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  • 1
    Electronic Resource
    Electronic Resource
    Orthosymplectic integration of linear Hamiltonian systems (1997)
    Springer
    Numerische Mathematik 77 (1997), S. 269-282 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The authors describe a continuous, orthogonal and symplectic factorization procedure for integrating unstable linear Hamiltonian systems. The method relies on the development of an orthogonal, symplectic change of variables to block triangular Hamiltonian form. Integration is thus carried out within the class of linear Hamiltonian systems. Use of an appropriate timestepping strategy ensures that the symplectic pairing of eigenvalues is automatically preserved. For long-term integrations, as are needed in the calculation of Lyapunov exponents, the favorable qualitative properties of such a symplectic framework can be expected to yield improved estimates. The method is illustrated and compared with other techniques in numerical experiments on the Hénon-Heiles and spatially discretized Sine-Gordon equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050286
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Asymptotic Error Analysis of the Adaptive Verlet Method (1999)
    Springer
    BIT 39 (1999), S. 25-33 
    Details
    ISSN: 1572-9125
    Keywords: Adaptive Verlet method ; time-reversible variable stepsizes ; Hamiltonian systems ; Sundman time-transformations ; backward error analysis ; asymptotic expansions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Adaptive Verlet method and variants are time-reversible schemes for treating Hamiltonian systems subject to a Sundman time transformation. These methods have been observed in computer experiments to exhibit superior numerical stability when implemented in a counterintuitive “reciprocal” formulation. Here we give a theoretical explanation of this behavior by examining the leading terms of the modified equation (backward error analysis) and those of the asymptotic error expansion. With this insight we are able to improve the algorithm by simply correcting the starting stepsize.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022313123291
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  • 3
    Electronic Resource
    Electronic Resource
    Time-transformations for reversible variable stepsize integration (1998)
    Springer
    Numerical algorithms 19 (1998), S. 55-71 
    Details
    ISSN: 1572-9265
    Keywords: adaptive Verlet method ; time-reversible methods ; adaptive timestepping ; variable stepsize methods ; N-body mechanical problems ; scattering ; Verlet ; 70-08 ; 70F05 ; 70F07 ; 70F10 ; 70H05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The development of a Sundman-type time-transformation for reversible variable stepsize integration of few-body problems is discussed. While a time-transformation based on minimum particle separation is suitable if the collisions only occur pairwise and isolated in time, the control of stepsize is typically much more difficult for a three-body close approach. Nonetheless, we find that a suitable choice of time-transformation based on particle separation can work quite well for certain types of three-body simulations, particularly those involving very steep repulsive walls. We confirm these observations using numerical examples from Lennard-Jones scattering.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019127111709
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  • 4
    Electronic Resource
    Electronic Resource
    Algorithms for constrained molecular dynamics (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Journal of Computational Chemistry 16 (1995), S. 1192-1209 
    Details
    ISSN: 0192-8651
    Keywords: Computational Chemistry and Molecular Modeling ; Biochemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Computer Science
    Notes: In molecular dynamics simulations, the fastest components of the potential field impose severe restrictions on the stability and hence the speed of computational methods. One possibility for treating this problem is to replace the fastest components with algebraic length constraints. In this article the resulting systems of mixed differential and algebraic equations are studied. Commonly used discretization schemes for constrained Hamiltonian systems are discussed. The form of the nonlinear equations is examined in detail and used to give convergence results for the traditional nonlinear solution technique SHAKE iteration and for a modification based on successive overrelaxation (SOR). A simple adaptive algorithm for finding the optimal relaxation parameter is presented. Alternative direct methods using sparse matrix techniques are discussed. Numerical results are given for the new techniques, which have been implemented in the molecular modeling software package CHARMM and show as much as twofold improvement over SHAKE iteration. © 1995 John Wiley & Sons, Inc.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/jcc.540161003
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  • 5
    Book
    Book
    New algorithms for macromolecular simulation /; 49 (2006)
    Berlin [u.a.] :Springer,
    Details
    Title: New algorithms for macromolecular simulation /; 49
    Contributer: Leimkuhler, Benedict , Algorithms for Macromolecular Modelling Meeting 〈4, 2004, Leicester〉 , Algorithms for Macromolecular Modelling Meeting Leicester 2004.08.
