisymplectic field theory and multisymplectic-energy-momentum integrators
    • 10.1023/A:1020269203008 0377-9017
    • Chen J-B 2002 Total variation in discrete multisymplectic field theory and multisymplectic-energy-momentum integrators Lett. Math. Phys. 61 63-73
    • (2002) Lett. Math. Phys. , vol.61 , Issue.1 , pp. 63-73
    • Chen, J.-B.1
  • 28
    • 0742333387 scopus 로고    scopus 로고
    • Multisymplectic geometry, local conservation laws and a multisymplectic integrator for the Zakharov-Kuznetsov equation
    • 10.1023/A:1023067332646 0377-9017
    • Chen J-B 2003 Multisymplectic geometry, local conservation laws and a multisymplectic integrator for the Zakharov-Kuznetsov equation Lett. Math. Phys. 63 115-24
    • (2003) Lett. Math. Phys. , vol.63 , Issue.2 , pp. 115-124
    • Chen, J.-B.1
  • 29
    • 18244380317 scopus 로고    scopus 로고
    • Variational formulation for multisymplectic Hamiltonian systems
    • 10.1007/s11005-005-1813-9 0377-9017
    • Chen J-B 2005 Variational formulation for multisymplectic Hamiltonian systems Lett. Math. Phys. 71 243-53
    • (2005) Lett. Math. Phys. , vol.71 , Issue.3 , pp. 243-253
    • Chen, J.-B.1
  • 30
    • 0037396154 scopus 로고    scopus 로고
    • Total variation in Hamiltonian formalism and symplectic-energy conservation
    • 10.1063/1.1559642 0022-2488
    • Chen J-B, Guo H Y and Wu K 2003 Total variation in Hamiltonian formalism and symplectic-energy conservation J. Math. Phys. 44 1688-702
    • (2003) J. Math. Phys. , vol.44 , Issue.4 , pp. 1688-1702
    • Chen, J.-B.1    Guo, H.Y.2    Wu, K.3
  • 31
    • 0005727359 scopus 로고    scopus 로고
    • Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation
    • Chen J-B and Qin M Z 2001 Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation Electron. Trans. Numer. Anal. 12 193-204
    • (2001) Electron. Trans. Numer. Anal. , vol.12 , pp. 193-204
    • Chen, J.-B.1    Qin, M.Z.2
  • 32
    • 0036644593 scopus 로고    scopus 로고
    • A multisymplectic variational integrator for the nonlinear Schrödinger equation
    • 10.1002/num.10021 0749-159X
    • Chen J-B and Qin M Z 2002 A multisymplectic variational integrator for the nonlinear Schrödinger equation Numer. Methods Partial Diff. Eqns 18 523-36
    • (2002) Numer. Methods Partial Diff. Eqns , vol.18 , Issue.4 , pp. 523-536
    • Chen, J.-B.1    Qin, M.Z.2
  • 33
    • 33646246487 scopus 로고    scopus 로고
    • Multisymplectic composition integrators of high order
    • 0254-9409
    • Chen J-B and Qin M Z 2003 Multisymplectic composition integrators of high order J. Comput. Math. 21 647-56
    • (2003) J. Comput. Math. , vol.21 , pp. 647-656
    • Chen, J.-B.1    Qin, M.Z.2
  • 34
    • 0036532037 scopus 로고    scopus 로고
    • Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation
    • 10.1016/S0898-1221(02)80015-3 0898-1221
    • Chen J-B, Qin M Z and Tang Y-F 2002 Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation Comput. Math. Appl. 43 1095-106
    • (2002) Comput. Math. Appl. , vol.43 , Issue.8-9 , pp. 1095-1106
    • Chen, J.-B.1    Qin, M.Z.2    Tang, Y.-F.3
  • 35
    • 8144219868 scopus 로고    scopus 로고
    • Adiabatic invariance and applications: From molecular dynamics to numerical weather prediction
    • 10.1023/B:BITN.0000046816.68632.49 0006-3835
    • Cotter C J and Reich S 2004 Adiabatic invariance and applications: from molecular dynamics to numerical weather prediction BIT 44 439-55
    • (2004) BIT , vol.44 , Issue.3 , pp. 439-455
    • Cotter, C.J.1    Reich, S.2
  • 36
    • 0005914572 scopus 로고    scopus 로고
    • Cotter C J and Reich S 2005 Semi-geostrophic particle motion and exponentially accurate normal forms Technical Report Universität Potsdam, Institut für Mathematik
    • (2005) Technical Report
    • Cotter, C.J.1    Reich, S.2
  • 38
    • 33646267274 scopus 로고    scopus 로고
    • Symplectic time-stepping for particle methods
    • Dixon M and Reich S 2004 Symplectic time-stepping for particle methods GAMM Mitt. 27 9-24
    • (2004) GAMM Mitt. , vol.27 , pp. 9-24
    • Dixon, M.1    Reich, S.2
  • 41
    • 3042529565 scopus 로고    scopus 로고
    • Nonsmooth Lagrangian mechanics and variational collision integrators
    • 10.1137/S1111111102406038 1111-1111
    • Fetecau R, Marsden J E, Ortiz M and West M 2003 Nonsmooth Lagrangian mechanics and variational collision integrators SIAM J. Appl. Dyn. Syst. 2 381-416
    • (2003) SIAM J. Appl. Dyn. Syst. , vol.2 , Issue.3 , pp. 381-416
    • Fetecau, R.1    Marsden, J.E.2    Ortiz, M.3    West, M.4
  • 43
    • 1042263828 scopus 로고    scopus 로고
    • Geometric space-time integration of ferromagnetic materials
    • 10.1016/j.apnum.2003.11.003 0168-9274
    • Frank J 2004 Geometric space-time integration of ferromagnetic materials Appl. Numer. Math. 48 307-22
    • (2004) Appl. Numer. Math. , vol.48 , Issue.3-4 , pp. 307-322
    • Frank, J.1
  • 44
    • 33646246609 scopus 로고    scopus 로고
    • Conservation of wave action under multisymplectic discretizations
    • Frank J 2006 Conservation of wave action under multisymplectic discretizations J. Phys. A: Math. Gen. this volume
    • (2006) J. Phys. A: Math. Gen.
