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The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform
On the accuracy of invariant numerical schemes
1. | LEPTIAB, Université de La Rochelle, Avenue Michel Crépeau, 17000 La Rochelle, France, France |
References:
[1] |
M. I. Bakirova, V. A. Doronitsyn and R. V. Kozlov, Symmetry-preserving differences schemes for some heat transfer equations, J. Phys. A: Math. Gen., 30 (1997), 8139-8155. |
[2] |
T. J. Bridges, Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147-190.
doi: doi:10.1017/S0305004196001429. |
[3] |
C. J. Budd and V. A. Dorodnitsyn, Symmetry adapted moving mesh schemes for the nonlinear Schrödinger equation, J. Phys., 34 (2001), 10387-10400. |
[4] |
C. J. Budd and M. Piggott, Geometric integration and its applications, in "Handbook of Numerical Analysis," North-Holland, (2000), 35-139. |
[5] |
C. J. Budd and G. J. Collins, Symmetry based numerical methods for partial differential equations, Numerical analysis, Notes Math., 380 (1998), 16-36. |
[6] |
M. P. Calvo and E. Hairer, Accurate long-term integration of dynamical systems, Journal of Applied Mathematics, 18 (1995), 95-105. |
[7] |
É. Cartan, "La Méthode du Repère Mobile, La Théorie des Groupes Continues," et Les Espaces Généralisés, Hermann, 5 1935, Exposés de Géométrie. |
[8] |
J-B. Chen, H. Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics, Applied Mathematics and Computation, 177 (2006), 226-234.
doi: doi:10.1016/j.amc.2005.11.002. |
[9] |
J-B. Chen and M. Z. Qin, Multisymplectic composition integrators of high orders, J. Comput. Math, 21 (2003), 647-656. |
[10] |
J-B. Chen, M. Z. Qin and Y-F Tang, Symplectic and multisymplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 43 (2002), 1095-1106.
doi: doi:10.1016/S0898-1221(02)80015-3. |
[11] |
J-B. Chen, Total variation in discrete multisymplectic field and multisymplectic-energy-momentum integrators, Letters in Mathematical Physics, 61 (2002), 63-73.
doi: doi:10.1023/A:1020269203008. |
[12] |
M. Chhay and A. Hamdouni, A new construction for invariant numerical schemes using moving frames, C. R. Acad. Sci. Mecanique, 338 (2010), 97-101. |
[13] |
G. Cicogna, A discussion on the different Notions of symmetry of differential equations, Proceedings of Institute of Mathematics of NAS of Ukraine, 50 (2004), 77-84. |
[14] |
A. S. Dawes, Invariant numerical methods, Int. Journ. for Numer. Meth. in Fluids, 56 (2008), 1185-1191.
doi: doi:10.1002/fld.1749. |
[15] |
V. A. Dorodnitsyn, Finite-difference models entirely inheriting continuous symmetry of original differential equations, International Journal of Modern Physics C, (Physics and Computers), 5 (1994), 723-734, |
[16] |
V. A. Dorodnitsyn, Some new invariant difference equations on evolutionary grids, IMACS World Congress of Computational and Applied Mathematics, 1 (1994), {Proceedings of 14-th}. |
[17] |
V. A. Dorodnitsyn, Continious symmetries of finite-difference evolution equations and grids, Proceedings of Workshop on Symmetries and Integrability of Difference Equations, CRM, 9 (1996), 103-112. |
[18] |
V. A. Dorodnitsyn, Group theoretical methods for finite difference modeling, Proceedings of the First World Congress of Nonlinear Analysts, (1996), 979-990. |
[19] |
M. Fels and P. J. Olver, Moving coframes I. a practical algorithm, Acta Appl. Math., 51 (1998), 161-213.
doi: doi:10.1023/A:1005878210297. |
[20] |
M. Fels and P. J. Olver, Moving coframes II. regularization and theoritical foundations, Acta Appl. Math., 55 (1999), 127-208.
doi: doi:10.1023/A:1006195823000. |
[21] |
Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Physics Letters A, 133 (1988), 134-139.
doi: doi:10.1016/0375-9601(88)90773-6. |
[22] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism, Mechanics, Analysis and Geometry: 200 Years after Lagrange, (1991), 203-235. |
[23] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations II: Space + Time decomposition, Diff. Geom. Appl., 1 (1991), 375-390.
