Semi-implicit integrators

For this discussion we consider a Hamiltonian system with Hamiltonian

$$H(x,p)=\frac{1}{2}p^{2}+U(x),$$

where we assume unit mass for simplicity, and $U(x)$ is a potential energy function. Then the family of symplectic integrators depending on $\alpha$ is defined by:

$$x_{n+1/2} =x_{n}+\frac{\Delta t}{2}p_{n},$$

$$F_{n+1/2} =-U^{\prime}(x_{n+1/2}+\alpha\Delta t^{2}F_{n+1/2}),$$

$$p_{n+1} =p_{n}+\Delta tF_{n+1/2},$$

$$x_{n+1} =x_{n+1/2}+\frac{\Delta t}{2}p_{n+1,}$$

where $\Delta t$ is the time step size. Note that for $\alpha \neq 0,$ this is an implicit method formulated as an implicit equation for the forces. The motivation for the use of implicit methods is to extend the stability range for MD. Nonlinear stability analysis of this family of integrators and experimental confirmation [79] have shown that nonlinear instabilities are eliminated by the LIM2 method ( $\alpha=1/2$) for simple molecular systems. These implicit methods have been proposed before, but discarded for MD, because they were too expensive. Even when the time step was extended significantly the speedup was small. For example, Zhang and Schlick [100] applied an implicit Euler method called LIN to the nucleic acid component deoxycytidine with timesteps of up to 1000 fs. However, computational performance is competitive only at very large time steps: a gain factor of 3-4 is obtained for runs with 1000 fs time steps. The main reason is the cost of solving the highly nonlinear system for the forces of MD.

Cheap implicit methods were proposed in [99]. The idea is to use mixed implicit-explicit methods that are implicit in only the fastest motions, and explicit on the rest. These ideas have been successfully applied in the solution of reaction-diffusion partial differential equations. In [99] the methods were tested in a simple nonlinear model problem, and the idea to create MTS mixed implicit-explicit methods is proposed but not implemented there. Our goal is to use a mixed implicit-explicit scheme with quick convergence on the fastest motions using the implicit part, and quick convergence on the slowest motions using the explicit part.