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''$S_n$-method''

One of the numerical methods for solving the kinetic equation of neutron transport in nuclear reactors. The first version of the method for a spherically symmetric geometry, proposed by B. Carlson in 1953, was based on piecewise-linear representation of the neutron flow as a function of the cosine of the angle between the velocity vector of the neutron and the radius vector. After integration with respect to the angular variable over an elementary cell one obtains a system of equations in each of which only two directions of the velocity of the neutron are involved, provided the collision integral (calculated by the trapezium rule) be known from the previous approximation. In other words, as in the [[Vladimirov method|Vladimirov method]], the solution of the transport equation is carried out by the method of successive approximation in terms of the collision integral. At each approximation the system splits into separate equations, provided that a supplementary equation is introduced for the radial direction, the latter being integrated from the outer boundary of the ball to the centre of the ball. Each subsequent direction in the equations of the system will now be related to the previous direction, for which the unknown function is already defined. For negative values of the cosine defining the direction, the integration is carried out from the outer boundary to the centre, and for positive values of the cosine, from the centre to the boundary.

The non-monotone character of the solution of the transport equation by the Carlson method (the possibility of oscillations and negative values of the neutron flow) has led to the necessity of a further development of the Carlson method. The discrete Carlson method, the $DS_n$-method is widely employed. In this method, difference equations are deduced from physical considerations by the method of particle balance in a cell in phase space. For low-order approximations, the $DS_n$-method does not guarantee sufficient accuracy in the multi-dimensional case. One way out of this situation is to add extra terms to the system of equations of the $DS_n$-method, in such a way that by a linear transformation of the unknowns the system is converted into a system of equations which occurs in the method of spherical harmonics (cf. [[Spherical harmonics, method of|Spherical harmonics, method of]]).

A large number of computations by the Carlson method (and modifications of it) for nuclear reactors give results in good agreement with the results by other numerical methods for solving equations of neutron transport.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.I. Marchuk,   V.I. Lebedev,   "Numerical methods in the theory of neutron transport" , Harwood  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Greenspan (ed.)  C.N. Kelber (ed.)  D. Okrent (ed.) , ''Computing methods in reactor physics'' , Gordon &amp; Breach  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.J. Bell,   S. Glasstone,   "Nuclear reactor theory" , v. Nostrand-Reinhold  (1971)</TD></TR></table>