doi: doi:10.1016/0926-2245(91)90014-Z. |
[24] |
E. Hairer, Variable time step integration with symplectic methods, Applied Numerical Mathematics: Transactions of IMACS, 25 (1997), 219-227.
doi: doi:10.1016/S0168-9274(97)00061-5. |
[25] |
E. Hairer and C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations," Springer-Verlag, 2002. |
[26] |
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations," Springer-Verlag - 2nd Ed., 1993. |
[27] |
E. Hoarau, C. David, P. Sagaut and T. H. Le, Lie group stability of finite difference schemes, Discrete and continuous dynamical systems, (2007), 1-10. |
[28] |
N. H. Ibragimov, "CRC Handbook of Lie Group Analysis of Differential Equations, Vol. I: Symmetries, Exact Solutions, and Conservation Laws," CRCC Press: Boca Raton, 1994. |
[29] |
C. Kane, J. E. Marsden and M. Ortiz, Symplectic energy-momentum integrators, J. Math. Phys., 40 (1999), 3353-3371.
doi: doi:10.1063/1.532892. |
[30] |
C. Kane, J. E. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, International Journal for Numerical Methods in Engineering, 49 (2000), 1295-1325.
doi: doi:10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W. |
[31] |
P. Kim, Invariantization of numerical schemes using moving frames, BIT Numerical Mathematics, 47 (2007), 525-546.
doi: doi:10.1007/s10543-007-0138-8. |
[32] |
P. Kim, Invariantization of the Crank Nicolson method for Burgers equation, Physica D: Nonlinear Phenomena, 237 (2008), 243-254.
doi: doi:10.1016/j.physd.2007.09.001. |
[33] |
A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators, Archive for Rational Mechanics and Analysis, 2 (2003), 85-146.
doi: doi:10.1007/s00205-002-0212-y. |
[34] |
A. Lew, J. E. Marsden, M. Ortiz and M. West, "Variational Time Integrators," International Journal for Numerical Methods in Engineering, 2003, |
[35] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 17, Springer-Verlag, Texts in Apllied Mathematics 2nd, 1999. |
[36] |
J. E. Marsden, G. W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Communication in Mathematical Physics, 199(1998), 351-395.
doi: http://dx.doi.org/10.1007/s002200050505. |
[37] |
J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575.
doi: doi:10.1017/S0305004198002953. |
[38] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), Cambridge University Press 357-515. |
[39] |
J. E. Marsden, S. Pekarsky, S. Shkoller and M. West, Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. Phys., 38 (2001), 253-284.
doi: doi:10.1016/S0393-0440(00)00066-8. |
[40] |
B. E. Moore and S. Reich, Multi-symplectic integration methods for Hamiltonian PDEs, Future Gener. Comput. Syst., 19 (2003), 395-402.
doi: doi:10.1016/S0167-739X(02)00166-8. |
[41] |
P. J. Olver, Moving frames, J. Symb. Comp., 3 (2003), 501-512.
doi: doi:10.1016/S0747-7171(03)00092-0. |
[42] |
P. J. Olver, The concise handbook of algebra Lie groups and differential equations, Kluwer Acad. Publ. 2002, Dordrecht, Netherlands, 92-97. |
[43] |
P. J. Olver., "Applpications of Lie Groups to Differential Equations," 2nd, Springer-Verlag., 1993. |
[44] |
G. R. W. Quispel and C. Dyt, Solving ODE's numerically while preserving symmetries, Hamiltonian structure, phase space volume or first integrals, Proceedings IMALS, 2 (1997), 601-607. |
[45] |
J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta Numerica, 1 (1992), 243-286.
doi: doi:10.1017/S0962492900002282. |
[46] |
J. C. Simo and N. Tarnow and K. K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Computer Methods in Applied Mechanics and Engineering, 10 (1992), 63-116.
