Modified Euler method

The modified Euler method needs two initial values y and y, and is given by... 

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

For the modified Euler method, we expand the Taylor series as... 

There are two methods of obtaining a curve of r vs x from Eqs. (B) and (D). The first approach is to write Eq. (B) in difference form for a small change in conversion. Ax, and solve by stepwise numerical integration. As an illustration let us follow through three incremental calculations using the modified Euler method. We write Eq. (B) as... 

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. 

Figure 4. Plot of the relative error (in percentage) as a function of the potential for (a) the modified Euler method and (b) the fourth-order Runge-Kutta method. (Voltammograms simulated for k=10 S i, scan rate = 200 mV/s and stq) size = 2 mV).

Using the modified Euler method, find an approximate value of y when X = 0.3 for the following differential equation ... 

USE MODIFIED EULER METHOD TO CALCULATE CONCENTRATION CHANGES DUE TO SOLUTION CHEMICAL REACTIONS. 

Further reduction of the time truncation error is obtained by using the modified Euler method for solving Eqs. B-20a and B-20b. Let x, y ) be the position of the point at the beginning of the small time interval 8t, The first approximation to the solution is... 

In this equation, we have introduced the time constant of the system, which is the inverse of the rate constant k, i.e. x = /k. This simple problem involves a linear differential equation that allows us to investigate how the implicit and the explicit methods behave as the step size is modified. Recall that the explicit Euler method is given by y +i = y +, y ). This means that... 

This combination of steps is known as the Euler predictor-corrector (or modified Euler) method, whose application is demonstrated graplhcaliy in Fig. 5.4. Correction by Eq. (5.67) may be applied more than once until the corrected value converges, that is, the difference between the two con.secutive corrected values becomes less than the convergence criterion. However, not much more accuracy is achieved after the second application of the corrector. 

MEuler.m-The Euler predictor-corrector (modified Euler) method This function solves the set of differentia] equations based on Eqs. (5.66) and (5.67). 

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... 

We present how to treat the polarization effect on the static and dynamic properties in molten lithium iodide (Lil). Iodide anion has the biggest polarizability among all the halogen anions and lithium cation has the smallest polarizability among all the alkaline metal cations. The mass ratio of I to Li is 18.3 and the ion size ratio is 3.6, so we expect the most drastic characteristic motion of ions is observed. The softness of the iodide ion was examined by modifying the repulsive term in the Born-Mayer-Huggins type potential function in the previous workL In the present work we consider the polarizability of iodide ion with the dipole rod method in which the dipole rod is put at the center of mass and we solve the Euler-Lagrange equation. This method is one type of Car-Parrinello method. 

When the EKR system is enhanced by the addition of an acid, these equations should be adequately modified in the same manner as was done for the onedimensional model. Rnally, the new transient values of the concentrations at the time t + At can be calculated using a numerical solution of differential equations Uke Euler s method ... 

Mazur et al. [103, 104] demonstrated the conformational dynamics of biomacromolecules. However, their method scaled exponentially with size and relied on an expensive expression for the inter-atomic potentials in internal coordinates. Subsequently, our group pioneered the development of internal coordinate constrained MD methods, based on ideas initially developed by the robotics community [102, 105-107], reaching 0(n) serial implementations, using the Newton-Euler Inverse Mass Operator or NEIMO [108-110] and Comodyn [111] based on a variant of the Articulated Body Inertia algorithm [112], as well as a parallel implementation of 0(log n) in 0(n) processors using the Modified Constraint Force Algorithm... 

Ccxnputation times also follow the expected evolution, increasing with the niunber of function evaluations per step required by the method. The Adams-Moulton scheme, however, which requires only two e uations per step, like the modified Euler scheme, was only slightly faster than the fourth-order Runge-Kutta due to the amount of computation involved in the predictor and corrector formulas. The former also provided extremely low errors, when applicable, but showed a tendency to become unstable at higher step sizes. 

Figure 3- Comparison of the average error versus computation time, for the modified Euler, 3 d 4 order Runge-Kutta, and Adams-Moulton integration methods, applied to an ECEE scheme [(a) k = 10 s l, scan rate = 200 mV/s (b) k =100 s l, scan rate =1000 mV/s].

The mathematical equations used for each numerical int ration method [7, 8] are summarized in this section, where y stands for a nonlinear ordinary differential equation and h for the step size. Modified Euler integration formula ... 

This is the modified Euler s method, which is a relatively simple predictor-corrector method. 

Different choices of b yield different second-order Runge-Kutta methods. If 6 = the method is called the improved Euler s or Heun s method. If 6 = 1, the method is called the improved polygon or modified Euler s method. As demonstrated by this development, Runge-Kutta methods are not unique since they involve the choice of an arbitrary constant. All second-order methods involve the evaluation of two slopes ki and A)2 (equation (3.1.38) and equation (3.1.39)) and the value of is a weighted average of these two slopes (equations (3.1.35) and (3.1.36)). 

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