Accuracy of the Diffusion Approximation in Solving Model Problems of Radiation Transport in a Medium Using Bicompact Schemes
摘要
Solutions to radiation transport problems in a medium obtained in the diffusion approximation (P1 approximation) and in the full kinetic formulation are compared. The problems are solved numerically using bicompact schemes, which are constructed by applying the method of lines on a minimal two-point stencil. The schemes have the fourth order of approximation in space. Time stepping is based on the L-stable Runge–Kutta method of the third order for radiation transport problems in a homogeneous medium and on the implicit Euler method of the first order for radiation transport problems in a heterogeneous medium. The convergence of the iterative processes is accelerated by applying a HOLO algorithm, namely, V.Ya. Gol’din’s quasi-diffusion method. In this case, high-order (HO) and low-order (LO) equations are solved jointly. The dimension of the original kinetic problem is reduced twice by averaging over angular and energy variables. The schemes for the low-order equations are kinetically consistent with the scheme for the high-order transport equation. The schemes are implemented and used to solve the first and second Fleck problems in a multigroup approximation. A scheme monotonization procedure is proposed and tested. It hybridizes solutions to equations obtained at different stages of the HOLO algorithm by applying high- and low-order approximation schemes. The solutions produced by the full HOLO algorithm, which includes solving the transport equation, are compared with those obtained using the diffusion approximation. The Fleck problems have no line spectrum, and the difference between the group Planck and Rosseland absorption coefficients typically does not exceed one order of magnitude. The difference can reach several orders of magnitude for problems with a line spectrum. It is shown that the differences in the solutions of Fleck problems can be noticeable at small calculation time. The differences in the solutions decrease as the calculation time increases. The steady-state solutions produced by the two methods are nearly identical for the problems under study. The largest differences arise at short times in an optically dense region. The use of the diffusion approximation is justified if a small loss of numerical accuracy is acceptable and there is no need to reproduce in detail the development of processes at short times.