doi: doi:10.1016/0045-7825(92)90115-Z. |
[47] |
Y. I. Shokin, "The Method of Differential Approximation," Springer-Verlag, 1983. |
[48] |
N. N. Yanenko and Yu. Shokin, Group classification of difference schemes for a system of one-dimensional equations of gaz dynamics, Amer. Math. Soc. Transl., 104 (1976), 259-265. |
show all references
References:
[1] |
M. I. Bakirova, V. A. Doronitsyn and R. V. Kozlov, Symmetry-preserving differences schemes for some heat transfer equations, J. Phys. A: Math. Gen., 30 (1997), 8139-8155. |
[2] |
T. J. Bridges, Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147-190.
doi: doi:10.1017/S0305004196001429. |
[3] |
C. J. Budd and V. A. Dorodnitsyn, Symmetry adapted moving mesh schemes for the nonlinear Schrödinger equation, J. Phys., 34 (2001), 10387-10400. |
[4] |
C. J. Budd and M. Piggott, Geometric integration and its applications, in "Handbook of Numerical Analysis," North-Holland, (2000), 35-139. |
[5] |
C. J. Budd and G. J. Collins, Symmetry based numerical methods for partial differential equations, Numerical analysis, Notes Math., 380 (1998), 16-36. |
[6] |
M. P. Calvo and E. Hairer, Accurate long-term integration of dynamical systems, Journal of Applied Mathematics, 18 (1995), 95-105. |
[7] |
É. Cartan, "La Méthode du Repère Mobile, La Théorie des Groupes Continues," et Les Espaces Généralisés, Hermann, 5 1935, Exposés de Géométrie. |
[8] |
J-B. Chen, H. Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics, Applied Mathematics and Computation, 177 (2006), 226-234.
doi: doi:10.1016/j.amc.2005.11.002. |
[9] |
J-B. Chen and M. Z. Qin, Multisymplectic composition integrators of high orders, J. Comput. Math, 21 (2003), 647-656. |
[10] |
J-B. Chen, M. Z. Qin and Y-F Tang, Symplectic and multisymplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 43 (2002), 1095-1106.
doi: doi:10.1016/S0898-1221(02)80015-3. |
[11] |
J-B. Chen, Total variation in discrete multisymplectic field and multisymplectic-energy-momentum integrators, Letters in Mathematical Physics, 61 (2002), 63-73.
doi: doi:10.1023/A:1020269203008. |
[12] |
M. Chhay and A. Hamdouni, A new construction for invariant numerical schemes using moving frames, C. R. Acad. Sci. Mecanique, 338 (2010), 97-101. |
[13] |
G. Cicogna, A discussion on the different Notions of symmetry of differential equations, Proceedings of Institute of Mathematics of NAS of Ukraine, 50 (2004), 77-84. |
[14] |
A. S. Dawes, Invariant numerical methods, Int. Journ. for Numer. Meth. in Fluids, 56 (2008), 1185-1191.
doi: doi:10.1002/fld.1749. |
[15] |
V. A. Dorodnitsyn, Finite-difference models entirely inheriting continuous symmetry of original differential equations, International Journal of Modern Physics C, (Physics and Computers), 5 (1994), 723-734, |
[16] |
V. A. Dorodnitsyn, Some new invariant difference equations on evolutionary grids, IMACS World Congress of Computational and Applied Mathematics, 1 (1994), {Proceedings of 14-th}. |
[17] |
V. A. Dorodnitsyn, Continious symmetries of finite-difference evolution equations and grids, Proceedings of Workshop on Symmetries and Integrability of Difference Equations, CRM, 9 (1996), 103-112. |
[18] |
V. A. Dorodnitsyn, Group theoretical methods for finite difference modeling, Proceedings of the First World Congress of Nonlinear Analysts, (1996), 979-990. |
[19] |
M. Fels and P. J. Olver, Moving coframes I. a practical algorithm, Acta Appl. Math., 51 (1998), 161-213.
doi: doi:10.1023/A:1005878210297. |
[20] |
M. Fels and P. J. Olver, Moving coframes II. regularization and theoritical foundations, Acta Appl. Math., 55 (1999), 127-208.
doi: doi:10.1023/A:1006195823000. |
[21] |
Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Physics Letters A, 133 (1988), 134-139.
doi: doi:10.1016/0375-9601(88)90773-6. |
[22] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism, Mechanics, Analysis and Geometry: 200 Years after Lagrange, (1991), 203-235. |
[23] |
M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations II: Space + Time decomposition, Diff. Geom. Appl., 1 (1991), 375-390.
