Numerical analysis Runge-Kutta method

Petzold, L. R., Order results for implicit Runge-Kutta methods applied to differential/algebraic," SIAM Journal on Numerical Analysis, No. 4, pp. 837-852 (1986). 

The set of non-dimensional model equations (eqs. 8.9-10 to 8.9-13) is solved numerically by a combination of the orthogonal collocation method and the Runge-Kutta method. Details of the collocation analysis are given in Appendix 8.9. A programming code ADSORB5A is provided with this book to give the reader a... 

To appreciate some of the similarities between the Taylor and Runge-Kutta methods we will state the associated difference equations without the detailed derivations. Full derivations and algorithms can be found in many texts on numerical analysis [5,9,18,22-24]. 

The major computational effort in applying the Rxmge-Kutta methods occurs in the evaluation of /. For second-order methods, two functional evaluations per step are required, and for the fourth-order method, four evaluations per step are required. As a consequence, the lower-order methods with smaller step size may be less costly than the higher-order methods using a larger step size. Relationships between evaluations per step and the order of local tmncation error can be found in the literature [24]. Also available in the numerical analysis literature [15] are developments that combine the best features of the Euler and a second-order Runge-Kutta method. 

Cameron IT. Solution of Differential-Algebraic Systems using Diagonally Implicit Runge-Kutta Methods. IMA J of Numerical Analysis 1983 3 273 289. 

Arn98] Arnold M. (1998) Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2. BIT, Numerical Analysis, to appear. 

All the methods presented so far, e.g. the Euler and the Runge-Kutta methods, are examples of explicit methods, as the numerical solution aty +i has an explicit formula. Explicit methods, however, have problems with stability, and there are certain stability constraints that prevent the explicit methods from taking very large time steps. Stability analysis can be used to show that the explicit Euler method is conditionally stable, i.e. the step size has to be chosen sufficiently small to ensure stability. This conditional stability, i.e. the existence of a critical step size beyond which numerical instabilities manifest, is typical for all explicit methods. In contrast, the implicit methods have much better stability properties. Let us introduce the implicit backward Euler method. 

Calculating an accurate value off is the most challenging part of solving differential equations numerically. Textbooks on numerical analysis discuss this issue in detail. For many problems, the fourth-order Runge-Kutta method can be used. In this method. 

The following details establish reactor performance, considers the overall fractional yield, and predicts the concentration profiles with time of complex reactions in batch systems using the Runge-Kutta numerical method of analysis. 

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. 

In ref 171 new Runge-Kutta-Nystrom methods for the numerical solution of periodic initial value problems are obtained. These methods are able to integrate exactly the harmonic oscillator. In this paper the analysis of the production of an embedded 5(3) RKN pair with four stages is presented. 

J. R. Cash and S. Girdlestone, Variable Step Runge-Kutta-Nystr5m Methods for the Numerical Solution of Reversible Systems, Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM), 2006, 1(1), 59-80. 

Phase-lag Analysis of the Runge-Kutta-Nystrom Methods. - For the numerical solution of the problem (129), the m-stage explicit Runge-Kutta-Nystrom (RKN) method shown in Table 6 can be used. Application of this method to the scalar test equation (130) produces the numerical solution... 

In [211] the authors obtained a new embedded 4(3) pair explicit four-stage fourth-order Runge-Kutta-Nystrom (RKN) method to integrate second-order differential equations with oscillating solutions. The proposed method has high phase-lag order with small principal local truncation error coefficient. The authors given the stability analysis of the proposed method. Numerical comparisons of this new obtained method to problems with oscillating and/or periodical behavior of the solution show the efficiency of the method. 

But87] Butcher J. C. (1987) The Numerical Analysis of Ordinary Differential Equations Runge-Kutta and General Linear Methods. Wiley, Chichester. 

Long before electronic computers were invented, it was realized that mathematical sophistication could be introduced into numerical integration in order to save computational elTort and improve accuracy. Textbooks of numerical analysis are full of ways to do this. The most popular of them, the Runge-Kutta and predictor-corrector algorithms, once were standard methods for numerical solution of the initial value problems of chemical kinetics. They have been replaced, however, by more suitable methods invented for the specific purpose of dealing with chemical kinetics problems. 

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