A Second-Order Finite Volume Discretization of the Time-Dependent Transport Equation on Arbitrary Quadrilaterals in R-Z Geometry

A second order, semi-implicit numerical method for solving the multigroup nonstationary transport equation and corresponding code is developed in two-dimensional R-Z geometry. Finite difference meshes are formed by arbitrary convex quadrangles. The conservative finite difference scheme is derived by the integro-interpolation method. The balance equation is augmented by linear approximations. The proposed additional relationships provide the second order of approximation at any side-visible cases using a corresponding choice of the weights of scheme. The number of additional relationships in spatial variables, as well as their form, depends on how many visible sides are under consideration. The additional relationships in time and angle variables are diamond-difference-like approximations relating the edge values to the cell-centered values.

An analytical test problem is used to demonstrate the second order of spatial approximation of the proposed method. To test the algorithm for solving the stationary transport equation, we compare the numerical results, obtained by the developed technique, with the results produced by one-dimensional (1-D) codes such as KIN1D (The Keldysh Institute of Applied Mathematics, Russia) and ANISN (U.S.) by using spherical symmetrical 1-D problems. Special analytical benchmarks are developed to test the nonstationary technique. The tests have shown good agreement of the results.