    • Frank, J.1
  • 46
    • 0031139085 scopus 로고    scopus 로고
    • Geometric integrators for classical spin systems
    • 10.1006/jcph.1997.5672 0021-9991
    • Frank J, Huang W and Leimkuhler B 1997 Geometric integrators for classical spin systems J. Comput. Phys. 133 160-72
    • (1997) J. Comput. Phys. , vol.133 , Issue.1 , pp. 160-172
    • Frank, J.1    Huang, W.2    Leimkuhler, B.3
  • 47
    • 33646259795 scopus 로고    scopus 로고
    • Linear PDEs and numerical methods that preserve a multi-symplectic conservation law
    • Frank J, Moore B E and Reich S 2006 Linear PDEs and numerical methods that preserve a multi-symplectic conservation law SIAM J. Sci. Comput. at press
    • (2006) SIAM J. Sci. Comput.
    • Frank, J.1    Moore, B.E.2    Reich, S.3
  • 48
    • 4043081955 scopus 로고    scopus 로고
    • Conservation properties of smoothed particle hydrodynamics applied to the shallow-water equations
    • 10.1023/A:1023620100065 0006-3835
    • Frank J and Reich S 2003 Conservation properties of smoothed particle hydrodynamics applied to the shallow-water equations BIT 43 40-54
    • (2003) BIT , vol.43 , Issue.1 , pp. 41-54
    • Frank, J.1    Reich, S.2
  • 49
    • 27544460086 scopus 로고    scopus 로고
    • The Hamiltonian particle-mesh method for the spherical shallow water equations
    • 10.1002/asl.70 1530-261X
    • Frank J and Reich S 2004 The Hamiltonian particle-mesh method for the spherical shallow water equations Atmos. Sci. Lett. 5 89-95
    • (2004) Atmos. Sci. Lett. , vol.5 , Issue.5 , pp. 89-95
    • Frank, J.1    Reich, S.2
  • 50
    • 0005914572 scopus 로고    scopus 로고
    • Frank J and Reich S 2005 Energy-conserving semi-discretizations and spurious numerical reflections Technical Report Universität Potsdam and CWI Amsterdam
    • (2005) Technical Report
    • Frank, J.1    Reich, S.2
  • 51
    • 27544492167 scopus 로고    scopus 로고
    • Analysis of a regularized, time-staggered discretization method and its link to the semi-implicit method
    • 10.1002/asl.97 1530-261X
    • Frank J, Reich S, Staniforth A, White A and Nigel N 2005 Analysis of a regularized, time-staggered discretization method and its link to the semi-implicit method Atmos. Sci. Lett. 6 97-104
    • (2005) Atmos. Sci. Lett. , vol.6 , Issue.2 , pp. 97-104
    • Frank, J.1    Reich, S.2    Staniforth, A.3    White, A.4    Nigel, N.5
  • 52
    • 85183430449 scopus 로고
    • Smoothed particle hydrodynamics: Theory and application to non-spherical stars
    • 0035-8711
    • Gingold R A and Monaghan J J 1977 Smoothed particle hydrodynamics: theory and application to non-spherical stars Mon. Not. R. Astron. Soc. 181 375-89
    • (1977) Mon. Not. R. Astron. Soc. , vol.181 , Issue.2 , pp. 375-389
    • Gingold, R.A.1    Monaghan, J.J.2
  • 54
    • 0003014117 scopus 로고    scopus 로고
    • Time integration and discrete Hamiltonian systems
    • 10.1007/s003329900018 0938-8974
    • Gonzalez O 1996 Time integration and discrete Hamiltonian systems J. Nonlinear Sci. 6 449-67
    • (1996) J. Nonlinear Sci. , vol.6 , Issue.5 , pp. 449-467
    • Gonzalez, O.1
  • 55
    • 0030217555 scopus 로고    scopus 로고
    • On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry
    • 10.1016/0045-7825(96)01009-2 0045-7825
    • Gonzalez O and Simo J C 1996 On the stability of symplectic and energy-momentum algorithms for nonlinear Hamiltonian systems with symmetry Comput. Methods Appl. Mech. Eng. 134 197-222
    • (1996) Comput. Methods Appl. Mech. Eng. , vol.134 , Issue.3-4 , pp. 197-222
    • Gonzalez, O.1    Simo, J.C.2
  • 56
    • 0002956967 scopus 로고
    • A multisymplectic framework for classical field theory and the calculus of variations: I. Covariant Hamiltonian formalism
    • Gotay M J 1991 A multisymplectic framework for classical field theory and the calculus of variations: I. Covariant Hamiltonian formalism Mechanics, Analysis and Geometry, 200 Years After Lagrange ed N Francaviglia (Amsterdam: North-Holland) pp 203-35
    • (1991) Mechanics, Analysis and Geometry, 200 Years after Lagrange , pp. 203-235
    • Gotay, M.J.1
  • 58
    • 0345149792 scopus 로고    scopus 로고
    • On variations in discrete mechanics and field theory
    • 10.1063/1.1621058 0022-2488
    • Guo H-Y and Wu K 2003 On variations in discrete mechanics and field theory J. Math. Phys. 44 5978-6004
    • (2003) J. Math. Phys. , vol.44 , Issue.12 , pp. 5978-6004
    • Guo, H.-Y.1    Wu, K.2
  • 59
    • 0031505057 scopus 로고    scopus 로고
    • The life-span of backward error analysis for numerical integrators
    • 10.1007/s002110050271 0029-599X
    • Hairer E and Lubich Ch 1997 The life-span of backward error analysis for numerical integrators Numer. Math. 76 441-62
    • (1997) Numer. Math. , vol.76 , Issue.4 , pp. 441-462
    • Hairer, E.1    Ch, L.2
