TY - GEN A1 - Deuflhard, Peter A1 - Reich, Sebastian T1 - 2nd International Symposium Algorithms for Macromolecular Modelling N2 - The report contains the program and the collected abstracts of the 2nd International Symposium ``Algorithms for Macromolecular Modelling''. T3 - ZIB-Report - TR-97-03 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5455 ER - TY - GEN A1 - Reich, Sebastian T1 - Backward Error Analysis for Numerical Integrators N2 - We consider backward error analysis of numerical approximations to ordinary differential equations, i.e., the numerical solution is formally interpreted as the exact solution of a modified differential equation. A simple recursive definition of the modified equation is stated. This recursion is used to give a new proof of the exponentially closeness of the numerical solutions and the solutions to an appropriate truncation of the modified equation. We also discuss qualitative properties of the modified equation and apply these results to the symplectic variable step-size integration of Hamiltonian systems, the conservation of adiabatic invariants, and numerical chaos associated to homoclinic orbits. T3 - ZIB-Report - SC-96-21 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2320 ER - TY - GEN A1 - Reich, Sebastian T1 - Smoothed Langevin dynamics of highly oscillatory systems N2 - In this paper we generalize a result by Rubin and Ungar on Hamiltonian systems containing a strong constraining potential to Langevin dynamics. Such highly oscillatory systems arise, for example, in the context of molecular dynamics. We derive constrained equations of motion for the slowly varying solution components. This includes in particular the derivation of a correcting force-term that stands for the coupling of the slow and fast degrees of motion. We will identify two limiting cases: (i) the correcting force becomes, over a finite interval of time, almost identical to the force term suggested by Rubin and Ungar (weak thermal coupling) and (ii) the correcting force can be approximated by the gradient of the Fixman potential as used in statistical mechanics (strong thermal coupling). The discussion will shed some light on the question which of the two correcting potentials is more appropriate under which circumstances for molecular dynamics. In Sec.~7, we also discuss smoothing in the context of constant temperature molecular dynamics. T3 - ZIB-Report - SC-96-04 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2157 ER - TY - GEN A1 - Reich, Sebastian T1 - Enhanced long-term simultation of Hamiltonian systems containing a strong non-convex potential N2 - Many physical systems exhibit rapid motion coupled to a slowly varying motion. Often the rapid motion is associated with a stiff contribution in the potential energy function. In this context, the situation typically considered in the literature is the one with a strictly convex potential. Under some technical assumptions, one can then show that the slow motion is reproduced by a properly constrained system. In this paper we are concerned with a different situation: Often different time-scales can be found because of many local minima and barrier crossing between these minima. We suggest here to replace the detailed motion in the minima and the local barrier crossings by a statistical model which is then coupled to the slow equations of motion over long periods of time. This leads to Langevin type equations of motion subject to an appropriate time transformation. T3 - ZIB-Report - SC-95-32 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1984 ER - TY - GEN A1 - Reich, Sebastian T1 - Modified potential energy functions for constrained molecular dynamics N2 - In molecular dynamics the highly oscillatory vibrations in the chemical bonds are often replaced by holonomic constraints that freeze the bond length/angle to its equilibrium value. In some cases this approach can be justified if the force constants of the bond vibrations are sufficiently large. However, for moderate values of the force constant, the constrained system might lead to a dynamical behavior that is too ``rigid'' compared to the flexible model. To compensate for this effect, the concept of soft constraints was introduced [Reich 1997, Reich 1995, Reich 1996]. However, its implementation is rather expensive. In this paper, we suggest an alternative approach that modifies the force field instead of the constraint functions. This leads to a more efficient method that avoids the resonance induced instabilities of multiple-time-stepping [Bishop 1996] and the above described effect of standard constrained dynamics. T3 - ZIB-Report - SC-97-49 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3189 ER - TY - GEN A1 - Ascher, Uri A1 - Reich, Sebastian T1 - On some difficulties in integrating highly oscillatory Hamiltonian systems N2 - The numerical integration of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or Hamiltonian partial differential equations, is a challenging task. Various methods have been suggested to overcome the step-size restrictions of explicit methods such as the Verlet method. Among these are multiple-time-stepping, constrained dynamics, and implicit methods. In this paper, we investigate the suitability of time-reversible, semi-implicit methods. Here semi-implicit means that only the highly oscillatory part is integrated by an implicit method such as the midpoint method or an energy-conserving variant of it. The hope is that such methods will allow one to use a step-size $k$ which is much larger than the period $\varepsilon$ of the fast oscillations. However, our results are discouraging. Even in the absence of resonance-type instabilities, we show that in general one must require that $k^2/\varepsilon$ be small enough. Otherwise the method might become unstable and/or it might lead to a wrong approximation of the slowly varying solution components. The latter situation might, in some cases, even require that $k/\varepsilon$ be small in order to avoid this danger. While certain (semi-implicit) energy conserving methods prove to be robust for some model problems, they may also yield deceptively-looking, wrong solutions for other simple model problems, in circumstances where the corresponding constrained dynamics formulation may not be easily derived and used. T3 - ZIB-Report - SC-97-50 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3193 ER - TY - GEN A1 - Ascher, Uri A1 - Reich, Sebastian T1 - The midpoint scheme and variants for Hamiltonian systems: advantages and pitfalls N2 - The (implicit) midpoint