Bicompact schemes for solving a steady-state transport equation by the quasi-diffusion method

Abstract

Accurate difference schemes (of up to the fourth order of accuracy) are constructed for the numerical solutions of a neutron-transport equation and a system of quasi-diffusion equations (a low order consequences of a transport equation) used to accelerate iterations on scattering. The considered difference schemes are based on the common principles of the compact (in the context of a single cell) approximation. This enables one to make accurate resolution for contact discontinuities in the medium. The fourth order of approximation on a minimum two-point stencil is attained by widening the list of unknowns and including on it (apart from the nodal values of an unknown function) additional unknowns. As these unknowns, an integral cell-average value or a value at a half-integer node can be taken. To connect these values, Simpson quadrature formulas are used. Equations for determining additional unknowns are constructed by using the Euler–Maclaurin formulas. Computations for several onedimensional test problems have been performed; here, the high actual accuracy of the constructed difference schemes is demonstrated. The schemes are naturally generalized to the two- and three-dimensional cases. Due to their high accuracy, monotonicity, efficiency, and compactness, the proposed schemes are very attractive for engineering computations (computations of nuclear reactors, etc.).