  • 60
    • 0042453539 scopus 로고    scopus 로고
    • Energy conservation by Störmer-type numerical integrators
    • Hairer E and Lubich Ch 2000 Energy conservation by Störmer-type numerical integrators Numerical Analysis 1999 ed G F Griffiths and G A Watson (Boca Raton, FL: CRC Press) pp 169-90
    • (2000) Numerical Analysis 1999 , pp. 169-190
    • Hairer, E.1    Ch, L.2
  • 62
    • 0000391648 scopus 로고
    • Theorie zur Errechnung des Wasserstandes und der Strömung in Randmeeren nebst Anwendungen
    • 0040-2826
    • Hansen W 1956 Theorie zur Errechnung des Wasserstandes und der Strömung in Randmeeren nebst Anwendungen Tellus 8 289-300
    • (1956) Tellus , vol.8 , pp. 289-300
    • Hansen, W.1
  • 63
    • 85095827382 scopus 로고    scopus 로고
    • Finite elements in computational electromagnetism
    • 10.1017/S0962492902000041 0962-4929
    • Hiptmair R 2002 Finite elements in computational electromagnetism Acta Numer. 11 237-339
    • (2002) Acta Numer. , vol.11 , Issue.0 , pp. 237-339
    • Hiptmair, R.1
  • 65
    • 0040951221 scopus 로고    scopus 로고
    • Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion
    • 10.1016/S0167-2789(99)00093-7 0167-2789 D
    • Holm D D 1999 Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion Physica D 133 215-69
    • (1999) Physica , vol.133 , Issue.1-4 , pp. 215-269
    • Holm, D.D.1
  • 66
    • 26944495302 scopus 로고    scopus 로고
    • Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations
    • 10.1016/j.jcp.2005.06.001 0021-9991
    • Hong J and Li C 2006 Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations J. Comput. Phys. 211 448-72
    • (2006) J. Comput. Phys. , vol.211 , Issue.2 , pp. 448-472
    • Hong, J.1    Li, C.2
  • 67
    • 16244399744 scopus 로고    scopus 로고
    • Hong J, Liu H and Sun G 2004 The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs Technical Report (Beijing: Chinese Academy of Sciences)
    • (2004) Technical Report
    • Hong, J.1    Liu, H.2    Sun, G.3
  • 68
    • 84867955067 scopus 로고    scopus 로고
    • A novel numerical approach to simulating nonlinear Schrödinger equations with varying coefficients
    • 10.1016/S0893-9659(03)00079-X 0893-9659
    • Hong J and Liu Y 2003 A novel numerical approach to simulating nonlinear Schrödinger equations with varying coefficients Appl. Math. Lett. 16 759-65
    • (2003) Appl. Math. Lett. , vol.16 , Issue.5 , pp. 759-765
    • Hong, J.1    Liu, Y.2
  • 69
    • 3543095217 scopus 로고    scopus 로고
    • Multisymplecticity of the centred box scheme for a class of Hamiltonian PDEs and an application to quasi-periodically solitary waves
    • 10.1016/j.mcm.2003.07.012 0895-7177
    • Hong J and Liu Y 2004 Multisymplecticity of the centred box scheme for a class of Hamiltonian PDEs and an application to quasi-periodically solitary waves Math. Comput. Modelling 39 1035-47
    • (2004) Math. Comput. Modelling , vol.39 , Issue.9-10 , pp. 1035-1047
    • Hong, J.1    Liu, Y.2
  • 70
    • 31244436460 scopus 로고    scopus 로고
    • Multisymplecticity of the centred box discretization for Hamiltonian PDEs with m ≥ 2 space dimensions
    • 10.1016/S0893-9659(02)00077-0 0893-9659
    • Hong J and Qin M Z 2002 Multisymplecticity of the centred box discretization for Hamiltonian PDEs with m ≥ 2 space dimensions Appl. Math. Lett. 15 1005-11
    • (2002) Appl. Math. Lett. , vol.15 , Issue.8 , pp. 1005-1011
    • Hong, J.1    Qin, M.Z.2
  • 71
    • 33646242111 scopus 로고    scopus 로고
    • A new multi-symplectic scheme for nonlinear 'good' Boussinesq equation
    • 0254-9409
    • Huang L-Y, Zeng W-P and Qin M Z 2003 A new multi-symplectic scheme for nonlinear 'good' Boussinesq equation J. Comput. Math. 21 703-14
    • (2003) J. Comput. Math. , vol.21 , pp. 703-714
    • Huang, L.-Y.1    Zeng, W.-P.2    Qin, M.Z.3
  • 72
    • 22244479327 scopus 로고    scopus 로고
    • Multisymplectic conservation laws for differential and differential-difference equations
    • 10.1098/rspa.2004.1444 1364-5021 A
    • Hydon P E 2005 Multisymplectic conservation laws for differential and differential-difference equations Proc. R. Soc. A 461 1627-37
    • (2005) Proc. R. Soc. , vol.461 , Issue.2058 , pp. 1627-1637
    • Hydon, P.E.1
  • 73
    • 0031071242 scopus 로고    scopus 로고
    • Natural discretizations for the divergence, gradient, and curl of logically rectangular grids
    • 10.1016/S0898-1221(97)00009-6 0898-1221
    • Hyman J M and Shashkov M 1997 Natural discretizations for the divergence, gradient, and curl of logically rectangular grids Comput. Math. Appl. 33 81-104
    • (1997) Comput. Math. Appl. , vol.33 , Issue.4 , pp. 81-104
    • Hyman, J.M.1    Shashkov, M.2
  • 74
    • 0035841060 scopus 로고    scopus 로고
    • Geometric integrators for the nonlinear Schrödinger equation
