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

On the accuracy of invariant numerical schemes

Marx Chhay 1, and  Aziz Hamdouni 1,

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.
Keywords: order of accuracy, parametrized scheme, geometric integration., Invariant methods, Lie symmetry, moving frames.
Mathematics Subject Classification: Primary: 58J70, 35A35; Secondary: 65D3.
Citation: Marx Chhay, Aziz Hamdouni. On the accuracy of invariant numerical schemes. Communications on Pure & 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. Google Scholar

[2]

T. J. Bridges, Multi-symplectic structures and wave propagation,, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147. doi: doi:10.1017/S0305004196001429. Google Scholar

[3]

C. J. Budd and V. A. Dorodnitsyn, Symmetry adapted moving mesh schemes for the nonlinear Schrödinger equation,, J. Phys., (2001), 10387. Google Scholar

[4]

C. J. Budd and M. Piggott, Geometric integration and its applications,, in, (2000), 35. Google Scholar

[5]

C. J. Budd and G. J. Collins, Symmetry based numerical methods for partial differential equations,, Numerical analysis, 380 (1998), 16. Google Scholar

[6]

M. P. Calvo and E. Hairer, Accurate long-term integration of dynamical systems,, Journal of Applied Mathematics, 18 (1995), 95. Google Scholar

[7]

É. Cartan, "La Méthode du Repère Mobile, La Théorie des Groupes Continues,", et Les Espaces G\'en\'eralis\'es, 5 (1935). Google Scholar

[8]

J-B. Chen, H. Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics,, Applied Mathematics and Computation, (2006), 226. doi: doi:10.1016/j.amc.2005.11.002. Google Scholar

[9]

J-B. Chen and M. Z. Qin, Multisymplectic composition integrators of high orders,, J. Comput. Math, (2003), 647. Google Scholar

[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. doi: doi:10.1016/S0898-1221(02)80015-3. Google Scholar

[11]

J-B. Chen, Total variation in discrete multisymplectic field and multisymplectic-energy-momentum integrators,, Letters in Mathematical Physics, (2002), 63. doi: doi:10.1023/A:1020269203008. Google Scholar

[12]

M. Chhay and A. Hamdouni, A new construction for invariant numerical schemes using moving frames,, C. R. Acad. Sci. Mecanique, 338 (2010), 97. Google Scholar

[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. Google Scholar

[14]

A. S. Dawes, Invariant numerical methods,, Int. Journ. for Numer. Meth. in Fluids, 56 (2008), 1185. doi: doi:10.1002/fld.1749. Google Scholar

[15]

V. A. Dorodnitsyn, Finite-difference models entirely inheriting continuous symmetry of original differential equations,, International Journal of Modern Physics C, 5 (1994), 723. Google Scholar

[16]

V. A. Dorodnitsyn, Some new invariant difference equations on evolutionary grids,, IMACS World Congress of Computational and Applied Mathematics, (1994). Google Scholar

[17]

V. A. Dorodnitsyn, Continious symmetries of finite-difference evolution equations and grids,, Proceedings of Workshop on Symmetries and Integrability of Difference Equations, (1996), 103. Google Scholar

[18]

V. A. Dorodnitsyn, Group theoretical methods for finite difference modeling,, Proceedings of the First World Congress of Nonlinear Analysts, (1996), 979. Google Scholar

[19]

M. Fels and P. J. Olver, Moving coframes I. a practical algorithm,, Acta Appl. Math., 51 (1998), 161. doi: doi:10.1023/A:1005878210297. Google Scholar

[20]

M. Fels and P. J. Olver, Moving coframes II. regularization and theoritical foundations,, Acta Appl. Math., 55 (1999), 127. doi: doi:10.1023/A:1006195823000. Google Scholar

[21]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,, Physics Letters A, (1988), 134. doi: doi:10.1016/0375-9601(88)90773-6. Google Scholar

[22]

M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism,, Mechanics, (1991), 203. Google Scholar

[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. doi: doi:10.1016/0926-2245(91)90014-Z. Google Scholar

[24]

E. Hairer, Variable time step integration with symplectic methods,, Applied Numerical Mathematics: Transactions of IMACS, 25 (1997), 219. doi: doi:10.1016/S0168-9274(97)00061-5. Google Scholar

[25]

E. Hairer and C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations,", Springer-Verlag, (2002). Google Scholar

[26]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations,", Springer-Verlag - 2nd Ed., (1993). Google Scholar

[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. Google Scholar

[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). Google Scholar

[29]

C. Kane, J. E. Marsden and M. Ortiz, Symplectic energy-momentum integrators,, J. Math. Phys., (1999), 3353. doi: doi:10.1063/1.532892. Google Scholar

[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, (2000), 1295. doi: doi:10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W. Google Scholar

[31]

