Time-stepping algorithms

Since many systems of interest in chemistry have intrinsic multiple time scales it is important to use integrators that deal efficiently with the multiple time scale problem. Since our multiple time step algorithm, the so-called reversible Reference System Propagator Algorithm (r-RESPA) [17, 24, 18, 26] is time reversible and symplectic, they are very useful in combination with HMC for constant temperature simulations of large protein systems. 

With the propagator written in this way, the equation of motion can be integrated by a multiple time step algorithm in Cartesian coordinates because At and At are different integration time steps (At > At when n> 1). As an example, the force terms are separated into two components... 

To carry out a numerical solution, a single strip of quadrilateral elements is placed along the x-axis, and all nodal temperatures are set Initially to zero. The right-hand boundary is then subjected to a step Increase in temperature (T(H,t) - 1.0), and we seek to compute the transient temperature variation T(x,t). The flow code accomplishes this by means of an unconditionally stable time-stepping algorithm derived from "theta" finite differences a solution of ten time steps required 22 seconds on a PC/AT-compatible microcomputer operating at 6 MHz. 

Muradoglu, M. and S. B. Pope (2002). Local time stepping algorithm for solving PDF turbulence model equations. AIAA Journal 40, 1755-1763. 

TIME Time interval for which moles of reaction are calculated in rate programs, automatically set in the time-step algorithm of the numerical integration method... 

The code can treat transient problems by means a two-point "theta-method" time-stepping algorithm. The dynamic algorithm is also useful in nonlinear problems, in which the final fluid state may be approached dynamically from an estimated initial state. 

M. Tuckermann, B.J. Berne and G.J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys., 97 (1992) 1990 P. Procacci and B.J. Berne, Computer simulation of solid C60 using multiple time-step algorithms, J. Chem. Phys., 101 (1994) 2421. 

C. J. Mundy, J. I. Siepmann, and M. L. Klein, J. Chetn. Phys., 102, 3376 (1995). Calculation of the Shear Viscosity of Decane Using a Reversible Multiple Time-Step Algorithm. 

To generate the trajectories that result from stochastic equations of motion (14.39) and (14.40) one needs to be able to properly address the stochastic input. For Eqs (14.39) and (14.40) we have to move the particle Linder the influence of the potential T(.v), the friction force—yvm and a time-dependent random force R(t). The latter is obtained by generating a Gaussian random variable at each time step. Algorithms for generating realizations of such variables are available in the applied mathematics or numerical methods hterature. The needed input for these algorithms are the two moments, (2J) and In our case (7 ) = 0, and (cf. Eq. (8.19)) = liiiyk/jT/At. where Ai is the time interval... 

This system of coupled equations must be solved by integrating forward in time starting from the initial conditions mk iO). Using an explicit time-stepping algorithm, Eq. (8.72) leads to... 

The combination of the multiple time step algorithm and PME[27] makes the simulation of large size biomolecular systems such as membrane proteins extremely efficient and affordable even for long time spans. Furthermore, it does not involve any uncontrolled approximation and is entirely consistent with periodic boundary conditions. 

Pearce L L and S C Harvey 1993. Langevin Dynamics Studies of Unsaturated Phospholipids in a Membrane Environment. Biophysical Journal 65 1084-1092 Procacci P and B Berne 1994 Computer Simulation of Solid C o Using Multiple Time-step Algorithms Journal of Chemical Physics 101-2421-2431. 

The computational requirements of the time stepping algorithm are concentrated in the solution of linear systems with the matrices and. For large-scale and accurate models, the dimension of these matrices will be very large. Therefore, it will be advantageous to solve them iteratively. Then we have to answer the following questions... 

Leimkuhler, B., Margul, D., Tuckerman, M. Stochastic, resonance-free multiple time-step algorithm for molecular dynamics with very large time steps. Mol. Phys. Ill, 3579-3594 (2013). doi 10.1080/00268976.2013.844369... 

The hybrid algorithm we use in the actual case is a combination of two sub-models Conventional SIMPLE approach together with a, k — e model and elliptic velocity-composition joint PDF scheme [6]. The sub-models interact as follows The CFD model supplies mean velocity fields, V(p and arrays of turbulent kinetic energy and dissipation rates as input for the PDF part. Having obtained these quantities as input, the fractional time step algorithm provides scalar composition and density as final output. The averaged density-field is finally handed back to the CFD sub-model. 

All these considerations lead to the need of a new B-bar operator if the fundamental conservation laws of energy and momentum are to be preserved. The new operator needs to account not only for the discrete finite element interpolations in space, but also the discrete structure in time of the EDMC time-stepping algorithms, as presented in this paper. 

Armero, F. 2006. Energy-dissipative momentum-conserving time-stepping algorithms for finite strain multiplicative plasticity , Comp. Meth. Appl. Mech. Eng., 195 4862 889. 

Armero, F. Sc Romero, I. 2001. On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II Second order methods, Computer Methods in Applied Mechanics and Engineering, 190, 6783-6824. 

IV. LIOUVILLE FORMULATION OF EQUATIONS OF MOTION—MULTIPLE TIME STEP ALGORITHMS... 

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