Stability of generalized Runge–Kutta methods for stiff kinetics coupled differential equations

Abstract

A stability and efficiency improved class of generalized Runge-Kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and/or nonlinear coupled differential equations. The determination of the coefficients required by the method is precisely obtained from the so-called equations of condition which in turn are derived by an approach based on Butcher series. Since the equations of condition are fewer in number, free parameters can be chosen for optimizing any desired feature of the process. A further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced. In addition, the A(α) stability properties of the method are investigated. Implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback.

A brief history of Accident Chernobyl: Simulation of the influence of Neutron Absorbing Poisons and temperature feedback effects by Point Kinetics Equations

“…In the second simulation interval, about 11 hours of reactor operation at half the power capacity, the reactor reaches a power of 720𝑀𝑊𝑡. In Figure 3 In the third interval, according to the reference of Silva [25], the reactor reaches a thermal power of only 30𝑀𝑊𝑡 caused by poisoning. In Fig.…”

Generalized and Stability Rational Functions for Dynamic Systems of Reactor Kinetics

“…where is the temperature increment of the reactor. After the reactivity 0 is inserted into the reactor, the power responds quickly and the adiabatic model can be used for the calculation of reactor temperature [41,42]. Then the derivative of temperature to time can be given as follows:…”

An Algorithm to Optimize the Calculation of the Fourth Order Runge-Kutta Method Applied to the Numerical Integration of Kinetics Coupled Differential Equations

“…Along these lines, Hagebeuk and Kivits have presented an algorithm in the form of expansion of a very small parameter 1 1 P  to overcome the problem of stiffness [35]. More recently, a new improved class of RK methods of the fourth order were applied with success in several sample problems [36][37][38]. Nevertheless, the problem of finding the smallest number of steps in which the solutions of RK meth-ods are stable, numerically meaningful and efficiently calculated still remains a subject of great interest [39][40][41][42].…”