Vladimirov method

One of the most accurate numerical methods for solving the kinetic equation of neutron transfer in nuclear reactors, based on integration along characteristics. Proposed in 1952 by V.S. Vladimirov for the solution of integro-differential kinetic equations in the case of spherically-symmetric reactors. The principle of the method may be illustrated by taking a subcritical reactor with a neutron source as an example. For a one-dimensional spherically-symmetric geometry and a single velocity, the equation of the neutron flux $ \phi ( r, \mu ) $( where $ r $ is the radius, $ 0 \leq r \leq R $, and $ \mu $ is the cosine of the angle between the vector of the neutron velocity and the radius) has the form

$$ \tag{1 } \mu \frac{\partial \phi }{\partial r } + \frac{1 - \mu ^ {2} }{r} \frac{\partial \phi }{\partial \mu } + \Sigma ( r) \phi = $$

$$ = \ \frac{\Sigma _ {s} ( r) }{2} \int\limits _ { - 1} ^ { + 1} \phi ( r, \mu ^ \prime ) d \mu ^ \prime + f ( r) $$

with the boundary condition

$$ \tag{2 } \phi ( R, \mu ) = 0 \ \ \textrm{ for } \mu \leq 0, $$

meaning that neutrons do not impinge from outside on the external boundary $ r = R $ of the system; $ \Sigma ( r) $, $ \Sigma _ {s} ( r) $ and $ f( r) $ are given piecewise-continuous functions of $ r $. The substitution

$$ \tag{3 } x = r \mu ; \ \ y = r \sqrt {1 - \mu ^ {2} } $$

yields the equation

$$ \tag{4 } \frac{\partial \psi }{\partial x } + \Sigma \psi ( x, y) = \ \frac{\Sigma _ {s} }{2} \int\limits_{-1}^{+1} \phi d \mu + f, $$

where $ \psi ( x, y) = \phi ( \sqrt {x ^ {2} + y ^ {2} } , x/ \sqrt {x ^ {2} + y ^ {2} } ) $. This equation may be readily solved as an ordinary first-order differential equation, and

$$ \tag{5 } \psi ( x, y) = \ \int\limits _ {- \sqrt {R ^ {2} - y ^ {2} } } ^ { x } \left [ \frac{\Sigma _ {s} }{2} \int\limits_{-1}^{+1} \phi d \mu + f \right ] e ^ {- \int\limits _ {x ^ \prime } ^ { x } \Sigma dx ^ {\prime\prime} } dx ^ \prime . $$

For each characteristic $ y = y _ {i} $ of the differential part of the kinetic equation (1) one chooses a specific system of nodes $ x _ {ki} = \sqrt {r _ {k} ^ {2} - y _ {i} ^ {2} } $, where $ r _ {k} $ is the grid radius chosen. The solution of equation (5) is conducted by the method of successive approximation (cf. Sequential approximation, method of), beginning from the initial approximation of the function:

$$ \tag{6 } Q ( r) = \frac{\Sigma _ {s} }{2} \int\limits_{-1}^{+1} \phi d \mu + f. $$

Here

$$ \phi ( r _ {k} , \mu _ {ki} ) = \ \psi ( x _ {ki} , y _ {i} ) $$

(where $ \mu _ {ki} = x _ {ki} / r _ {k} $) is readily found with the aid of (5) at all nodes of the grid, after having replaced the integrals in (5) by sums and having obtained an expression interconnecting the values of $ \phi $ and $ Q $ at two neighbouring points on the characteristic. In order to obtain the value of $ Q( r) $ in the following approximation, one must compute $ \int _{-1}^ {+1} \phi d \mu $, which is done using the quadrature formula involving the points of the circle $ x ^ {2} + y ^ {2} = r _ {k} ^ {2} $. The rate of convergence of the successive approximations is determined by the size and the physical characteristics of the reactor.

The eigenvalue problem (the determination of the critical parameters of the reactor) is solved in a similar manner.

Vladimirov's method can be generalized to include problems involving several velocities and multi-dimensional problems, and can be readily programmed in a computer. Unlike the Carlson method, Vinogradov's method employs $ \mu $- variable grids for different $ r $' s, thus enhancing the computational accuracy on the reactor-vacuum boundary (close to $ r = R $), as compared with the regions in the vicinity of $ r = 0 $, where the neutron flux is near-isotropic.

References

[1]	G.I. Marchuk, "Methods of calculation of nuclear reactors" , Moscow (1961) (In Russian)