    • 10.1006/jcph.2001.6854 0021-9991
    • Islas A L, Karpeev D A and Schober C M 2001 Geometric integrators for the nonlinear Schrödinger equation J. Comput. Phys. 173 116-48
    • (2001) J. Comput. Phys. , vol.173 , Issue.1 , pp. 116-148
    • Islas, A.L.1    Karpeev, D.A.2    Schober, C.M.3
  • 75
    • 3242702916 scopus 로고    scopus 로고
    • On the preservation of phase space structure under multisymplectic discretization
    • 10.1016/j.jcp.2003.12.010 0021-9991
    • Islas A L and Schober C M 2004 On the preservation of phase space structure under multisymplectic discretization J. Comput. Phys. 197 585-609
    • (2004) J. Comput. Phys. , vol.197 , Issue.2 , pp. 585-609
    • Islas, A.L.1    Schober, C.M.2
  • 76
    • 19044372348 scopus 로고    scopus 로고
    • Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs
    • 10.1016/j.matcom.2005.01.006 0378-4754
    • Islas A L and Schober C M 2005 Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs Math. Comput. Simul. 69 290-303
    • (2005) Math. Comput. Simul. , vol.69 , Issue.3-4 , pp. 290-303
    • Islas, A.L.1    Schober, C.M.2
  • 77
    • 0002999318 scopus 로고
    • Derivation of the discrete conservation laws for a family of finite difference schemes
    • 10.1016/0096-3003(94)90137-6 0096-3003
    • Jiménez S 1994 Derivation of the discrete conservation laws for a family of finite difference schemes Appl. Math. Comput. 64 13
    • (1994) Appl. Math. Comput. , vol.64 , Issue.1 , pp. 13
    • Jiménez, S.1
  • 78
    • 0033243778 scopus 로고    scopus 로고
    • Symplectic-energy-momentum preserving variational integrators
    • 10.1063/1.532892 0022-2488
    • Kane C, Marsden J E and Ortiz M 1999 Symplectic-energy-momentum preserving variational integrators J. Math. Phys. 40 3353-71
    • (1999) J. Math. Phys. , vol.40 , Issue.7 , pp. 3353-3371
    • Kane, C.1    Marsden, J.E.2    Ortiz, M.3
  • 80
    • 0003034563 scopus 로고
    • The symplectic method for the computation of Hamiltonian equations
    • Keng F and Qin M Z 1987 The symplectic method for the computation of Hamiltonian equations Lecture Notes in Mathematics vol 1297 ed Y L Zhu and B Y Guo (Berlin: Springer) pp 1-37
    • (1987) Lecture Notes in Mathematics , vol.1297 , pp. 1-37
    • Keng, F.1    Qin, M.Z.2
  • 81
    • 0043124933 scopus 로고    scopus 로고
    • A variational approach to second-order multisymplectic field theories
    • 10.1016/S0393-0440(00)00012-7 0393-0440
    • Kournabaeva S and Shkoller S 2000 A variational approach to second-order multisymplectic field theories J. Geom. Phys. 35 333-66
    • (2000) J. Geom. Phys. , vol.35 , Issue.4 , pp. 333-366
    • Kournabaeva, S.1    Shkoller, S.2
  • 82
    • 0034165330 scopus 로고    scopus 로고
    • A frame bundle generalization of multisymplectic geometry
    • 10.1016/S0034-4877(00)89031-X 0034-4877
    • Lawson J K 2000 A frame bundle generalization of multisymplectic geometry Rep. Math. Phys. 45 183-205
    • (2000) Rep. Math. Phys. , vol.45 , Issue.2 , pp. 183-205
    • Lawson, J.K.1
  • 83
    • 10644245860 scopus 로고
    • Can time be a discrete dynamical variable?
    • 10.1016/0370-2693(83)90687-1 0370-2693 B
    • Lee T D 1982 Can time be a discrete dynamical variable? Phys. Lett. B 122 217-20
    • (1982) Phys. Lett. , vol.122 , Issue.3-4 , pp. 217-220
    • Lee, T.D.1
  • 84
    • 0001529171 scopus 로고
    • Difference equations and conservation laws
    • 10.1007/BF01011145 0022-4715
    • Lee T D 1987 Difference equations and conservation laws J. Stat. Phys. 46 843-60
    • (1987) J. Stat. Phys. , vol.46 , Issue.5-6 , pp. 843-860
    • Lee, T.D.1
  • 86
    • 0344511753 scopus 로고    scopus 로고
    • Variational integrators and time-dependent Lagrangian systems
    • 10.1016/S0034-4877(02)80017-9 0034-4877
    • De Len M and de Diego D M 2002 Variational integrators and time-dependent Lagrangian systems Rep. Math. Phys. 49 183-92
    • (2002) Rep. Math. Phys. , vol.49 , Issue.2-3 , pp. 183-192
    • De Len, M.1    De Diego, D.M.2
  • 88
  • 90
    • 0000301299 scopus 로고
    • A symplectic difference scheme for the infinite dimensional Hamiltonian system
    • 0377-0427
    • Li C W and Qin M Z 1988 A symplectic difference scheme for the infinite dimensional Hamiltonian system J. Comput. Appl. Math. 6 164-74
    • (1988) J. Comput. Appl. Math. , vol.6 , pp. 164-174
    • Li, C.W.1    Qin, M.Z.2
  • 91
    • 84965060858 scopus 로고
    • Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation
    • 10.1137/0732083 0036-1429
    • Li S and Vu-Quoc L 1995 Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation SIAM J. Numer. Anal. 32 1839-75
    • (1995) SIAM J. Numer. Anal. , vol.32 , Issue.6 , pp. 1839-1875
    • Li, S.1    Vu-Quoc, L.2
  • 92
    • 0035981746 scopus 로고    scopus 로고
    • Multisymplectic geometry and multisymplectic Preissman scheme for the KP equation
