Time Integration Methods

Abstract

With a discretization in time like the Störmer-Verlet method from Section 3.1, the solution of the continuous Newton’s equations is computed only approximately at selected points along the time axis. There, approximations to the values at later points in time are computed from the values of the approximations at previous points in time in an incremental fashion. Now, we consider time integration methods in a more general framework. First, we discuss local and global error estimates. It will turn out that one has to be careful in the interpretation of results from integrations over long times with many time steps. We thus study the conservation properties of time integration schemes and introduce so-called symplectic integrators. Then, we give techniques to construct more general schemes. Finally, we discuss possibilities to speed up the time integration. To this end, we focus on three approaches:

• — Higher order schemes. Their errors show a better order of convergence with respect to the time step. Thus, for a given (small enough) time step, the use of a higher order scheme decreases the error in the simulation. Alternatively, also a larger time step can be selected to obtain a solution of the same accuracy but now with a smaller number of time steps.

• — Methods with multiple time steps. Especially for molecular problems, the different length scales for the bond, angle, torsion angle, Lennard-Jones, and Coulomb parts of the overall potential function imply different time scales in the problem. With an appropriate separation, these different time scales can be treated individually using time steps of different size. A typical representative of such methods is the impulse or r-Respa method [274, 275, 621, 623] presented in Section 6.3.

• — Freezing high frequency modes by enforcing additional constraints. If the bond and angle forces are kept fixed, a larger time step can be used without compromising the stability of the time integration scheme. This approach is presented in more detail in Section 6.4, for which the SHAKE and RATTLE methods [43, 534] give an example.