Cheap enhancement of symplectic integrators. (English) Zbl 0843.65050
Griffiths, D. F. (ed.) et al., Numerical analysis 1995. Proceedings of the 16th Dundee conference on numerical analysis, University of Dundee, UK, June 27-30, 1995. Harlow: Longman. Pitman Res. Notes Math. Ser. 344, 107-122 (1996).
We show how to increase the accuracy of symplectic integrations of Hamiltonian systems by cheaply processing the results given by the numerical method being used.
We discuss the modified equation approach to error analysis and its relation to processing, both in the case of general and Hamiltonian systems. Thereafter we consider only Hamiltonian problems. We assume that the (Hamiltonian) problem being solved is linear. Our interest is in symplectic methods of orders 2 and 4 that are time-reversible. We show how our pre- and postprocessors may be approximately computed at virtually no cost. We demonstrate that these processors apply also to nonlinear problems. To illustrate our general methodology, we look at a family of integrators that includes the standard Verlet method used in molecular dynamics and also a method of effective order 4.
Some numerical experiments for a highly nonlinear molecular dynamics problem are presented. It turns out that, at essentially no cost, processing very significantly enhances the accuracy of the numerical methods studied. Finally, we discuss how to evaluate the potential gradient and potential Hessian required by related methods when applied to molecular dynamics problems involving two-body interactions.
For the entire collection see [Zbl 0837.00017].
We discuss the modified equation approach to error analysis and its relation to processing, both in the case of general and Hamiltonian systems. Thereafter we consider only Hamiltonian problems. We assume that the (Hamiltonian) problem being solved is linear. Our interest is in symplectic methods of orders 2 and 4 that are time-reversible. We show how our pre- and postprocessors may be approximately computed at virtually no cost. We demonstrate that these processors apply also to nonlinear problems. To illustrate our general methodology, we look at a family of integrators that includes the standard Verlet method used in molecular dynamics and also a method of effective order 4.
Some numerical experiments for a highly nonlinear molecular dynamics problem are presented. It turns out that, at essentially no cost, processing very significantly enhances the accuracy of the numerical methods studied. Finally, we discuss how to evaluate the potential gradient and potential Hessian required by related methods when applied to molecular dynamics problems involving two-body interactions.
For the entire collection see [Zbl 0837.00017].
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 81V55 | Molecular physics |
