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March  2011, 10(2): 761-783. doi: 10.3934/cpaa.2011.10.761

On the accuracy of invariant numerical schemes

1. 

LEPTIAB, Université de La Rochelle, Avenue Michel Crépeau, 17000 La Rochelle, France, France

Received  March 2010 Revised  May 2010 Published  December 2010

In this paper we present a method of construction of invariant numerical schemes for partial differential equations. The resulting schemes preserve the Lie-symmetry group of the continuous equation and they are at least as accurate as the original scheme. The improvement of the numerical properties thanks to the Lie-symmetry preservation is illustrated on the example of the Burgers equation.
Citation: Marx Chhay, Aziz Hamdouni. On the accuracy of invariant numerical schemes. Communications on Pure and Applied Analysis, 2011, 10 (2) : 761-783. doi: 10.3934/cpaa.2011.10.761
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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