Dispersion Runge-Kutta

The next example illustrates the use of reverse shooting in solving a problem in nonisothermal axial dispersion and shows how Runge-Kutta integration can be applied to second-order ODEs. 

When the axial dispersion terms are present, D > Q and E > Q, Equations (9.14) and (9.24) are second order. We will use reverse shooting and Runge-Kutta integration. The Runge-Kutta scheme (Appendix 2) applies only to first-order ODEs. To use it here. Equations (9.14) and (9.24) must be converted to an equivalent set of first-order ODEs. This can be done by defining two auxiliary variables ... 

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. 

In Section 3 we present the Dispersion and Dissipation properties for Runge-Kutta methods. Based on these properties we have constructed ... 

A dispersive-fitted and dissipative-fitted explicit Runge-Kutta... 

A Runge-Kutta method with minimal dispersion error... 

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

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 ... 

An Explicit Runge-Kutta with Minimal Dispersive or Dissipative Error. Here we give a brief explanation on the procedure followed in order to derive methods with maximum finite order and constant coefficients. We describe the procedure ... 

The produced methods are give by the following formulae Runge-Kutta with Maxiimin Dispersion Order... 

We compare these two methods to the corresponding method with infinite order of dispersion and dissipation and to some classical Runge-Kutta methods. 

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. 

The New developed Runge-Kutta method with minimal dispersion is the most efficient for small stepsizes... 

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. 

Appendix B Maple Program for the development of Dispersive-fitted and dissipative-fitted explicit Runge-Kutta method 216... 

Equations (10.103) and (10.106) are solved using the Runge-Kutta method. The required coefficients are determined from the remaining relationships indicated above. The parameters are chosen on the basis of the information from Table 10.6, the results of the dispersion calculations are presented in Table 10.7 and Fig. 10.26. 

K. Tselios and T. E. Simos, Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. Journal of Computational and Applied Mathematics, 2005, 175(1), 173-181. 

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