Numerical methods Runge-Kutta integration

Thus we find that the choice of quaternion variables introduces barriers to efficient symplectic-reversible discretization, typically forcing us to use some off-the-shelf explicit numerical integrator for general systems such as a Runge-Kutta or predictor-corrector method. 

This improved procedure is an example of the Runge-Kutta method of numerical integration. Because the derivative was evaluated at two points in the interval, this is called a second-order Runge-Kutta process. We chose to evaluate the mean derivative at points Pq and Pi, but because there is an infinite number of points in the interval, an infinite number of choices for the two points could have been made. In calculating the average for such choices appropriate weights must be assigned. 

Numerical integration of sets of differential equations is a well developed field of numerical analysis, e.g. most engineering problems involve differential equations. Here, we only give a very brief introduction to numerical integration. We start with the Euler method, proceed to the much more useful Runge-Kutta algorithm and finally demonstrate the use of the routines that are part of the Matlab package. 

The fourth order Runge-Kutta method is the workhorse for the numerical integration of ODEs. Elaborate routines with automatic step-size control are available in Matlab. 

This system of equations can be integrated numerically by a Runge-Kutta method, and the charge transfer probability is then obtained as P oo) = lim riooit). The quantities >j ,(t) can be related to a /t) in (21) by writing... 

Table III presents the values of the constants used in the calculations. The t/o data have been obtained from the variation of Pq with Z by numerical integration of the Gibbs-Duhem equation using the Runge-Kutta method (22,23). The comparison of the Pq values of Table I with those obtained by some previous workers (24, 25) shows that our results are at most higher by 0.5-1 Torr.

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... 

Eqns (16), (20) and (22) were integrated numerically to obtain the separation performance of the one- and two-column processes The GEAR package (13) was used for the integration after determining that it was faster than, say, Runge-Kutta methods For all calculations N 50 and e - 0.40 Dimensionless parameters varied were 8, ph pL Yf and, for the two-column process, H. Combinations of the parameters of 6 and Pr/Pl were chosen to correspond to the me thane-helium system on BPL carbon Adsorption isotherm data for methane at 25°C (14) were represented by... 

Starting with an initial value of cA and given c,(t), Eq. (8-4) can be solved for cA(t + At). Once cA(t + At) is known, the solution process can be repeated to calculate cA(t + 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. As discussed in Sec. 3, 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 ... 

Numerical integration was applied to the model. The local fluxes in the extractor and back extractor were first calculated from Eqs. 5 and 10, respectively. The solute flux / in Eq. 5 was determined using Fortran IMSL subroutine NEQNJ. The concentrations of lactic acid, amine, and the complex in the extractor and back extractor were then calculated from Eqs. 11-16, which were solved by using the Runge-Kutta method. Equations 17-20... 

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

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. 

Equation (6.120) is more suited for numerical than analytical integration, using the Runge-Kutta 4th method. However, Equation (6.120) can be rearranged as ... 

These three equations must be simultaneously solved using a suitable numerical method to give the variations of H and A2 with X. A simple program, TURBINRK, for obtaining such a solution based on the use of the Runge-Kutta method to numerically integrate the simultaneous differentia] equation is available as discussed in the Preface. This program is written in FORTRAN. 

Here zm denotes the valency of the counterions with the highest absolute charge. The function G is dependent upon the zeta potential of the particle as well as the bulk concentrations, valencies and diffusivities of the ions. The equilibrium electrical potential can be obtained using the Runge-Kutta method to solve the Poisson-Boltzmann equation numerically, and then the integral in Eq. (52) can be evaluated. 

When the quadrature of eq 2 cannot be performed analytically the integration should be carried out numerically by robust routines such as the Runge-Kutta, Adams-Moulton predictor-corrector or Bulirsch-Stoer methods with step size and error control [53, 55, 56], These routines can also be found in computer codings at Netlib and in standard books on computer codes [53]. 

The steady-state heat and mass balance equations of the different models were numerically integrated using a fourth-order Runge-Kutta-Gill method for the one-dimensional models, while the Crank-Nicholson finite differences method was used to solve the two-dimensional models. 

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