Predictor corrector

All these observations tend to favour the Verlet algoritlnn in one fonn or another, and we look closely at this in the following sections. For historical reasons only, we mention the more general class of predictor-corrector methods which have been optimized for classical mechanics simulations, [40, 4T] further details are available elsewhere [7, 42, 43]. 

Thus we find that the choice of quaternion variables introduces barriers to efficient symplectic-reversible discretization, typically forcing us to use some off-the-shelf explicit numerical integrator for general systems such as a Runge-Kutta or predictor-corrector method. 

There are many variants of the predictor-corrector theme of these, we will only mention the algorithm used by Rahman in the first molecular dynamics simulations with continuous potentials [Rahman 1964]. In this method, the first step is to predict new positions as follows ... 

This series expansion is truncated at a specified order and is probably most easily implemei ted within a predictor-corrector type of algorithm, where the higher-order terms are ahead computed. This method has been applied to relatively simple systems such as molecuh fluids [Streett et al. 1978] and alkane chain liquids [Swindoll and Haile 1984]. 

Our discussion so far has considered the use of SHAKE with the Verlet algorithm Versions have also been derived for other integration schemes, such as the leap-froj algorithm, the predictor-corrector methods and the velocity Verlet algorithm. In the cast of the velocity Verlet algorithm, the method has been named RATTLE [Anderson 1983]... 

Repeat steps 2 through 6 with a corrector step for the same time increment. Repeat again for any further predictor and/or predictor-corrector steps that may be advisable. Distefano (ibid.) discusses and compares a number of suitable explicit methods. 

If the stress is at the primary time step loeation and the veloeities are at the middle of the time step, then the resulting finite-difference equation is second-order accurate in space and time for uniform time steps and elements. If all quantities are at the primary time step, then a more complicated predictor-corrector procedure must be used to achieve second-order accuracy. A typical predictor-corrector scheme predicts the stresses at the middle of the time step and uses them to calculate the divergence of the stress tensor. 

MD runs for polymers typically exceed the stability Umits of a micro-canonical simulation, so using the fluctuation-dissipation theorem one can define a canonical ensemble and stabilize the runs. For the noise term one can use equally distributed random numbers which have the mean value and the second moment required by Eq. (13). In most cases the equations of motion are then solved using a third- or fifth-order predictor-corrector or Verlet s algorithms. 

One way to do this is afforded by the predictor-corrector method. We ignore terms higher than those shown explicitly, and calculate the predicted terms starting with bP(t). However, this procedure will not give the correct trajectory because we have not included the force law. This is done at the corrector step. We calculate from the new position rP the force at time t + St and hence the correct acceleration a (t -f 5t). This can be compared with the predicted acceleration aP(f -I- St) to estimate the size of the error in the prediction step... 

These values are now better approximations to the true position, velocity and so on, hence the generic term predictor-corrector for the solution of such differential equations. Values of the constants cq through C3 are available in the literature. 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

As can readily be observed, they are not monotone, thus causing some ripple . This obstacle can be avoided by refining some suitable grids in time. When solving equations of the form (13) with a weak quasilinearity for the coefficients k = k x,t), f = f u) and c = c x,t), common practice involves predictor-corrector schemes of accuracy 0(r" -f /r). Such a scheme for the choice c = k = 1, f = f u) is available now ... 

The temperature, pore width and average pore densities were the same as those used by Snook and van Megen In their Monte Carlo simulations, which were performed for a constant chemical potential (12.). Periodic boundary conditions were used In the y and z directions. The periodic length was chosen to be twice r. Newton s equations of motion were solved using the predictor-corrector method developed by Beeman (14). The local fluid density was computed form... 

It is also possible to use the information which has been stored to write programs for other tasks. A useful one, for example, keeps track of the stoichiometry (i.e. total atom counts) of the system. For a closed system, stoichiometry should be automatically maintained by linear predictor-corrector solvers, and the stoichiometry program provides a diagnostic of numerical errors (and others) which have accumulated. In other than closed systems, it gives an independent check on the sources and sinks which are being modeled. 

A predictor-corrector algorithm for automatic computer-assisted integration of stiff ordinary differential equations. This procedure carries the name of its originator. ... 

---