    • 10.1063/1.1487444 0022-2488
    • Liu T and Qin M Z 2002 Multisymplectic geometry and multisymplectic Preissman scheme for the KP equation J. Math. Phys. 43 4060-77
    • (2002) J. Math. Phys. , vol.43 , Issue.8 , pp. 4060-4077
    • Liu, T.1    Qin, M.Z.2
  • 93
    • 0001765873 scopus 로고
    • A numerical approach to the testing of the fission hypothesis
    • 10.1086/112164 0004-6256
    • Lucy L B 1977 A numerical approach to the testing of the fission hypothesis Astron. J. 82 1013-24
    • (1977) Astron. J. , vol.82 , Issue.12 , pp. 1013-1024
    • Lucy, L.B.1
  • 94
    • 0346946971 scopus 로고    scopus 로고
    • Verlet-I/r-RESPA is limited by nonlinear stability
    • 10.1137/S1064827501399833 1064-8275
    • Ma Q, Izaguirre J and Skeel R D 2003 Verlet-I/r-RESPA is limited by nonlinear stability SIAM J. Sci. Comput. 24 1951-73
    • (2003) SIAM J. Sci. Comput. , vol.24 , Issue.6 , pp. 1951-1973
    • Ma, Q.1    Izaguirre, J.2    Skeel, R.D.3
  • 95
    • 12844266060 scopus 로고    scopus 로고
    • A γ-convergence result for variational integrators of quadratic Lagrangians
    • 10.1051/cocv:2004025 1292-8119
    • Maggi F and Morini M 2004 A γ-convergence result for variational integrators of quadratic Lagrangians ESIAM: Control Optim. Calc. Var. 10 656-65
    • (2004) ESIAM: Control Optim. Calc. Var. , vol.10 , Issue.4 , pp. 656-665
    • Maggi, F.1    Morini, M.2
  • 96
    • 0007148604 scopus 로고
    • Resonance in the dynamics of chemical systems simulated by the implicit-midpoint scheme
    • 10.1016/0009-2614(95)00316-V 0009-2614
    • Mandziuk M and Schlick T 1995 Resonance in the dynamics of chemical systems simulated by the implicit-midpoint scheme Chem. Phys. Lett. 237 525-35
    • (1995) Chem. Phys. Lett. , vol.237 , Issue.5-6 , pp. 525-535
    • Mandziuk, M.1    Schlick, T.2
  • 98
    • 0032476963 scopus 로고    scopus 로고
    • Multi-symplectic geometry, variational integrators, and nonlinear PDEs
    • 10.1007/s002200050505 0010-3616
    • Marsden J E, Patrick G P and Shkoller S 1999 Multi-symplectic geometry, variational integrators, and nonlinear PDEs Commun. Math. Phys. 199 351-95
    • (1999) Commun. Math. Phys. , vol.199 , Issue.2 , pp. 351-395
    • Marsden, J.E.1    Patrick, G.P.2    Shkoller, S.3
  • 99
    • 0000140852 scopus 로고    scopus 로고
    • Variational methods, multisymplectic geometry and continuum mechanics
    • 10.1016/S0393-0440(00)00066-8 0393-0440
    • Marsden J E, Pekarsky S, Shkoller S and West M 2001 Variational methods, multisymplectic geometry and continuum mechanics J. Geom. Phys. 38 253-84
    • (2001) J. Geom. Phys. , vol.38 , Issue.3-4 , pp. 253-284
    • Marsden, J.E.1    Pekarsky, S.2    Shkoller, S.3    West, M.4
  • 101
    • 0013039699 scopus 로고    scopus 로고
    • Multi-symplectic geometry, covariant Hamiltonians and water waves
    • 10.1017/S0305004198002953 0305-0041
    • Marsden J E and Shkoller S 1999 Multi-symplectic geometry, covariant Hamiltonians and water waves Math. Proc. Camb. Phil. Soc. 125 553-75
    • (1999) Math. Proc. Camb. Phil. Soc. , vol.125 , Issue.3 , pp. 553-575
    • Marsden, J.E.1    Shkoller, S.2
  • 102
    • 84959193128 scopus 로고    scopus 로고
    • Discrete mechanics and variational integrators
    • 10.1017/S096249290100006X 0962-4929
    • Marsden J E and West M 2001 Discrete mechanics and variational integrators Acta Numer. 10 357-514
    • (2001) Acta Numer. , vol.10 , Issue.0 , pp. 357-514
    • Marsden, J.E.1    West, M.2
  • 103
    • 0001312358 scopus 로고
    • Explicit Lie-Poisson integration and the Euler equations
    • 10.1103/PhysRevLett.71.3043 0031-9007
    • McLachlan R I 1993 Explicit Lie-Poisson integration and the Euler equations Phys. Rev. Lett. 71 3043-6
    • (1993) Phys. Rev. Lett. , vol.71 , Issue.19 , pp. 3043-3046
    • McLachlan, R.I.1
  • 104
    • 34249766017 scopus 로고
    • Symplectic integration of Hamiltonian wave equations
    • 10.1007/BF01385708 0029-599X
    • McLachlan R I 1994 Symplectic integration of Hamiltonian wave equations Numer. Math. 66 465-92
    • (1994) Numer. Math. , vol.66 , Issue.1 , pp. 465-492
    • McLachlan, R.I.1
  • 105
    • 85095839561 scopus 로고    scopus 로고
    • Splitting methods
    • 10.1017/S0962492902000053 0962-4929
    • McLachlan R I and Quispel G R W 2002 Splitting methods Acta Numer. 11 341-434
    • (2002) Acta Numer. , vol.11 , Issue.0 , pp. 341-434
    • McLachlan, R.I.1    Quispel, G.R.W.2
  • 106
    • 0348229409 scopus 로고    scopus 로고
    • Geometric integration using discrete gradients
    • 10.1098/rsta.1999.0363 1364-503X A
    • McLachlan R I, Quispel G R W and Robidoux N 1999 Geometric integration using discrete gradients Phil. Trans. R. Soc. A 357 1021-46
    • (1999) Phil. Trans. R. Soc. , vol.357 , Issue.1754 , pp. 1021-1046
