| Abstract: |
| We present a positive and asymptotic preserving numerical scheme for solving linear transport equations.
The proposed scheme is developed based on a standard spectral angular discretization and classical micro-macro decomposition.
The three main ingredients are a semi-implicit temporal discretization, a dedicated finite difference spatial discretization, and positivity limiters in the angular discretization.
We show that the proposed scheme is asymptotic preserving in the sense that when the mean-free-path of the particles goes to zero, the scheme achieves a standard numerical scheme for the limiting diffusion equation.
We also prove that the proposed scheme preserves positivity of the spatial particle concentration in the solution, which fixes a common defect of spectral angular discretizations.
The proposed scheme is tested on two benchmark problems, one with uniform material and one with two types of material embedded as a checkerboard.
We also perform a space-time accuracy test and compare the numerical results with theoretical estimates. |
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