Explicit Runge-Kutta Methods

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

Equation (2.53) can be solved numerically by the Runge-Kutta method (see Appendix 2A). Explicit solutions of (2.53) are given here. 

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. 

The method is one way to handle a stiff set of odes, and is an extension of fourth-order explicit Runge-Kutta. The function to be solved is approximated over the next time interval by a combination of a linear function of the dependent variable and a quadratic function of time (assuming that it is strongly time-dependent) and this increases the accuracy and stability of the fourth-order Runge-Kutta method considerably. Today, however, we have other methods of dealing with stiff sets of odes, so this method might be said to have outlived its usefulness. 

Z. A. Anastassi and T. E. Simos, A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for the numerical solution of orbital problems, New Astronomy, 2004, 10(1), 31-37. 

Dispersion and Dissipation Properties for Explicit Runge-Kutta Methods... 

Basic Theory. - 3.1.1 Explicit Runge-Kutta Form-Order Conditions. An i-stage explicit Runge-Kutta method used for the computation of the approximation of yn+i(x) in problem (1), when y (x) is known, can be expressed by the following relations ... 

Definition 2 (see 5). In the explicit s- stage Runge-Kutta method, presented in... 

Construction of Runge-Kutta Methods which is Based on Dispersion and Dissipation Properties. - 3.2.1 A dispersive-fitted and dissipative-fitted explicit Runge-Kutta. We consider a 6-Stage explicit Runge-Kutta method ... 

Table 1 Order Equations for 6-Stage explicit Runge-Kutta method 1st Alg. Order (1 equation)...

In Appendix B we present a Maple programme for the development of Dispersive-fitted and dissipative-fitted explicit Runge-Kutta method. In Appendix C we present a Maple programme for the development of explicit Runge-Kutta method with minimal dispersion. In Appendix D we present a Maple programme for the development of explicit Runge-Kutta method with minimal dissipation. 

Explicit Runge-Kutta Methods for the Schrodinger Equation. - An 5- stage explicit Runge-Kutta method used for the computation of the approximation... 

Construction of Trigonometrically-Fitted Runge-Kutta Methods. Consider the explicit Runge-Kutta method Kutta-Nystrom 14, which has fifth algebraic order and six stages. The coefficients are shown in (70). 

In 40 the authors present a new explicit Runge-Kutta method with algebraic order four, minimum error of the fifth algebraic order (the limit of the error is zero, when the step-size tends to zero), infinite order of dispersion and eighth order of dissipation i.e. they present an optimized explicit Runge-Kutta method of fourth order. The efficiency of the newly developed method is shown through the numerical illustrations of a wide range of methods when these are applied to well-known periodic orbital problems. 

Maple Program for the development of Dispersive-fitted and dissipative-fitted explicit Runge-Kutta method... 

Coefficients of Explicit Runge-Kutta method of England... 

The basic idea of the Runge-Kutta methods is illustrated through a simple second order method that consists of two steps. The integration method is constructed by making an explicit Euler-like trail step to the midpoint of the time interval, and then using the values of t and tp at the midpoint to make the real step across the whole time interval ... 

For first-order equations or for first-order systems of equations we write the m-stage explicit Runge-Kutta method in the matrix form given in Table 5. 

Table 9 Modified five-stage Explicit Runge-Kutta method of order four derived by Simos and Williams106 ...

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