    • McLachlan, R.I.1    Quispel, G.R.W.2    Robidoux, N.3
  • 107
    • 33646239457 scopus 로고    scopus 로고
    • Moore B E 2003 Geometric Integration of multi-symplectic Hamiltonian PDEs PhD Thesis University of Surrey
    • (2003) PhD Thesis
    • Moore, B.E.1
  • 108
    • 0242339583 scopus 로고    scopus 로고
    • Backward error analysis for multi-symplectic integration methods
    • 10.1007/s00211-003-0458-9 0029-599X
    • Moore B E and Reich S 2003 Backward error analysis for multi-symplectic integration methods Numer. Math. 95 625-52
    • (2003) Numer. Math. , vol.95 , Issue.4 , pp. 625-652
    • Moore, B.E.1    Reich, S.2
  • 109
    • 0037400145 scopus 로고    scopus 로고
    • Multi-symplectic integrators for Hamiltonian PDEs
    • 10.1016/S0167-739X(02)00166-8 0167-739X
    • Moore B E and Reich S 2003 Multi-symplectic integrators for Hamiltonian PDEs Future Gener. Comput. Syst. 19 395-402
    • (2003) Future Gener. Comput. Syst. , vol.19 , Issue.3 , pp. 395-402
    • Moore, B.E.1    Reich, S.2
  • 110
    • 0032364496 scopus 로고    scopus 로고
    • Hamiltonian description of the ideal fluid
    • 10.1103/RevModPhys.70.467 0034-6861
    • Morrison P J 1998 Hamiltonian description of the ideal fluid Rev. Mod. Phys. 70 467-521
    • (1998) Rev. Mod. Phys. , vol.70 , Issue.2 , pp. 467-521
    • Morrison, P.J.1
  • 112
    • 84867951207 scopus 로고    scopus 로고
    • On the γ-convergence of discrete dynamics and variational integrators
    • 10.1007/s00332-004-0585-1 0938-8974
    • Müller S and Ortiz M 2004 On the γ-convergence of discrete dynamics and variational integrators J. Nonlinear Sci. 14 279-96
    • (2004) J. Nonlinear Sci. , vol.14 , Issue.3 , pp. 279-296
    • Müller, S.1    Ortiz, M.2
  • 113
    • 0021660205 scopus 로고
    • The separation of motions in systems with rapidly rotating phase
    • 10.1016/0021-8928(84)90078-9 0021-8928
    • Neishtadt A I 1984 The separation of motions in systems with rapidly rotating phase J. Appl. Math. Mech. 48 133-9
    • (1984) J. Appl. Math. Mech. , vol.48 , Issue.2 , pp. 133-139
    • Neishtadt, A.I.1
  • 114
    • 0000653129 scopus 로고
    • Generalized symplectic geometry on the frame bundle of a manifold
    • 0082-0717
    • Norris L K 1993 Generalized symplectic geometry on the frame bundle of a manifold Proc. Symp. Pure Math. 54 435-65
    • (1993) Proc. Symp. Pure Math. , vol.54 , pp. 435-465
    • Norris, L.K.1
  • 115
    • 2942627242 scopus 로고    scopus 로고
    • Approximate momentum conservation for spatial semidiscretizations of nonlinear wave equations
    • 10.1007/s00211-003-0488-3 0029-599X
    • Oliver M, West M and Wulff C 2004 Approximate momentum conservation for spatial semidiscretizations of nonlinear wave equations Numer. Math. 97 493-535
    • (2004) Numer. Math. , vol.97 , Issue.3 , pp. 493-535
    • Oliver, M.1    West, M.2    Wulff, C.3
  • 118
    • 0001444647 scopus 로고    scopus 로고
    • Backward error analysis for numerical integrators
    • 10.1137/S0036142997329797 0036-1429
    • Reich S 1999 Backward error analysis for numerical integrators SIAM J. Numer. Anal. 36 475-91
    • (1999) SIAM J. Numer. Anal. , vol.36 , Issue.5 , pp. 1549-491
    • Reich, S.1
  • 119
    • 0043187304 scopus 로고    scopus 로고
    • Finite volume methods for multi-symplectic PDEs
    • 10.1023/A:1022375915113 0006-3835
    • Reich S 2000 Finite volume methods for multi-symplectic PDEs BIT 40 559-82
    • (2000) BIT , vol.40 , Issue.3 , pp. 559-582
    • Reich, S.1
  • 120
    • 0034687898 scopus 로고    scopus 로고
    • Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations
    • 10.1006/jcph.1999.6372 0021-9991
    • Reich S 2000 Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations J. Comput. Phys. 157 473-99
    • (2000) J. Comput. Phys. , vol.157 , Issue.2 , pp. 473-499
    • Reich, S.1
  • 121
    • 0005914572 scopus 로고    scopus 로고
    • Reich S 2005 Linearly implicit time-stepping methods for numerical weather prediction Technical Report Universität Potsdam, Institut für Mathematik
    • (2005) Technical Report
    • Reich, S.1
  • 122
    • 0020515138 scopus 로고
    • Practical use of Hamilton's principle
    • 10.1017/S0022112083001706 0022-1120
    • Salmon R 1983 Practical use of Hamilton's principle J. Fluid Mech. 132 431-44
    • (1983) J. Fluid Mech. , vol.132 , pp. 431-444
    • Salmon, R.1
  • 124
    • 23444451442 scopus 로고    scopus 로고
    • A general method for conserving quantities related to potential vorticity in numerical models
    • 0951-7715 R01
    • Salmon R 2005 A general method for conserving quantities related to potential vorticity in numerical models Nonlinearity 18 R1-16
    • (2005) Nonlinearity , vol.18 , Issue.5 , pp. 1-16
    • Salmon, R.1
  • 126
    • 0000848080 scopus 로고