    Publisher: Berlin [u.a.] :Springer,
    Year of publication: 2006
    Pages: XVI, 362 S. : , graph. Darst.
    Series Statement: Lecture notes in computational science and engineering 49
    ISBN: 3-540-25542-7
    Type of Medium: Book
    Language: English
    URL: http://deposit.ddb.de/cgi-bin/dokserv?id=2660985&prov=M&dokÏvar=1&dokÏext=htm
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  • 6
    Book
    Book
    Simulating Hamiltonian dynamics /; 14 (2005)
    New York :Cambridge University Press,
    Details
    Title: Simulating Hamiltonian dynamics /; 14
    Author: Leimkuhler, Benedict
    Contributer: Reich, Sebastian
    Edition: 1
    Publisher: New York :Cambridge University Press,
    Year of publication: 2005
    Pages: XVI, 379 S.
    Series Statement: Cambridge monographs on applied & computational mathematics 14
    ISBN: 0-521-77290-7
    Type of Medium: Book
    Language: English
    URL: Publisher description
    URL: Table of contents
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  • 7
    Book
    Book
    Computational molecular dynamics: challenges, methods, ideas ; proceedings of the 2nd international symposium on algorithms for macromolecular modelling, Berlin 21-24 May, 1997; 4 (1999)
    Berlin u.a. :Springer,
    Details
    Title: Computational molecular dynamics: challenges, methods, ideas ; proceedings of the 2nd international symposium on algorithms for macromolecular modelling, Berlin 21-24 May, 1997; 4
    Contributer: Deuflhard, Peter , Hermans, Jan , Leimkuhler, Benedict , Mark, Alan E. , Reich, Sebastian , Skeel, Robert D.
    Publisher: Berlin u.a. :Springer,
    Year of publication: 1999
    Pages: 494 S.
    Series Statement: Lecture notes in computational science and engineering 4
    Type of Medium: Book
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  • 8
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    Explicit Variable Step-Size and Time-Reversible Integration (1998)
    Details
    Publication Date: 2014-02-26
    Description: A variable step-size, semi-explicit variant of the explicit Störmer-Verlet method has been proposed for the time-reversible integration of Newton's equations of motion by Huang & Leimkuhler. Here we propose a fully explicit version of this approach applicable to explicit and symmetric integration methods for general time-reversible differential equations. As applications, we discuss the variable step-size, time-reversible, and fully explicit integration of rigid body motion and reversible Nos\'e-Hoover dynamics.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 9
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    Computational Molecular Dynamics: Challenges, Methods, Ideas (1999)
    Details
    Publication Date: 2021-01-21
    Language: English
    Type: proceedings , doc-type:Other
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  • 10
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    The Numerical Solution of Constrained Hamiltonian Systems. (1992)
    Details
    Publication Date: 2021-01-21
    Description: A Hamiltonian system subject to smooth constraints can typically be viewed as a Hamiltonian system on a manifold. Numerical computations, however, must be performed in $ R^n$. In this paper, canonical transformations from ``Hamiltonian differential--algebraic equations'' to ODEs in Euclidean space are considered. In \S2, canonical parameterizations or local charts are developed and it is shown how these can be computed in a practical framework. In \S3 we consider the construction of unconstrained Hamiltonian ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an integral invariant and whose flow reduces to the flow of the constrained system along the manifold. It is shown that certain of these unconstrained Hamiltonian systems force Lyapunov stability of the constraint--invariants, while others lead to an unstable invariant. In \S4, we compare various projection techniques which might be incorporated to better insure preservation of the constraint--invariants in the context of numerical discretization. Numerical experiments illustrate the degree to which the constraint and symplectic invariants are maintained under discretization of various formulations. {\bf Keywords:} differential--algebraic equations, Hamiltonian systems, canonical discretization schemes. {\bf AMS(MOS):} subject classification 65L05.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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