doi: doi:10.1016/0926-2245(91)90014-Z. |
[24] |
E. Hairer, Variable time step integration with symplectic methods, Applied Numerical Mathematics: Transactions of IMACS, 25 (1997), 219-227.
doi: doi:10.1016/S0168-9274(97)00061-5. |
[25] |
E. Hairer and C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations," Springer-Verlag, 2002. |
[26] |
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations," Springer-Verlag - 2nd Ed., 1993. |
[27] |
E. Hoarau, C. David, P. Sagaut and T. H. Le, Lie group stability of finite difference schemes, Discrete and continuous dynamical systems, (2007), 1-10. |
[28] |
N. H. Ibragimov, "CRC Handbook of Lie Group Analysis of Differential Equations, Vol. I: Symmetries, Exact Solutions, and Conservation Laws," CRCC Press: Boca Raton, 1994. |
[29] |
C. Kane, J. E. Marsden and M. Ortiz, Symplectic energy-momentum integrators, J. Math. Phys., 40 (1999), 3353-3371.
doi: doi:10.1063/1.532892. |
[30] |
C. Kane, J. E. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, International Journal for Numerical Methods in Engineering, 49 (2000), 1295-1325.
doi: doi:10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W. |
[31] |
P. Kim, Invariantization of numerical schemes using moving frames, BIT Numerical Mathematics, 47 (2007), 525-546.
doi: doi:10.1007/s10543-007-0138-8. |
[32] |
P. Kim, Invariantization of the Crank Nicolson method for Burgers equation, Physica D: Nonlinear Phenomena, 237 (2008), 243-254.
doi: doi:10.1016/j.physd.2007.09.001. |
[33] |
A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators, Archive for Rational Mechanics and Analysis, 2 (2003), 85-146.
doi: doi:10.1007/s00205-002-0212-y. |
[34] |
A. Lew, J. E. Marsden, M. Ortiz and M. West, "Variational Time Integrators," International Journal for Numerical Methods in Engineering, 2003, |
[35] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," 17, Springer-Verlag, Texts in Apllied Mathematics 2nd, 1999. |
[36] |
J. E. Marsden, G. W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Communication in Mathematical Physics, 199(1998), 351-395.
doi: http://dx.doi.org/10.1007/s002200050505. |
[37] |
J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125 (1999), 553-575.
doi: doi:10.1017/S0305004198002953. |
[38] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), Cambridge University Press 357-515. |
[39] |
J. E. Marsden, S. Pekarsky, S. Shkoller and M. West, Variational methods, multisymplectic geometry and continuum mechanics, J. Geom. Phys., 38 (2001), 253-284.
doi: doi:10.1016/S0393-0440(00)00066-8. |
[40] |
B. E. Moore and S. Reich, Multi-symplectic integration methods for Hamiltonian PDEs, Future Gener. Comput. Syst., 19 (2003), 395-402.
doi: doi:10.1016/S0167-739X(02)00166-8. |
[41] |
P. J. Olver, Moving frames, J. Symb. Comp., 3 (2003), 501-512.
doi: doi:10.1016/S0747-7171(03)00092-0. |
[42] |
P. J. Olver, The concise handbook of algebra Lie groups and differential equations, Kluwer Acad. Publ. 2002, Dordrecht, Netherlands, 92-97. |
[43] |
P. J. Olver., "Applpications of Lie Groups to Differential Equations," 2nd, Springer-Verlag., 1993. |
[44] |
G. R. W. Quispel and C. Dyt, Solving ODE's numerically while preserving symmetries, Hamiltonian structure, phase space volume or first integrals, Proceedings IMALS, 2 (1997), 601-607. |
[45] |
J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta Numerica, 1 (1992), 243-286.
doi: doi:10.1017/S0962492900002282. |
[46] |
J. C. Simo and N. Tarnow and K. K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics, Computer Methods in Applied Mechanics and Engineering, 10 (1992), 63-116.
doi: doi:10.1016/0045-7825(92)90115-Z. |
[47] |
Y. I. Shokin, "The Method of Differential Approximation," Springer-Verlag, 1983. |
[48] |
N. N. Yanenko and Yu. Shokin, Group classification of difference schemes for a system of one-dimensional equations of gaz dynamics, Amer. Math. Soc. Transl., 104 (1976), 259-265. |
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