TY - JOUR A1 - Bornemann, Folkmar A. A1 - Nettesheim, Peter A1 - Schütte, Christof T1 - Quantum-classical molecular dynamics as an approximation to full quantum dynamics JF - J. Chem. Phys. Y1 - 1996 UR - http://publications.imp.fu-berlin.de/103/ U6 - https://doi.org/10.1063/1.471952 VL - 105 IS - 3 SP - 1074 EP - 1083 ER - TY - JOUR A1 - Nettesheim, Peter A1 - Bornemann, Folkmar A. A1 - Schmidt, Burkhard A1 - Schütte, Christof T1 - An Explicit and Symplectic Integrator for Quantum-Classical Molecular Dynamics JF - Chem. Phys. Lett. Y1 - 1996 UR - http://publications.imp.fu-berlin.de/101/ U6 - https://doi.org/10.1016/0009-2614(96)00471-X VL - 256 IS - 6 SP - 581 EP - 588 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - Adaptive Accuracy Control for Microcanonical Car-Parrinello Simulations N2 - The Car-Parrinello (CP) approach to ab initio molecular dynamics serves as an approximation to time-dependent Born-Oppenheimer (BO) calculations. It replaces the explicit minimization of the energy functional by a fictitious Newtonian dynamics and therefore introduces an artificial mass parameter $\mu$ which controls the electronic motion. A recent theoretical investigation shows that the CP-error, i.e., the deviation of the CP--solution from the BO-solution {\em decreases} like $\mu^{1/2}$ asymptotically. Since the computational effort {\em increases} like $\mu^{-1/2}$, the choice of $\mu$ has to find a compromise between efficiency and accuracy. The asymptotical result is used in this paper to construct an easily implemented algorithm which automatically controls $\mu$: the parameter $\mu$ is repeatedly adapted during the simulation by choosing $\mu$ as large as possible while pushing an error measure below a user-given tolerance. The performance and reliability of the algorithm is illustrated by a typical example. T3 - ZIB-Report - SC-96-20 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2311 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Krause, Rolf T1 - Classical and Cascadic Multigrid - A Methodical Comparison N2 - Using the full multigrid method {\em without} any coarse grid correction steps but with an a posteriori control of the number of smoothing iterations was shown by Bornemann and Deuflhard [1996] to be an optimal iteration method with respect to the energy norm. They named this new kind of multigrid iteration the {\em cascadic multigrid method}. However, numerical examples with {\em linear} finite elements raised serious doubts whether the cascadic multigrid method can be made optimal with respect to the {\em $L^2$-norm}. In this paper we prove that the cascadic multigrid method cannot be optimal for linear finite elements and show that the case might be different for higher order elements. We present a careful analysis of the two grid variant of the cascadic multigrid method providing a setting where one can understand the methodical difference between the cascadic multigrid method and the classical multigrid $V$-cycle almost immediately. As a rule of thumb we get that whenever the cascadic multigrid works the classical multigrid will work too but not vice versa. T3 - ZIB-Report - SC-96-25 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2368 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - A Mathematical Investigation of the Car-Parrinello Method N2 - The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the {\em quantitative} influence of the involved parameter $\mu$, the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order ${\rm O}(\mu^{1/2})$ --- provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain non-degeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general. Finally, we show how to use the gained quantitative insight for an automatic control of the unphysical ``fake'' kinetic energy of the method. T3 - ZIB-Report - SC-96-19 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2302 ER - TY - GEN A1 - Schütte, Christof A1 - Bornemann, Folkmar A. T1 - Homogenization Approach to Smoothed Molecular Dynamics N2 - {\footnotesize In classical Molecular Dynamics a molecular system is modelled by classical Hamiltonian equations of motion. The potential part of the corresponding energy function of the system includes contributions of several types of atomic interaction. Among these, some interactions represent the bond structure of the molecule. Particularly these interactions lead to extremely stiff potentials which force the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles) via increasing the stiffness of the strong part of the potential to infinity. However, the naive way of doing this via holonomic constraints mistakenly ignores the energy contribution of the fast oscillations. The paper presents a mathematically rigorous discussion of the limit situation of infinite stiffness. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given via a careful inspection of the limit energy of the fast oscillations. Unfortunately, the theory is valid only as long as the system does not run into certain resonances of the fast motions. Behind those resonances, there is no unique limit solution but a kind of choatic scenario for which the notion ``Takens chaos'' was coined. For demonstrating the relevance of this observation for MD, the theory is applied to a realistic, but still simple system: a single butan molecule. The appearance of ``Takens chaos'' in smoothed MD is illustrated and the consequences are discussed.} T3 - ZIB-Report - SC-96-31 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2410 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schemann, Martin T1 - Adaptive Rothe's Method for the 2D Wave Equation N2 - The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved {\em virtually} using an adaptive state-of-the-art integrator. The {\em actual} realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus {\em perturbing} the virtual function space method. The admissible size of that perturbation can be computed {\em a priori} and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle. T3 - ZIB-Report - SC-96-39 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2492 ER - TY - JOUR A1 - Bornemann, Folkmar A. A1 - Deuflhard, Peter T1 - The cascadic multigrid method for elliptic problems JF - Numer. Math. Y1 - 1996 VL - 75 SP - 135 EP - 152 ER - TY - CHAP A1 - Bornemann, Folkmar A. A1 - Deuflhard, Peter ED - Glowinski, J. R. ED - Widlund, Olof T1 - Cascadic Multigrid Methods T2 - Domain Decomposition Methods in Sciences and Engineering Y1 - 1996 SP - 205 EP - 212 PB - John Wiley & Sons Ltd ER -