P. Kim, Invariantization of numerical schemes using moving frames,, BIT Numerical Mathematics, 47 (2007), 525. doi: doi:10.1007/s10543-007-0138-8. Google Scholar

[32]

P. Kim, Invariantization of the Crank Nicolson method for Burgers equation,, Physica D: Nonlinear Phenomena, 237 (2008), 243. doi: doi:10.1016/j.physd.2007.09.001. Google Scholar

[33]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators,, Archive for Rational Mechanics and Analysis, (2003), 85. doi: doi:10.1007/s00205-002-0212-y. Google Scholar

[34]

A. Lew, J. E. Marsden, M. Ortiz and M. West, "Variational Time Integrators,", International Journal for Numerical Methods in Engineering, (2003). Google Scholar

[35]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", \textbf{17}, 17 (1999). Google Scholar

[36]

J. E. Marsden, G. W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs,, Communication in Mathematical Physics, (1998), 351. doi: http://dx.doi.org/10.1007/s002200050505. Google Scholar

[37]

J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves,, Math. Proc. Camb. Phil. Soc., (1999), 553. doi: doi:10.1017/S0305004198002953. Google Scholar

[38]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, (2001), 357. Google Scholar

[39]

J. E. Marsden, S. Pekarsky, S. Shkoller and M. West, Variational methods, multisymplectic geometry and continuum mechanics,, J. Geom. Phys., (2001), 253. doi: doi:10.1016/S0393-0440(00)00066-8. Google Scholar

[40]

B. E. Moore and S. Reich, Multi-symplectic integration methods for Hamiltonian PDEs,, Future Gener. Comput. Syst., 19 (2003), 395. doi: doi:10.1016/S0167-739X(02)00166-8. Google Scholar

[41]

P. J. Olver, Moving frames,, J. Symb. Comp., 3 (2003), 501. doi: doi:10.1016/S0747-7171(03)00092-0. Google Scholar

[42]

P. J. Olver, The concise handbook of algebra Lie groups and differential equations,, Kluwer Acad. Publ. 2002, (2002), 92. Google Scholar

[43]

P. J. Olver., "Applpications of Lie Groups to Differential Equations,", 2nd, (1993). Google Scholar

[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. Google Scholar

[45]

J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview,, Acta Numerica, 1 (1992), 243. doi: doi:10.1017/S0962492900002282. Google Scholar

[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, (1992), 63. doi: doi:10.1016/0045-7825(92)90115-Z. Google Scholar

[47]

Y. I. Shokin, "The Method of Differential Approximation,", Springer-Verlag, (1983). Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

T. J. Bridges, Multi-symplectic structures and wave propagation,, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147. doi: doi:10.1017/S0305004196001429. Google Scholar

[3]

C. J. Budd and V. A. Dorodnitsyn, Symmetry adapted moving mesh schemes for the nonlinear Schrödinger equation,, J. Phys., (2001), 10387. Google Scholar

[4]

C. J. Budd and M. Piggott, Geometric integration and its applications,, in, (2000), 35. Google Scholar

[5]

C. J. Budd and G. J. Collins, Symmetry based numerical methods for partial differential equations,, Numerical analysis, 380 (1998), 16. Google Scholar

[6]

M. P. Calvo and E. Hairer, Accurate long-term integration of dynamical systems,, Journal of Applied Mathematics, 18 (1995), 95. Google Scholar

[7]

É. Cartan, "La Méthode du Repère Mobile, La Théorie des Groupes Continues,", et Les Espaces G\'en\'eralis\'es, 5 (1935). Google Scholar

[8]

J-B. Chen, H. Y. Guo and K. Wu, Discrete total variation calculus and Lee's discrete mechanics,, Applied Mathematics and Computation, (2006), 226. doi: doi:10.1016/j.amc.2005.11.002. Google Scholar

[9]

J-B. Chen and M. Z. Qin, Multisymplectic composition integrators of high orders,, J. Comput. Math, (2003), 647. Google Scholar

[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. doi: doi:10.1016/S0898-1221(02)80015-3. Google Scholar

[11]

J-B. Chen, Total variation in discrete multisymplectic field and multisymplectic-energy-momentum integrators,, Letters in Mathematical Physics, (2002), 63. doi: doi:10.1023/A:1020269203008. Google Scholar

[12]

M. Chhay and A. Hamdouni, A new construction for invariant numerical schemes using moving frames,, C. R. Acad. Sci. Mecanique, 338 (2010), 97. Google Scholar

[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. Google Scholar

[14]

A. S. Dawes, Invariant numerical methods,, Int. Journ. for Numer. Meth. in Fluids, 56 (2008), 1185. doi: doi:10.1002/fld.1749. Google Scholar

[15]

V. A. Dorodnitsyn, Finite-difference models entirely inheriting continuous symmetry of original differential equations,, International Journal of Modern Physics C, 5 (1994), 723. Google Scholar

[16]

