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Center for Nonlinear Studies |
The $P_N$ equations describe neutron and photon transport through a material medium. In the limit where particle collisions become dominant, the solution to the $P_N$ equations can be approximated by the solution of a diffusion equation. Capturing this limit is difficult, as a conventional numerical scheme that performs well in weakly collisional regimes requires a small time step in collision-dominated regimes, due to the fact that it resolves the fast speeds associated with the collisions. In addition, standard numerical solvers use numerical dissipation to maintain stability around discontinuities. The numerical dissipation increases as collisions dominate due to the fast time scales in the system, and the numerical dissipation can dominate the actual physical diffusion in the system.
We use a parity form of the transport equation to suggest a splitting of the $P_N$ equations that separates the fast, collision-dominated behavior and the slow, macroscopic behavior, and provides structure for efficient numerical simulation. This splitting allows us to take larger time steps, under-resolving the fast scales of the system in the diffusive limit when such fast scales are not important. Furthermore, the scheme captures the proper diffusion limit in collision-dominated regimes. However, some subtleties in the splitting produce nonphysical oscillations near discontinuities. We investigate these oscillations and suggest some changes to the scheme to properly account for them.
Results are shown for the linear neutron transport equation and the nonlinear radiative transfer equation in one dimension.
Host: Cory Hauck, CCS-2 and CNLS