Algorithm predictor-corrector

Predictor-corrector algorithms (AUen and Tildesley 1987) constitute another commonly used class of method to integrate the equations of motion. The method consists of three steps  

Predictor From the positions and their time derivatives up to a certain order q, aU known at time f, one predicts the same quantities, such as accelerations, at time t + At, by means of a Taylor expansion. 

Force evaluation The force is computed, taking the gradient of the potential at the predicted positions. The resulting acceleration is in general different from the predicted acceleration. The difference between the two constitutes an error signal. 

Corrector This error signal is used to correct positions and their derivatives. AU the corrections are proportional to the error signal, the coefficient of proportionality being a magic number determined to maximize the stability of the algorithm. 

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

SERIES FIRST ORDER REACTION PULSE CHASE EXPERIMENTS Predictor-corrector algorithm,... 

As mentioned above, the goal of MD is to compute the phase-space trajectories of a set of molecules. We shall just say a few words about numerical technicalities in MD simulations. One of the standard forms to solve these ordinary differential equations i.s by means of a finite difference approach and one typically uses a predictor-corrector algorithm of fourth order. The time step for integration must be below the vibrational frequency of the atoms, and therefore it is typically of the order of femtoseconds (fs). Consequently the simulation times achieved with MD are of the order of nanoseconds (ns). Processes related to collisions in solids are only of the order of a few picoseconds, and therefore ideal to be studied using this technique. 

Potential energy 14, 16, 25 Potential energy minima 51 Potential energy surface 52, 54, 75 Predictor-corrector algorithm 63 Primary properties 266 Primitive GTO 164 Principal axes 268, 284, 317 Projection equation 207 Protein data Bank 178 Protein docking 56 Pseudo-orbital 172... 

Schrodinger s equation leads then to Nx + second-order coupled differential equations with first derivatives. These are solved transforming the set into 2(N + Ne) first-order coupled equations, and generating N2 + N linearly independent solutions by choosing suitable boundary conditions at small rAC. A Bashforth-Moulton fourth-order predictor-corrector algorithm was used in the integration. 

The other main class of algorithms are the predictor-corrector algorithms (Gear, 1971). New positions, velocities, accelerations and higher time derivatives of r at (n + l)At are predicted using Taylor expansions and the current values at nAt. But these are not correct, and will eventually fail, because the forces have not been updated. So the accelerations at (n + l)Af are now calculated using the predicted positions, and hence the forces at (n + l)Af, and... 

These correction terms involve numerical coefficients (Gear, 1971 see also Allen and Tildesley, 1987) chosen to give optimum stability and accuracy. Ideally, the corrector step would be repeated to improve the accuracy of the estimates at (n+ l)Af but each correction involves a new evaluation of the forces, which is the most time-consuming part of an MD simulation. So in practice just one or two corrector steps are carried out. Other forms of predictor-corrector algorithm exist. A discussion of various MD algorithms has been given by Berendsen and van Gunsteren (1986). 

The computation starts with a predictor-corrector algorithm for the determination of velocity field at to + At. In the predictor stage of the solution algorithm, the pressure is replaced by an arbitrary pseudo-pressure P (which in most cases is set equal to zero at full cells), and tentative velocities are then calculated. A pseudo-pressure boundary condition is applied in surface cells to satisfy the normal stress condition. Since pressure has been ignored in the full cells, the tentative velocity field does not satisfy the incompressible continuity equation. The deviation fi om incompressibility is used to calculate a pressure potential field [J/, which then is used to correct the velocity field. In the final steps, the velocity boundary conditions are calculated, the new location of free surface is determined by tracking the markers, and the velocity boundary conditions associated with the new fluid cells are assigned (Fig. 5). 

Keywords Ab initio molecular dynamics simulations Always stable predictor-corrector algorithm Associated liquids Basis set Bom-Oppenheimer molecular dynamics simulations Car-Parrinello molecular dynamics simulations Catalysis Collective variable Discrete variable representation Dispersion Effective core potential Enhanced sampling Fictitious mass First-principles molecular dynamics simulations Free energy surface Hartree-Fock exchange Ionic liquids Linear scaling Metadynamics Nudged elastic band Numerically tabulated atom-centered orbitals Plane waves Pseudopotential Rare event Relativistic electronic structure Retention potential Self consistent field SHAKE algorithm ... 

The predictor/corrector algorithm in Diva includes a stepsize control in order to minimize the number of predictor and corrector steps. Finally, the continuation package contains methods for the computation of the dominating eigenvalues of DAEs. This allows a stability analysis of the steady state solutions and a detection of local bifurcations for large sparse systems. As the continuation method is embedded into a dynamic simulator, the user has the opportunity to switch interactively from continuation to time integration. This allows additional investigations of transient behaviour or domains of attraction with the same simulation tool[2]. 

In principle, the predictor/corrector algorithm can also be applied to the continuation of periodic solutions, as will be explained for the example of a system of ordinary differential equations... 

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