The method of characteristics for the solution of the kinetic equation

Abstract

IN THIS paper we derive and investigate difference equations which arise in the solution of a single-speed kinetic equation using the method of characteristics. This method was first suggested for solving the kinetic equation by Vladimirov [1] and Neumann [2]. The description of some schemes of this method is given in [2], [3]. We maintain that tne method of cnaracteristics is in a sense more natural than the S n -method, since the integration involved in it is carried out along the trajectories of neutrons in coordinate space, and tne principle of the construction of the difference equations is the same for all geometries. Some of the earlier published difference equations of tne method of characteristics possess one or even a few of the following drawbacks: (1) To carry out calculations using these equations repeated calculation of the values of some transcendental functions is required, which increases the time required to solve the problem; (2) Because the step ħ of integration along the characteristic for proolems with central symmetry can be a quantity of order O(Δr 1 2 ) , where Δr is the step along the radius, for some schemes the order of approximation of the differential operator by the difference operator becomes of order O( Δr); (3) The non-monotonicity of the scheme: in deriving the difference equations an upper limit on the step ħ is not shown to be necessary, and without observing this restriction the solution of the difference problem can go off into tae negative region with positive sources. In this paper difference equations are presented which are free from these drawbacks. The paper is divided into two parts according to the approach and the choice of questions for investigation. In the first part we are interested in questions concerning the derivation of the difference equations for the general case and also for the most difficult one-dimensional case — that of the infinite cylinder. Here we present, in as much detail as possible, difference equations which are satisfied in the solution of problems by the KP-method [4]. Considerable practical testing of the solution of variants confirms the possibility of solving by these difference schemes problems for a reactor and for a nucleus with complicated distribution of the media and the properties of these media [5, 6]. In the second part we investigate the convergence of the KP-method, evaluate the error and determine the value of the “inexpensive” algorithm of the KP-method for the case of a plane periodic problem. Here we proceed to the restrictions on the co-efficients of the equation for detecting singularities, watch are spherical for the problem considered, and obtaining effective quantitative evaluations for certain quantities. It appears to us that some of the detected singularities are present in the general case also. The paper also verifies the truth of some assumptions used by the author in [4].