    • On the Legendre map in higher-order field theories
    • 0305-4470 016
    • Saunders D J and Crampin M 1990 On the Legendre map in higher-order field theories J. Phys. A: Math. Gen. 23 3169-82
    • (1990) J. Phys. A: Math. Gen. , vol.23 , Issue.14 , pp. 3169-3182
    • Saunders, D.J.1    Crampin, M.2
  • 127
    • 24744469688 scopus 로고    scopus 로고
    • An adaptive Rothe method for the wave equation
    • 10.1007/s007910050013 1432-9360
    • Schemann M and Bornemann F A 1998 An adaptive Rothe method for the wave equation Comput. Vis. Sci. 1 137-44
    • (1998) Comput. Vis. Sci. , vol.1 , Issue.3 , pp. 137-144
    • Schemann, M.1    Bornemann, F.A.2
  • 129
    • 0000024344 scopus 로고
    • Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics
    • 0065-2687
    • Shepherd T G 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics Adv. Geophys. 32 287-338
    • (1990) Adv. Geophys. , vol.32 , pp. 287-338
    • Shepherd, T.G.1
  • 130
    • 0026939543 scopus 로고
    • Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics
    • 10.1016/0045-7825(92)90115-Z 0045-7825
    • Simo J C, Tarnow N and Wong K K 1992 Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics Comput. Methods Appl. Mech. Eng. 100 63-116
    • (1992) Comput. Methods Appl. Mech. Eng. , vol.100 , Issue.1 , pp. 63-116
    • Simo, J.C.1    Tarnow, N.2    Wong, K.K.3
  • 131
    • 0042694406 scopus 로고    scopus 로고
    • Nonlinear stability analysis of area preserving integrators
    • 10.1137/S0036142998349527 0036-1429
    • Skeel R D and Srinivas K 2000 Nonlinear stability analysis of area preserving integrators SIAM J. Numer. Anal. 38 129-48
    • (2000) SIAM J. Numer. Anal. , vol.38 , Issue.1 , pp. 129-148
    • Skeel, R.D.1    Srinivas, K.2
  • 132
    • 33646243089 scopus 로고    scopus 로고
    • Staniforth A, Wood N and Reich S A time-staggered semi-Lagrangian discretization of the shallow-water equations Technical Report UK MetOffice and University of Potsdam submitted
    • Technical Report
    • Staniforth, A.1    Wood, N.2    Reich, S.3
  • 133
    • 0000765110 scopus 로고
    • Numerical solution of a nonlinear Klein-Gordon equation
    • 10.1016/0021-9991(78)90038-4 0021-9991
    • Strauss W and Vzquez L 1978 Numerical solution of a nonlinear Klein-Gordon equation J. Comput. Phys. 28 271
    • (1978) J. Comput. Phys. , vol.28 , Issue.2 , pp. 271
    • Strauss, W.1    Vzquez, L.2
  • 134
    • 33646230631 scopus 로고    scopus 로고
    • Su H and Qin M-Z 2005 Multisymplectic geometry method for Maxwell's equations and multisymplectic scheme Technical Report Preprint math-ph/0302058
    • (2005) Technical Report
    • Su, H.1    Qin, M.-Z.2
  • 135
    • 0142216144 scopus 로고    scopus 로고
    • Multi-symplectic methods for the coupled 1D nonlinear Schrödinger equation
    • 10.1016/S0010-4655(03)00285-6 0010-4655
    • Sun J-Q and Qin M Z 2003 Multi-symplectic methods for the coupled 1D nonlinear Schrödinger equation Comput. Phys. Commun. 155 221-35
    • (2003) Comput. Phys. Commun. , vol.155 , Issue.3 , pp. 221-235
    • Sun, J.-Q.1    Qin, M.Z.2
  • 136
    • 0042969142 scopus 로고    scopus 로고
    • Variational integrators for higher-order differential equations
    • 0254-9409
    • Sun Y and Qin M Z 2003 Variational integrators for higher-order differential equations J. Comput. Math. 21 135-44
    • (2003) J. Comput. Math. , vol.21 , Issue.2 , pp. 135-144
    • Sun, Y.1    Qin, M.Z.2
  • 137
    • 0034311551 scopus 로고    scopus 로고
    • Construction of multisymplectic schemes of any finite order for modified wave equations
    • 10.1063/1.1316062 0022-2488
    • Sun Y-J and Qin M Z 2000 Construction of multisymplectic schemes of any finite order for modified wave equations J. Math. Phys. 41 7854-68
    • (2000) J. Math. Phys. , vol.41 , Issue.11 , pp. 7854-7868
    • Sun, Y.-J.1    Qin, M.Z.2
  • 138
    • 3342956458 scopus 로고    scopus 로고
    • A multi-symplectic scheme for RLW equation
    • 0254-9409
    • Sun Y-J and Qin M Z 2004 A multi-symplectic scheme for RLW equation J. Comput. Math. 22 611-21
    • (2004) J. Comput. Math. , vol.22 , pp. 611-621
    • Sun, Y.-J.1    Qin, M.Z.2
  • 139
    • 84966238011 scopus 로고
    • Stability of finite-difference models containing two boundaries or interfaces
    • 0025-5718
    • Trefethen L N 1985 Stability of finite-difference models containing two boundaries or interfaces Math. Comput. 45 279-300
    • (1985) Math. Comput. , vol.45 , Issue.172 , pp. 279-300
    • Trefethen, L.N.1
  • 141
    • 0027643950 scopus 로고
    • Invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation
    • 10.1016/0045-7825(93)90073-7 0045-7825
    • Vu-Quoc L and Li S 1993 Invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation Comput. Methods Appl. Mech. Eng. 107 341-91
