On the accuracy of invariant numerical schemes

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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

Abstract
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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 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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