Runge-Kutta stages

In general, a Rosenbrock method consists of a number s of stages. At each stage, a Runge-Kutta-type ki value is calculated, from explicit rearrangement of implicit equations for these. At stage i, the equation is... 

We first obtain the mean flow by solving the coupled DDEs (6.3.11) and (6.3.13) by standard four-stage Runge-Kutta method. These equations have been solved by taking maximum similarity co-ordinate, r max = 12 equally divided into 4000 sub-intervals. For different Re and K, mean flow has been obtained here. Fixing K, instead of Gr, is motivated by our discussion in the introduction where we have noted that for instability of mixed-convection boundary layers, K is more relevant than Gr. As we have investigated the mixed convection problem in air, we have fixed the value of Pr = 0.7 for all cases. Obtained mean-field results for the non-dimensional velocity and temperature are shown in Fig. 6.1. 

In ref. 143 the authors develop a third-order 3-stage diagonally implicit Runge-Kutta-Nystrom method embedded in fourth-order 4-stage for solving special second-order initial value problems. The obtained method has been developed in order to have minimal local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The authors also study the stability of the method. The new proposed method is illustrated via a set of test problems. 

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. 

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)...

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 52 the author develops a symplectic exponentially fitted modified Runge-Kutta-Nystrom method. The method of development was based on the development of symplectic exponentially fitted modified Runge-Kutta-Ny-strom method by Simos and Vigo-Aguiar. The new method is a two-stage second-order method with FSAL-property (first step as last). 

In order to improve the time accuracy, a second-order, two-stage, strong stabilitypreserving Runge-Kutta (RK2SSP) scheme can be used (Vikas et at., 2011a). The updated moment sets are found from... 

RK2SSP second-order two-stage strong stability-preserving Runge-Kutta... 

This robust higher order finite-difference method, originally presented in [10,13,25], develops a seven-point spatial operator along with an explicit six-stage time-advancing technique of the Runge-Kutta form. For the former operator, two central-difference suboperators are required a) an antisymmetric... 

The temporal update through the single-stage ifth-order Runge-Kutta algorithms [33] is performed by... 

Theoretical and Numerical Solutions of Equation (2.108) by Four-Stage Runge-Kutta Method... 

The interval of absolute stability for four-stage Runge-Kutta is (-2.78,0). If the numerical solution is to converge to the theoretical solution, the step size h must be selected such that -2.78 monotonically decreasing function of x, the maximum value of y is y(0) = 1, and the minimum oih = -100ft < 2.78. For ft = 0.01 and 0.02, ft = -1, -2, respectively, which remain in the interval of absolute stability. For ft = 0.04, h = -A, which falls outside of the interval of absolute stability. Hence, the numerical solution diverges for ft = 0.04. 

There exists a family of Runge-Kutta methods that differs by the number of functional evaluations oiJ x, yd- A method that requires R functional evaluations is known as the R-stage Runge-Kutta method. 

Four-stage Fourth-order Runge-Kutta method... 

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. 

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... 

Table 6 m-stage explicit Runge-Kutta-Nystrom method... 

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. 

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

Some recent Runge-Kutta formulae are based on quadrature methods, that is, the points at which the intermediate stage approximations are taken are the same points used in integration with either Gauss or Lobatto or Radau rules (Chapter 1). For example, the Runge-Kutta method derived from the Lobatto quadrature with three points (also called the Cavalieri-Simpson rule) is... 

The more general family of Partitioned Runge-Kutta methods is defined by making use of a partitioning of the system and introducing combinations of a set of internal stages. This more general family of schemes is discussed in some detail in [326] (see also discussions of [164,227]). 

In [169] the authors studied the optimization of explicit Runge-Kutta methods. More specifically they studied the s-stage explicit Runge-Kutta method of the form ... 

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