scheme, like higher order Gauss-collocation schemes, is algebraically stable and symplectic, and it preserves quadratic integral invariants. It may appear particularly suitable for the numerical solution of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or structural mechanics, because there is no stability restriction when it is applied to a simple harmonic oscillator. Although it is well-known that the midpoint scheme may also exhibit instabilities in various stiff situations, one might still hope for good results when resonance-type instabilities are avoided. In this paper we investigate the suitability of the midpoint scheme for highly oscillatory, frictionless mechanical systems, where the step-size $k$ is much larger than the system's small parameter $\varepsilon$, in case that the solution remains bounded as $\varepsilon \rightarrow 0$. We show that in general one must require that $k^2/\varepsilon$ be small enough, or else, even the errors in slowly varying quantities like the energy may grow undesirably (especially when fast and slow modes are tightly coupled) or, worse, the computation may yield misleading information. In some cases this may already happen when $k = O(\varepsilon )$. The same holds for higher order collocation at Gaussian points. The encountered restrictions on $k$ are still better than the corresponding ones for explicit schemes. T3 - ZIB-Report - SC-97-51 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3201 ER - TY - GEN A1 - Reich, Sebastian T1 - A Free Energy Approach to the Torsion Dynamics of Macromolecules N2 - Based on the concept of free energy, we give a Hamiltonian formulation for the torsion dynamics of macromolecules. The appropriate reaction coordinates for the free energy calculations are defined in terms of soft constraints as introduced in Brooks, B.R., Zhou, J., and Reich, S., Elastic molecular dynamics with flexible constraints, in preparation and Reich, S., Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems, Physica D, to appear, 1995. We consider a few simplifications that allow one to calculate the free energy analytically and to write the corresponding equations of motion as a constrained Hamiltonian system. We also discuss a possible stochastic embedding of the reduced dynamics by means of a generalized Langevin approach. T3 - ZIB-Report - SC-95-17 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1839 ER - TY - GEN A1 - Leimkuhler, Benedict A1 - Reich, Sebastian T1 - The Numerical Solution of Constrained Hamiltonian Systems. N2 - 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. T3 - ZIB-Report - SC-92-16 KW - differential algebraic equations KW - Hamiltonian system KW - canonical discretization schemes Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-868 ER - TY - GEN A1 - Reich, Sebastian T1 - Smoothed Dynamics of Highly Oscillatory Hamiltonian Systems N2 - We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step-size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method. T3 - ZIB-Report - SC-94-28 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1564 ER - TY - GEN A1 - Reich, Sebastian T1 - Preservation of Adiabatic Invariants under Symplectic Discretization N2 - Symplectic methods, like the Verlet method, are a standard tool for the long term integration of Hamiltonian systems as they arise, for example, in molecular dynamics. One of the reasons for the popularity of symplectic methods is the conservation of energy over very long periods of time up to small fluctuations that scale with the order of the method. In this paper, we discuss a qualitative feature of Hamiltonian systems with separated time scales that is also preserved under symplectic discretization. Specifically, highly oscillatory degrees of freedom often lead to almost preserved quantities (adiabatic invariants). Using recent results from backward error analysis and normal form theory, we show that a symplectic method, like the Verlet method, preserves those adiabatic invariants. We also discuss step-size restrictions necessary to maintain adiabatic invariants in practical computations. T3 - ZIB-Report - SC-97-52 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3210 ER - TY - GEN A1 - Barth, Eric A1 - Leimkuhler, Benedict A1 - Reich, Sebastian T1 - A Time-Reversible Variable-Stepsize Integrator for Constrained Dynamics N2 - This article considers the design and implementation of variable-timestep methods for simulating holonomically constrained mechanical systems. Symplectic variable stepsizes are briefly discussed, we then consider time-reparameterization techniques employing a time-reversible (symmetric) integration method to solve the equations of motion. We give several numerical examples, including a simulation of an elastic (inextensible, unshearable) rod undergoing large deformations and collisions with the sides of a bounding box. Numerical experiments indicate that adaptive stepping can significantly smooth the numerical energy and improve the overall efficiency of the simulation. T3 - ZIB-Report - SC-97-53 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3220 ER - TY - GEN A1 - Nettesheim, Peter A1 - Reich, Sebastian T1 - Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics N2 - The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. T3 - ZIB-Report - SC-97-56 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3250 ER - TY - THES A1 - Reich, Sebastian T1 - Dynamical Systems, Numerical Integration, and Exponentially Small Estimates N2 - Dynamical systems with two well-separated time-scales are investigated using normal form theory. Exponential estimates for the normal form truncation error are derived and applied to the numerical integration of differential equations (backward error analysis) and the reduction of highly oscillatory Hamiltonian systems (constrained dynamics and correcting potentials). The theoretical results are used to formulate new algorithms for the time integration of conservative Hamiltonian systems (projected multiple time stepping, soft constraints, rigid bodies, symplectic variable step-size methods). T3 - ZIB-Report - SC-98-19 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3622 ER - TY - GEN A1 - Holder, Thomas A1 - Leimkuhler, Benedict A1 - Reich, Sebastian T1 - Explicit Variable Step-Size and Time-Reversible Integration N2 - 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. T3 - ZIB-Report - SC-98-17 KW - reversible differential equations KW - time integration KW - adaptive methods Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3607 ER -