Runge-Kutta integration algorithm

Listing 10.15. Code for implementing a fourth order Runge-Kutta integration algorithm. 

The Runge-Kutta integration scheme advances the solution by step size h — Az from time level n to n + 1. The algorithm is... 

An extended Runge-Kutta integration scheme from time tn to tn + At employs an iterative procedure during time advancement, with the notation Ynj = for the j — th iteration. With similar notation for other functions of time, and with At = h, the following algorithm can be used [21]... 

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]. 

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... 

Initial numerical simulations of population density dynamics incorporated experimental data of Figures 2 and 4 and Equations 16, 20, 23, and 27 into a Runge-Kutta-Gill integration algorithm (21). The constant k.. was manipulated to obtain an optimum fit, both with respect to sample time and to degree of polymerization. Further modifications were necessary to improve the numerical fit of the population density distribution surface. 

Runaway reactions, dangers of, 24 184 Runge-Kutta—Gill fourth-order-correct integration algorithm, 25 311 Runoff, herbicide, 13 308-309 Run-to-run control, 20 704-705 Rupture ductility, 13 476 Rupture testing, 13 474 environment for, 13 477 Rural wastewater disposal systems,... 

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. 

This equation is usually integrated using e.g a Runge-Kutta algorithm. The parameters are To, A1A2 and E1+E2 addition to the initial value 0a. 

The Runge-Kutta algorithm is entered into a spreadsheet, and the two Stefan-Maxwell equations (12.196 and 12.197) are integrated using a step size Z/200, with guessed values for the fluxes Ni, N2. The calculated mole fractions of species 1 and 2, Xj (Z) and X2 (Z) were used to define a residual r,... 

Calculated effective diffusivities are tabulated in Table II. This problem was solved on the computer by the "shooting method" for a range of temperatures. Integrations were by the 4th order Runge-Kutta algorithm. The system equations are summarized below. 

The initial value problem, Eqs. 1-3, can be integrated by any marching algorithm which is based on the Runge-Kutta or Adams-Moulton techniques. Based on the calculated space profiles of C,... 

EROS handles concurrent reactions with a kinetic modeling approach, where the fastest reaction has the highest probability to occur in a mixture. The data for the kinetic model are derived from relative or sometimes absolute reaction rate constants. Rates of different reaction paths are obtained by evaluation mechanisms included in the rule base that lead to partial differential equations for the reaction rate. Three methods are available that cover the integration of the differential equations the GEAR algorithm, the Runge-Kutta method, and the Runge-Kutta-Merson method [120,121], The estimation of a reaction rate is not always possible. In this case, probabilities for the different reaction pathways are calculated based on probabilities for individual reaction steps. 

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