SOME IMPLICIT AND EXPLICIT TIME DISCRETIZATION SCHEMES FOR STIFF SYSTEMS (NUCLEAR REACTOR KINETICS, HEAT TRANSFER)

Non-prompt-critical transients of the neutron population in a thermal nuclear reactor can be studied via a system of diffusion equations for neutrons coupled to a system of ordinary differential equations (ODEs) for precursors. When a "method of lines" approach is used, the original problem reduces to an initial value problem for ODEs; the number of ODEs is usually very large. This system exhibits "stiffness" which is due to the typical stiffness of space-discretized parabolic equations and also to the vastly different time constants of prompt neutrons and delayed neutron precursors.;In this work, the RRK methods have been studied theoretically, and tested extensively on problems originating from nuclear reactor kinetics and heat transfer. Several plausible variations have also been implemented. In addition some classical linear one-step schemes have been examined, and their performance compared to the performance of the RRK schemes.;We find that the non linear RRK schemes--if applied for problems such as dx(t)/dt = Ax(t) + u(t) under stiff conditions--perform satisfactorily provided the spread of the eigenvalues of the matrix A is small. The reasons are related to a specific "mode interaction" feature of the RRK schemes. Accordingly, the RRK schemes are usually not to be recommended for the treatment of space-time kinetics and heat transfer problems. Specific situations for which the RRK schemes are appropriate are indicated.;The main purpose of this research is to investigate the performance of some classes of procedures for the discretization of the time variable. It is well known that, for the integration of stiff initial value problems, one has to use extremely small step sizes unless one uses A-stable (or partially A-stable schemes). However, linear A-stable procedures are implicit and require matrix inversions, which are costly when the number of ODEs is large. In the literature, some non linear Rational Runge-Kutta (RRK) methods have been recently proposed: they are partially A-stable and explicit; their use has been recommended for parabolic problems.