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Volume 19, Issue 3
Quadratic Invariants and Symplectic Structure of General Linear Methods

Ai-Guo Xiao, Shou-Fu Li & Min Yang

J. Comp. Math., 19 (2001), pp. 269-280.

Published online: 2001-06

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In this paper, we present some invariants and conservation laws of general linear methods applied to differential equation systems. We show that the quadratic invariants and symplecticity of the systems can be extended to general linear methods by a tensor product, and show that general linear methods with the matrix M=0 inherit in an extended sense the quadratic invariants possessed by the differential equation systems being integrated and preserve in an extended sense the symplectic structure of the phase space in the integration of Hamiltonian systems. These unify and extend existing relevant results on Runge-Kutta methods, linear multistep methods and one-leg methods. Finally, as special cases of general linear methods, we examine multistep Runge-Kutta methods, one-leg methods and linear two-step methods in detail.  

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@Article{JCM-19-269, author = {Xiao , Ai-GuoLi , Shou-Fu and Yang , Min}, title = {Quadratic Invariants and Symplectic Structure of General Linear Methods}, journal = {Journal of Computational Mathematics}, year = {2001}, volume = {19}, number = {3}, pages = {269--280}, abstract = {

In this paper, we present some invariants and conservation laws of general linear methods applied to differential equation systems. We show that the quadratic invariants and symplecticity of the systems can be extended to general linear methods by a tensor product, and show that general linear methods with the matrix M=0 inherit in an extended sense the quadratic invariants possessed by the differential equation systems being integrated and preserve in an extended sense the symplectic structure of the phase space in the integration of Hamiltonian systems. These unify and extend existing relevant results on Runge-Kutta methods, linear multistep methods and one-leg methods. Finally, as special cases of general linear methods, we examine multistep Runge-Kutta methods, one-leg methods and linear two-step methods in detail.  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8979.html} }
TY - JOUR T1 - Quadratic Invariants and Symplectic Structure of General Linear Methods AU - Xiao , Ai-Guo AU - Li , Shou-Fu AU - Yang , Min JO - Journal of Computational Mathematics VL - 3 SP - 269 EP - 280 PY - 2001 DA - 2001/06 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8979.html KW - Quadratic invariants, Symplecticity, General linear methods, Hamiltonian systems. AB -

In this paper, we present some invariants and conservation laws of general linear methods applied to differential equation systems. We show that the quadratic invariants and symplecticity of the systems can be extended to general linear methods by a tensor product, and show that general linear methods with the matrix M=0 inherit in an extended sense the quadratic invariants possessed by the differential equation systems being integrated and preserve in an extended sense the symplectic structure of the phase space in the integration of Hamiltonian systems. These unify and extend existing relevant results on Runge-Kutta methods, linear multistep methods and one-leg methods. Finally, as special cases of general linear methods, we examine multistep Runge-Kutta methods, one-leg methods and linear two-step methods in detail.  

Ai-Guo Xiao, Shou-Fu Li & Min Yang. (1970). Quadratic Invariants and Symplectic Structure of General Linear Methods. Journal of Computational Mathematics. 19 (3). 269-280. doi:
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