Corrector solution

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. 

The OWL optical design is shown in Fig. 1. It is based on a spherical and flat folding secondary mirrors, with a four-element corrector providing for the compensation of spherical and field aberrations as well as advanced active optics and dual-conjugate adaptive optics. A complete discussion would exceed the scope of this report we shall however mention a few key arguments supporting this solution ... 

We solve the same equations as in Section 4.1.1. Since a is very close to the threshold value a = 2, the difference between the solution to the effective equation obtained by taking the simple mean, at one side, and the solutions to the original problem and to our upscaled equation, are spectacular. Our model approximates fairly well with the physical solution even without adding the correctors (Table 4). Parameters are given at Table 3. 

Adding correctors would get us even closer to the solution for the 2D problem. Figures 3, 4 and 5 show the simulation by FreeFm++ in the case from Section 4.2.1. Advantage of our approach is again fairly clear and the errors of the model obtained by taking a simple mean persist in time. 

M71 Solution of ordinary differential equations predictor-corrector method of Milne 7100 7188... 

SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS t 4 PREDICTOR-CORRECTOR HETHOD OF HILNE 4 444444444444444444444444444444444444444444444444444... 

An important question is the relative numerical efficiency of the two methods or, more generally, the two families of methods. At a fixed step size the predictor - corrector methods clearly require fewer function evaluations. This does not necessarily means, however, that the predictor - corrector methods are superior in every application. In fact, in our present example increasing the step size leaves the FTunge - Kutta solution almost unchanged, whereas the Milne solution is deteriorating as shown in Table 5.1. 

High-level DAE software (e.g., Dassl) makes a time-step selection based on an estimate of the local truncation error, which depends on the difference between a predictor and a corrector step [13,46]. If the difference is too great, the time step is reduced. In the limit of At 0, the predictor is just the initial condition. For the simple linear problem illustrated here, the corrector will always converge to the correct solution y2 = 1, independent of the time step. However, if the initial condition is y2 1, then there is simply no time step for which the predictor and corrector values will be sufficiently close, and the error estimate will always fail. Based on this simple problem, it may seem like a straightforward task to build software that identifies and avoids the problem, and there is current research on the subject [13], The problem is that in highly nonlinear, coupled, problems the inconsistent initial conditions can be extremely difficult to identify and fix in a general way. 

Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful. 

The solution procedure for the moving bed has been described in detail elsewhere (J9). The two point boundary value problem is solved by a predictor-corrector procedure on the missing boundary at the top of the reactor until agreement with the inlet gas composition at the base of the reactor is achieved. 

The three methods for lead in air are essentially identical however, one should use S-341 because this method has been validated unlike P CAM 155 or P CAM 173. Although all the methods recommend 2-3 ml of nitric acid for wet ashing, the final solutions differ in that P CAM 155 recommends 1% nitric, P CAM 173 recommends 1% HC1, and S-341 recommends 10% nitric with EDTA 0.1 M to suppress phosphate, carbonate, iodide, fluoride, and acetate ion that cause flame suppression. EDTA is suggested in P CAM 173 where interferences are anticipated. Both S-341 and P CAM 173 use the 217.0 nm line which is twice as sensitive as the 283.3 nm line. Strong nonatomic absorption found when high concentrations of dissolved solid are present requires use of the background corrector. These two methods differ from P CAM 155 and those for biological analysis,... 

G. Psihoyios and T. E. Simos, The numerical solution of the radial Schrodinger equation via a trigonometrically fitted family of seventh algebraic order Predictor-Corrector methods, J. Math. Chem., 2006, 40(3), 269-293. 

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. 

In 30 the authors have developed trigonometrically fitted Adams-Bashforth-Moulton predictor-corrector (P-C) methods. It is the first time in the literature that these methods are applied for the efficient solution of the resonance problem of the Schrodinger equation. The new trigonometrically fitted P-C schemes are based on the well known Adams-Bashforth-Moulton methods. In particular, they are based on the fourth order Adams-Bashforth scheme (as predictor) and on the fifth order Adams-Moulton scheme (as corrector). More... 

Predictor-Corrector methods have been constructed attempting to combine the best properties of the explicit and implicit methods. The multistep methods are using information at more than two points. The additional points are ones at which data has already been computed. In one view, Adams methods arise from underlying quadrature formulas that use data outside of specifically approximate solutions computed prior to t . 

Numerous predictor-corrector methods have been developed over the years. A simple second order predictor-corrector method can be constructed by first approximating the solution at the new time step using the first order explicit Euler method ... 

The formula is known both as the improved Euler formula and as Heun s method [170]. This method has roughly the stability of the explicit Euler method. Unfortunately, the stability may not be improved by iterating the corrector because this iteration procedure converges to the trapezoid rule solution only if At is small enough. 

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. 

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