V. A. Dorodnitsyn, Some new invariant difference equations on evolutionary grids,, IMACS World Congress of Computational and Applied Mathematics, (1994). Google Scholar

[17]

V. A. Dorodnitsyn, Continious symmetries of finite-difference evolution equations and grids,, Proceedings of Workshop on Symmetries and Integrability of Difference Equations, (1996), 103. Google Scholar

[18]

V. A. Dorodnitsyn, Group theoretical methods for finite difference modeling,, Proceedings of the First World Congress of Nonlinear Analysts, (1996), 979. Google Scholar

[19]

M. Fels and P. J. Olver, Moving coframes I. a practical algorithm,, Acta Appl. Math., 51 (1998), 161. doi: doi:10.1023/A:1005878210297. Google Scholar

[20]

M. Fels and P. J. Olver, Moving coframes II. regularization and theoritical foundations,, Acta Appl. Math., 55 (1999), 127. doi: doi:10.1023/A:1006195823000. Google Scholar

[21]

Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,, Physics Letters A, (1988), 134. doi: doi:10.1016/0375-9601(88)90773-6. Google Scholar

[22]

M. J. Gotay, A multisymplectic framework for classical field theory and the calculus of variations I: Covariant Hamiltonian formalism,, Mechanics, (1991), 203. Google Scholar

[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. doi: doi:10.1016/0926-2245(91)90014-Z. Google Scholar

[24]

E. Hairer, Variable time step integration with symplectic methods,, Applied Numerical Mathematics: Transactions of IMACS, 25 (1997), 219. doi: doi:10.1016/S0168-9274(97)00061-5. Google Scholar

[25]

E. Hairer and C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-preserving Algorithms for Ordinary Differential Equations,", Springer-Verlag, (2002). Google Scholar

[26]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations,", Springer-Verlag - 2nd Ed., (1993). Google Scholar

[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. Google Scholar

[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). Google Scholar

[29]

C. Kane, J. E. Marsden and M. Ortiz, Symplectic energy-momentum integrators,, J. Math. Phys., (1999), 3353. doi: doi:10.1063/1.532892. Google Scholar

[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, (2000), 1295. doi: doi:10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W. Google Scholar

[31]

P. Kim, Invariantization of numerical schemes using moving frames,, BIT Numerical Mathematics, 47 (2007), 525. doi: doi:10.1007/s10543-007-0138-8. Google Scholar

[32]

P. Kim, Invariantization of the Crank Nicolson method for Burgers equation,, Physica D: Nonlinear Phenomena, 237 (2008), 243. doi: doi:10.1016/j.physd.2007.09.001. Google Scholar

[33]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators,, Archive for Rational Mechanics and Analysis, (2003), 85. doi: doi:10.1007/s00205-002-0212-y. Google Scholar

[34]

A. Lew, J. E. Marsden, M. Ortiz and M. West, "Variational Time Integrators,", International Journal for Numerical Methods in Engineering, (2003). Google Scholar

[35]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", \textbf{17}, 17 (1999). Google Scholar

[36]

J. E. Marsden, G. W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs,, Communication in Mathematical Physics, (1998), 351. doi: http://dx.doi.org/10.1007/s002200050505. Google Scholar

[37]

J. E. Marsden and S. Shkoller, Multisymplectic geometry, covariant Hamiltonians and water waves,, Math. Proc. Camb. Phil. Soc., (1999), 553. doi: doi:10.1017/S0305004198002953. Google Scholar

[38]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, (2001), 357. Google Scholar

[39]

J. E. Marsden, S. Pekarsky, S. Shkoller and M. West, Variational methods, multisymplectic geometry and continuum mechanics,, J. Geom. Phys., (2001), 253. doi: doi:10.1016/S0393-0440(00)00066-8. Google Scholar

[40]

B. E. Moore and S. Reich, Multi-symplectic integration methods for Hamiltonian PDEs,, Future Gener. Comput. Syst., 19 (2003), 395. doi: doi:10.1016/S0167-739X(02)00166-8. Google Scholar

[41]

P. J. Olver, Moving frames,, J. Symb. Comp., 3 (2003), 501. doi: doi:10.1016/S0747-7171(03)00092-0. Google Scholar

[42]

P. J. Olver, The concise handbook of algebra Lie groups and differential equations,, Kluwer Acad. Publ. 2002, (2002), 92. Google Scholar

[43]

P. J. Olver., "Applpications of Lie Groups to Differential Equations,", 2nd, (1993). Google Scholar

[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. Google Scholar

[45]

J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview,, Acta Numerica, 1 (1992), 243. doi: doi:10.1017/S0962492900002282. Google Scholar

[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, (1992), 63. doi: doi:10.1016/0045-7825(92)90115-Z. Google Scholar

[47]

Y. I. Shokin, "The Method of Differential Approximation,", Springer-Verlag, (1983). Google Scholar

[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. Google Scholar

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