标题： 辐射传输混合方法的数值分析 显示英文标题

标题： Numerical analysis of a hybrid method for radiation transport

摘要： 在本工作中，我们为求解时间依赖的辐射传输方程（RTE）而引入的一种混合方法提供了严格的误差估计。该方法依赖于将辐射的动量分布函数分为未碰撞和碰撞部分。使用高分辨率方法（在角度上）来近似未碰撞部分，而使用低分辨率方法来近似碰撞部分。在每个时间步之后，动量分布被重新初始化为完全未碰撞。为了集中分析，我们考虑一个周期性域上的单能问题，具有任意大小的常数材料截面。为了聚焦分析，我们假设未碰撞方程被精确求解，而碰撞部分通过球面谐波展开（$\text{P}_N$方法）在角度上进行近似。 使用非标准的半范数集合，我们得到如下形式的估计 $C(\varepsilon,\sigma,\Delta t)N^{-s}$ ，其中 $s\geq 1$ 表示解在角度上的正则性， $\varepsilon$ 和 $\sigma$ 是散射参数， $\Delta t$ 是重初始化前的时间步长，而 $C$ 是 $\varepsilon$、 $\sigma$ 和 $\Delta t$ 的复杂函数。 这些估计涉及对多尺度 RTE 的分析，该分析包括但不仅限于常规的谱分析。 我们还计算了在混合方法中碰撞部分相同分辨率下的整体$\text{P}_N$方法的误差估计。 我们的结果突显了混合方法在高度散射和流 regimes 中相对于整体离散化的优点。 显示英文摘要

摘要： 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.