Predictor-corrector scheme

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. 

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 ... 

This predictor-corrector scheme or double-step scheme is done in two steps ... 

This paper deals with thermal wave behavior during frmisient heat conduction in a film (solid plate) subjected to a laser heat source with various time characteristics from botii side surfaces. Emphasis is placed on the effect of the time characteristics of the laser heat source (constant, pulsed and periodic) on tiiermal wave propagation. Analytical solutions are obtained by memis of a numerical technique based on MacCormack s predictor-corrector scheme to solve the non-Fourier, hyperbolic heat conduction equation. 

Heat waves have been theoretically studied in a very thin film subjected to a laser heat source and a sudden symmetric temperature change at two side walls. The non-Fourier, hyperbolic heat conduction equation is solved using a numerical technique based on MacCormak s predictor-corrector scheme. Results have been obtained for ftie propagation process, magnitude and shape of thermal waves and the range of film ftiickness Mid duration time wiftiin which heat propagates as wave. 

The Adam-Bashforth methods are frequently used as predictors and the Adam-Moulton methods are often used as correctors. The combination of the two formulas results in predictor-corrector schemes. 

Systematic errors associated with the predictor-corrector scheme of numerical integration... 

In a very recent study we examined theoretically the structure of water inside the water cylinder of inverse hexagonal mesophase (Hu) of monoolein (GMO). using methods of Molecular Dynamics. Due to the complex geometry of the system, a novel method for obtaining the distribution of water at the initial moment of calculation process is developed and applied. The initial density of water was obtained by fitting the final results within predictor-corrector scheme. 

One of the most common numerical methods used in molecular dynamics to solve Newton s equations of motions is the Velocity Verlet integrator. This is typically implemented as a second order method, and we find that it can become numerically tmstable dtuing the course of hyperthermal collision events, where the atom velocities are often far from equilibrium. As an alternative, we have implemented a fifth/ sixth order predictor-corrector scheme for our calculations. Specifically, the driver we chose utilizes the Adams-Bashforth predictor method together with the Adams-Moulton corrector method for approximating the solution to the equations of motion. 

The tangent matrix is then used to find increments to the accelerations (8ii) and Lagrange multipliers (8L) in a Newmark predictor-corrector scheme. 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

The Adams-Mouhon method, being a predictor-corrector scheme, involved the use of the corrector formula until the difference in two successive iterations was below a predetermined value, but in most cases it only used the corrector formula once. 

G. Psiho ios and T. E. Simos, A fourth algebraic order trigonometrically fitted predictor-corrector scheme for IVPs with oscillating solutions. Journal of Computational and Applied Mathematics, 2005, 175(1), 137-147. 

The time step for the convection-diffusion equation (Eq. 7) is done in the second stage, using the values obtained for the velocity components. We use a predictor corrector scheme with approximation of the convective terms against the flow [25] for this purpose. 

This results in the Adams Predictor-Corrector scheme ... 

In this formula no information about the function / is incorporated. It is only useful as a predictor in a predictor-corrector scheme. 

Since the governing equations are nonlinear, the implementation of the implicit algorithm requires an iterative method. A direct iterative scheme based on Newton s method is computationally too demanding. Instead, as an alternative, a predictor-corrector scheme can be used with the corrector to be iteratively applied until a convergence criterion is met. To this end, in Eqs. (1.19) and (1.20) + Peff,... 

Prominent representatives of the first class are predictor-corrector schemes, the Runge-Kutta method, and the Bulir-sch-Stoer method. Among the more specific integrators we mention, apart from the simple Taylor-series expansion of the exponential in equation (57), the Cayley (or Crank-Nicholson) scheme, finite differencing techniques, especially those of second or fourth order (SOD and FOD, respectively) the split-operator, method and, in particular, the Chebychev and the shoit-time iterative Lanczos (SIL) integrators. Some of the latter integration schemes are norm-conserving (namely Cayley, split-operator, and SIL) and thus accumulate only... 

The constants depend on the order of the integration scheme. For a fourth-order predictor corrector scheme, Allen and Tildesley calculate the constants to have the following values ... 

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