Time integrals, Runge-Kutta method

Figure 4 shows the output of this program, which consists of concentrations of o-methylol, p-methylol and methylene ether groups at various reaction times. Although many integration routines can be used in CSMP calculations, the variable interval Runge-Kutta method was used in this case since that is the option selected when no other method is specified. 

With hmax = ma,x(xi+i — x/), the global error order of the classical Runge-Kutta method is of order 4, or 0(h/nax), provided that the solution function y of (1.13) is 5 times continuously differentiable. The global error order of a numerical integrator measures the maximal error committed in all approximations of the true solution y(xi) in the computed y values y. Thus if we use a constant step of size h = 10 3 for example and the classical Runge-Kutta method for an IVP that has a sufficiently often differentiable solution y, then our global error satisfies... 

First, without explaining the details [15], we will develop an Excel spreadsheet for the numerical integration of the reaction mechanism 2as seen in Figure 7.13. The fourth-order Runge-Kutta method requires four evaluations of concentrations and derivatives per step. This appears to be a serious disadvantage, but as it turns out, significantly larger step sizes can be taken for the same accuracy, and the overall computation times are much shorter. We will comment on the choice of appropriate step sizes after this description. 

Starting with an initial value of Ca and knowing C (t), Eq. (8-4) can be solved for cjt + At). Once CA(t -I- At) is known, the solution process can be repeated to calculate c it -I- 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Euler approach, one often needs to take small steps in time, At. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations ... 

Equation (45) and Eq. (40) both contain the unknown quantities X (t) and CB(t). However, because Cg(t) cannot be solved with Eq. (40) so that it can be inserted in Eq. (45), the two equations have to be solved simultaneously and numerically by an iteration process (e.g., Runge-Kutta method). The integral term in Eq. (40) is also evaluated numerically. At the end of each infinitesimal time interval. At, the resulting value of Cp is used to calculate c (t) with Eqs. (42) and (44). 

Several standard procedures are available numerically to integrate these equations and thus calculate the time variations of the different concentrations. The Runge-Kutta method (Vetterling et al, 1986) is generally advised. 

This resulting integral may be evaluated to obtain the time required to achieve an X = 0.5. One procedure would be to employ the trapezoidal rule. However, the Runge-Kutta method would produce more accurate results. 

These results were obtained by numerically integrating the system with a Runge-Kutta method. The time step was 0.25, and 15,000 points were computed. Grassberger and Procaccia also report that the convergence was rapid the correlation dimension could be estimated to within 5 percent using only a few thousand points. ... 

The basic idea of the Runge-Kutta methods is illustrated through a simple second order method that consists of two steps. The integration method is constructed by making an explicit Euler-like trail step to the midpoint of the time interval, and then using the values of t and tp at the midpoint to make the real step across the whole time interval ... 

A combination of the Runge Kutta method and methods of non-linear regression allows a parameter identification from the time-course data. This technique starts with a given set of parameters, performs the numeric integration of the rate equation... 

The KPS calculation was based on integrating Hamilton s equations of motion for the time evolution of the Cartesian components of the Jacobi coordinates that describe the three-atom system. A fourth-order Runge-Kutta method was used for the numerical integration, and the computations were done using an IBM 7090-4 computer at the IBM Watson Research Center and the Columbia Computing Center. The computation time per trajectory was listed as 10 s using a time step of 0.025 fs. 

In our early work on multimode vibronic dynamics, a fourth-order predictor-corrector method has been used to integrate the time-dependent Schrodinger equation. Later, FOD schemes and a fourth-order Runge-Kutta method have also been employed. These techniques proved to be superior to the predictor-corrector method for example, the FOD scheme was found to be 3-5 times faster than the SOD integrator (the latter... 

The explicit integration methods, such as leapfrog, prediction-correction or Runge-Kutta methods, are usually used to integrate SPH equations for fluid flows. The explicit time integration is conditionally stable. The time step should satisfy the convective stabihty constraint, i.e., the so-caUed Courant-Friedrichs-Lewy (CFL) condition,... 

The algorithm must not require an expensively large number of force evaluations per integration time step. Many common techniques for the solution of ordinary differential equations (such as the fourth-order Runge-Kutta method) become inappropriate, since they do not fulfill this criterion. 

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