    • (1993) Comput. Methods Appl. Mech. Eng. , vol.107 , Issue.3 , pp. 341-391
    • Vu-Quoc, L.1    Li, S.2
  • 142
    • 20444496902 scopus 로고    scopus 로고
    • High-order multi-symplectic schemes for the nonlinear Klein-Gordon equation
    • 10.1016/j.amc.2004.07.007 0096-3003
    • Wang Y and Wang B 2005 High-order multi-symplectic schemes for the nonlinear Klein-Gordon equation Appl. Math. Comput. 166 608-32
    • (2005) Appl. Math. Comput. , vol.166 , Issue.3 , pp. 608-632
    • Wang, Y.1    Wang, B.2
  • 143
    • 0348170697 scopus 로고    scopus 로고
    • Numerical implementation of the multisymplectic Preissman scheme and its equivalent schemes
    • 10.1016/S0096-3003(03)00080-8 0096-3003
    • Wang Y, Wang B and Qin M Z 2004 Numerical implementation of the multisymplectic Preissman scheme and its equivalent schemes Appl. Math. Comput. 149 299-326
    • (2004) Appl. Math. Comput. , vol.149 , Issue.2 , pp. 299-326
    • Wang, Y.1    Wang, B.2    Qin, M.Z.3
  • 144
    • 20144377606 scopus 로고    scopus 로고
    • An artificial boundary condition for the multisymplectic Preissman scheme
    • 10.1143/JPSJ.73.1457 0031-9015
    • Wang Y, Wang B, Yang H and Wang Y 2004 An artificial boundary condition for the multisymplectic Preissman scheme J. Phys. Soc. Japan 73 1457-63
    • (2004) J. Phys. Soc. Japan , vol.73 , Issue.6 , pp. 1457-1463
    • Wang, Y.1    Wang, B.2    Yang, H.3    Wang, Y.4
  • 145
    • 0035609802 scopus 로고    scopus 로고
    • Multisymplectic geometry and multisymplectic scheme for the nonlinear Klein-Gordon equation
    • 10.1143/JPSJ.70.653 0031-9015
    • Wang Y S and Qin M Z 2001 Multisymplectic geometry and multisymplectic scheme for the nonlinear Klein-Gordon equation J. Phys. Soc. Japan 70 653-61
    • (2001) J. Phys. Soc. Japan , vol.70 , Issue.3 , pp. 653-661
    • Wang, Y.S.1    Qin, M.Z.2
  • 146
    • 0037195882 scopus 로고    scopus 로고
    • Multisymplectic schemes for the nonlinear Klein-Gordon equation
    • 10.1016/S0895-7177(02)00250-9 0895-7177
    • Wang Y S and Qin M Z 2002 Multisymplectic schemes for the nonlinear Klein-Gordon equation Math. Comput. Modelling 36 963-77
    • (2002) Math. Comput. Modelling , vol.36 , Issue.9-10 , pp. 963-977
    • Wang, Y.S.1    Qin, M.Z.2
  • 148
    • 11844306011 scopus 로고    scopus 로고
    • The maximum allowable time step for the shallow water α model and its relation to time-implicit differencing
    • 10.1175/MWR2816.1 0027-0644
    • Wingate B A 2004 The maximum allowable time step for the shallow water α model and its relation to time-implicit differencing Mon. Weather Rev. 132 2719-31
    • (2004) Mon. Weather Rev. , vol.132 , Issue.12 , pp. 2719-2731
    • Wingate, B.A.1
  • 149
    • 33646174704 scopus 로고    scopus 로고
    • An improved regularization for time-staggered discretization and its link to the semi-implicit method
    • 1530-261X
    • Wood N, Staniforth A and Reich S 2006 An improved regularization for time-staggered discretization and its link to the semi-implicit method Atmos. Sci. Lett. at press
    • (2006) Atmos. Sci. Lett.
    • Wood, N.1    Staniforth, A.2    Reich, S.3
  • 150
    • 84894021661 scopus 로고
    • Numerical solution of initial boundary value problems involving Maxwell's equations in isentropic coordinates
    • 10.1109/TAP.1966.1138693 0018-926X
    • Yee K S 1966 Numerical solution of initial boundary value problems involving Maxwell's equations in isentropic coordinates IEEE Trans. Antennas Propag. 14 302-7
    • (1966) IEEE Trans. Antennas Propag. , vol.14 , Issue.3 , pp. 302-307
    • Yee, K.S.1
  • 151
    • 13444303238 scopus 로고
    • Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure
    • 10.1016/0167-2789(91)90152-Y 0167-2789 D
    • Zeitlin V 1991 Finite-mode analogs of 2D ideal hydrodynamics: coadjoint orbits and local canonical structure Physica D 49 353-62
    • (1991) Physica , vol.49 , Issue.3 , pp. 353-362
    • Zeitlin, V.1
  • 152
    • 0034640067 scopus 로고    scopus 로고
    • Multisymplectic geometry and multisymplectic Preissman scheme for the KdV equation
    • 0305-4470 308
    • Zhao P-F and Qin M-Z 2000 Multisymplectic geometry and multisymplectic Preissman scheme for the KdV equation J. Phys. A: Math. Gen. 33 3613-26
    • (2000) J. Phys. A: Math. Gen. , vol.33 , Issue.18 , pp. 3613-3626
    • Zhao, P.-F.1    Qin, M.-Z.2
  • 153
    • 0043164835 scopus 로고    scopus 로고
    • Symplectic and multisymplectic schemes with the simple finite element method
    • 10.1016/S0375-9601(03)00955-1 0375-9601 A
    • Zhen L, Bai Y, Li Q and Wu K 2003 Symplectic and multisymplectic schemes with the simple finite element method Phys. Lett. A 314 443-55
    • (2003) Phys. Lett. , vol.314 , Issue.5-6 , pp. 443-455
    • Zhen, L.1    Bai, Y.2    Li, Q.3    Wu, K.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.