Numerical methods Runge-Kutta

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. 

If we put Eq.(9) into Eq.(8), we get two differential equations for shear stress and driving air velocity. We have solved them by Runge-Kutta numerical method. The results are the air... 

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 ... 

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. 

The method of characteristics, the distance method of lines (continuous-time discrete-space), and the time method of lines (continuous-space discrete-time) were used to solve the solids stream partial differential equations. Numerical stiffness was not considered a problem for the method of characteristics and time method of lines calculations. For the distance method of lines, a possible numerical stiffness problem was solved by using a simple sifting procedure. A variable-step fifth-order Runge-Kutta-Fehlberg method was used to integrate the differential equations for both the solids and the gas streams. 

In ref 156 the author studies the stability properties of a family of exponentially fitted Runge-Kutta-Nystrom methods. More specifically the author investigates the P-stability which is a very important property usually required for the numerical solution of stiff oscillatory second-order initial value problems. In this paper P-stable exponentially fitted Runge-Kutta-Nystrom methods with arbitrary high order are developed. The results of this paper are proved based on a S5unmetry argument. 

In ref 164 new and elficient trigonometrically-fitted adapted Runge-Kutta-Nystrom methods for the numerical solution of perturbed oscillators are obtained. These methods combine the benefits of trigonometrically-fitted methods with adapted Runge-Kutta-Nystrom methods. The necessary and sufficient order conditions for these new methods are produced based on the linear-operator theory. 

In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. 

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. 

T. E. Simos and P. S. Williams, A new Runge-Kutta-Nystrom method with phase-lag of order infinity for the numerical solution of the Schrodinger equation, MATCH Commun. Math. Comput. Chem., 2002, 45, 123-137. 

Z. Kalogiratou and T. E. Simos, Construction of trigonometrically and exponentially fitted Runge-Kutta-Nystrom methods for the numerical solution of the SchrSdinger equation and related problems a method of 8th algebraic order, J. Math. Chem., 2002, 31(2), 211-232. 

P. J. Van Der Houwen and B. P. Sommeijer, Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions, SIAM J. Numer. Anal., 1987, 24(3), 595-617. 

J. M. Franco, Runge-Kutta-Nystrom methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Commun., 2002, 147, 770-787. 

Hans Van de Vyver, An embedded exponentially fitted Runge-Kutta-Nystrom method for the numerical solution of orbital problems. New Astronomy, 2006, 11, 577-587. 

In 27 the authors have developed trigonometrically-fitted Runge-Kutta method for the numerical integration of orbital problems. The developed method is based on the Runge-Kutta Zonneveld method. More specifically is based on a modification of the Butcher table for the Runge-Kutta methods showed below ... 

Runge-Kutta-Gill method provides an efficient algorithm for solving a system of first-order differential equations and makes use of much less computer memory when compared with other numerical methods. 

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... 

Based on the results presented in the relative papers and based on some numerical tests made for this review, the most efficient Runge-Kutta method for specific Schrodinger equations is the one developed by Simos and Williams106 with seven stages while the Runge-Kutta-Nystrom method developed by Simos, Dimas and Sideridis107 gives similar results in accuracy and computational efficiency. 

By simply knowing the phase equilibrium behavior and the composition within the beaker at the start of the experiment (x ), one can easily construct a residue curve by integrating Equation 2.8. Such integration is usually performed with the use of a numerical integration method (see later, Section 2.5.3), such as Runge Kutta type methods, remembering that at each function evaluation, a bubble point calculation must be performed in order to determine y(x). 

As mentioned previously, numerical integration of the residue curve equation can be done with Runge Kutta type methods. Formnately, mathematical software packages... 

In the same formula, h is the step size of the integration and n is the number of steps, i.e. is the approximation of the solution in the point x and Xn = XQ + n h and xq is the initial value point. Using the above methods and based on the theory of phase-lag and amplification error for the Runge-Kutta-Nystrom methods, the authors have derived two fourth algebraic order Runge-Kutta-Nystrom methods with phase-lag of order four and amplification error of order five. For both of methods the authors have obtained the stability regions. The efficiency of the produced methods is proved via numerical experiments. 

The following (m + l)-stage modified Runge-Kutta-Nystrom method has been introduced in order to solve numerically the above problem ... 

A more accurate technique used for numerical integration is the classical fourth-order Runge-Kutta (RK) method. In this... 

T. E. Simos, E. Dimas and A. B. Sideridis, A Runge-Kutta-Nystrom Method for the Numerical-Integration of Special 2nd-order Periodic Initial-Value Problems, Journal of Computational and Applied Mathematics, 1994, 51(3), 317-326. 

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