A Corrected $H^T_N$-UGKS for Boundary Layer Problems of the Gray Radiative Transfer Equations

Abstract

In this paper, the corrected method to the original $H^T_N$-unified gas kinetic scheme ($H^T_N$-UGKS) is developed in order to solve the nonlinear radiative transfer equations with boundary layers. The $H^T_N$-UGKS is an asymptotic preserving (AP) scheme that uses UGKS for spatial discretization and the hybrid $H^T_N$ method for angular discretization which is constructed in the paper (Li et al. in Nucl. Sci. Eng. 198(5): 993–1020, 2024). First, the correction idea in Mieussens (J. Comput. Phys. 253: 138–156, 2013) is adopted, such that $H^T_N$-UGKS can correctly simulate the linear radiative transfer equation with boundary layers. Then, for the nonlinear radiative transfer equations with boundary layers, the transformation from the implicit Monte Carlo (IMC) method is introduced to rewrite the nonlinear transfer equations into a linearized system. It is the key point in the construction of the current scheme to use this linearized system to construct the numerical boundary fluxes. In this way, the boundary density is included in the numerical fluxes, and consequently, the modification method for the linear radiative transfer equation can be used to deal with the nonlinear problem studied in this paper. A number of numerical examples are presented to demonstrate the accuracy and effectiveness of the current scheme for resolving boundary layers in both linear and nonlinear radiative transfer problems.

Funding

National Key R &D Program(2020YFA0712200)

Sino-German Science Center(GZ 1465)

NSFC(12292981,12292982)

Beijing Natural Science Foundation(Z230003)