Nonlinear Finite-Difference Scheme with a Limiter for Numerical Modeling of Radiation Transport Processes

Numerical modeling of the nonstationary radiation transport process in the kinetic model is a very labor-intensive task. The complexity is due to the large dimensionality of the problem and, additionally, for problems of radiant energy transfer, strong nonlinearity. For deterministic approaches based on discretization of the particle flight direction, it is necessary to solve a system of hyperbolic equations of large dimension. Accordingly, it is desirable that the methods used for numerical modeling be economical both in terms of memory use and calculation time and show acceptable results for a wide range of Courant numbers. In the case of radiant transport, the situation is aggravated by the strong nonlinearity of the problem being solved, which leads to a significant change in the properties of the medium at time steps. This imposes increased requirements for monotonicity of the schemes with a change in optical thickness. According to Godunov’s theorem, among two-layer linear schemes in time, there are no monotonic schemes of a higher approximation order. One of the directions of solving this problem is the development of NFC (Nonlinear Flux Correction) schemes of end-to-end counting, in which an increased order of accuracy on smooth solutions and monotonicity are achieved owing to nonlinear correction of flows. The numerical solution is monotonized using a special algorithm in the vicinity of large gradients of the exact solution. The paper provides a brief overview and characteristics of the finite-difference scheme developed and successfully used for many years at RFNC VNIITF to solve radiation transport problems. The Total Variation Diminishing (TVD) technique is used to monotonize the scheme.