DANGERS OF MULTIPLE TIME-STEP METHODS

In this work we demonstrate two potential dangers of multiple time step techniques for numerical integration of differential equations. This idea of using different time steps for different interactions was proposed in 1978 for molecular dynamics but undoubtedly has other useful applications. The use of multiple time stepping with the popular Verlet method was proposed in a 1991 paper. However, the method advocated in this paper does not retain the symplectic (or canonical) property of the Verlet method, which is an abstract property satisfied by the flow of any Hamiltonian system, of which molecular dynamics is an example. Recent work reported in the literature suggests that this property is important for the long-time integration of Hamiltonian dynamical systems. We perform experiments on linear and nonlinear problems comparing symplectic and nonsymplectic multiple time stepping extensions of the Verlet method. We observe that in the nonsymplectic case either instability or dissipation becomes evident after a long integration. However, our experiments also indicate that it is quite possible to obtain an artificial “resonance” for the symplectic method that is much worse than that for the nonsymplectic methods. © 1993 Academic Press, Inc.