Runge-Kutta-4 Integrator

Numerical Integration Runge-Kutta 4th Order Method... 

The IE and IM methods described above turn out to be quite special in that IE s damping is extreme and IM s resonance patterns are quite severe relative to related symplectic methods. However, success was not much greater with a symplectic implicit Runge-Kutta integrator examined by Janezic and coworkers [40],... 

D. Janezic and B. Orel. Implicit Runge-Kutta method for molecular dynamics integration. J. Chem. Info. Comp. Sd., 33 252-257, 1993. 

Janezic, D., Orel, B. Implicit Runge-Kutta Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 33 (1993) 252-257 Janezic, D., Orel, B. Improvement of Methods for Molecular Dynamics Integration. Int. J. Quant. Chem. 51 (1994) 407-415... 

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. 

The reaction rate equations give differential equations that can be solved with methods such as the Runge-Kutta [14] integration or the Gear algorithm [15]. 

A popular fourth-order Runge-Kutta method is the Runge-Kutta-Feldberg formulas (Ref. Ill), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method. 

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

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. 

More than two points can be used in the Runge-Kutta method, and the fourth-order Runge-Kutta integration is commonly employed. Obviously computers are... 

Two schemes for second-order Runge-Kutta numerical integration can be presented as follows ... 

One can narrow the interval size h for greater precision, or increase it as the reaction proceeds for greater efficiency. A Runge-Kutta solution to Scheme I, Eq. (5-54), is of some interest, since this result could not have been obtained by integration. The val-... 

The computer model consists of the numerical integration of a set of differential equations which conceptualizes the high-pressure polyethylene reactor. A Runge-Kutta technique is used for integration with the use of an automatically adjusted integration step size. The equations used for the computer model are shown in Appendix A. 

Example 2.14 Use fourth-order Runge-Kutta integration to solve the following set of ODEs ... 

Solution The coding is left to the reader, but if you really need a worked example of the Runge-Kutta integration, check out Example 6.4. The following are detailed results for At= 1.0, which means that only one step was taken to reach the answer. 

Extrapolation can reduce computational effort by a large factor, but computation is cheap. The value of the computational reduction will be trivial for most problems. Convergence acceleration can be useful for complex problems or for the inside loops in optimization studies. For such cases, you should also consider more sophisticated integration schemes such as Runge-Kutta. It too can be extrapolated, although the extrapolation rule is different. The extrapolated factor for Runge-Kutta integration is based on the series... 

The next example treats isothermal and adiabatic PFRs. Newton s method is used to determine the throughput, and Runge-Kutta integration is used in the Reactor subroutine. (The analytical solution could have been used for the isothermal case as it was for the CSTR.) The optimization technique remains the random one. 

Adiabatic version of PFR equations solved by Runge-Kutta integration... 

The next example illustrates the use of reverse shooting in solving a problem in nonisothermal axial dispersion and shows how Runge-Kutta integration can be applied to second-order ODEs. 

When the axial dispersion terms are present, D > Q and E > Q, Equations (9.14) and (9.24) are second order. We will use reverse shooting and Runge-Kutta integration. The Runge-Kutta scheme (Appendix 2) applies only to first-order ODEs. To use it here. Equations (9.14) and (9.24) must be converted to an equivalent set of first-order ODEs. This can be done by defining two auxiliary variables ... 

There are four equations in four dependent variables, a, d, T, and T. They can be integrated using the Runge-Kutta method as outlined in Appendix 2. Note that they are integrated in the reverse direction e.g., a = at) — similarly for 2 and in Equations (2.47). 

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