In this work, we prove rigorous error estimates for a hybrid method introduced in [15] for solving the time-dependent radiation transport equation (RTE). The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domains, with constant material cross-sections of arbitrary size. To focus the analysis, we assume the uncollided equation is solved exactly and the collided part is approximated in angle via a spherical harmonic expansion ($\text{P}_N$ method). Using a non-standard set of semi-norms, we obtain estimates of the form $C(\varepsilon,\sigma,\Delta t)N^{-s}$ where $s\geq 1$ denotes the regularity of the solution in angle, $\varepsilon$ and $\sigma$ are scattering parameters, $\Delta t$ is the time-step before reinitialization, and $C$ is a complicated function of $\varepsilon$, $\sigma$, and $\Delta t$. These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic $\text{P}_N$ method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes.

翻译：本文对文献[15]中提出的求解时变辐射传输方程（RTE）的混合方法建立了严格的误差估计。该方法基于将辐射的动力学分布函数分裂为未碰撞分量和碰撞分量：采用高分辨率（角度方向）方法近似未碰撞分量，采用低分辨率方法近似碰撞分量。在每个时间步后，动力学分布被重新初始化为完全未碰撞状态。本分析考虑周期域上的单能问题，材料截面为任意大小的常数。为聚焦分析，假设未碰撞方程被精确求解，碰撞部分通过球谐展开（$\text{P}_N$方法）在角度方向上进行近似。通过使用一组非标准半范数，我们得到形如$C(\varepsilon,\sigma,\Delta t)N^{-s}$的估计，其中$s\geq 1$表示解在角度方向的正则性，$\varepsilon$和$\sigma$为散射参数，$\Delta t$为重新初始化前的时间步长，$C$为关于$\varepsilon$、$\sigma$和$\Delta t$的复杂函数。这些估计涉及对包含但超越常规谱分析的多尺度RTE的分析。我们还计算了与混合方法中碰撞部分具有相同分辨率的整体$\text{P}_N$方法的误差估计。结果凸显了混合方法在高散射和流占优两种情况下相对于整体离散化的优势。