轴对称星周包层内连续辐射转移问题的间断Galerkin有限元方法
Discontinuous Galerkin finite element method for the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes
AI总结：
本文研究了间断Galerkin有限元方法（DGFEM）求解轴对称星周包层内频率相关二维辐射转移方程的准确性，发现其温度分布误差在1%以内，SED和图像误差分别在5%和10%以内，为辐射转移方程提供了可靠的替代数值解法。
摘要：
研究星周包层内的连续辐射转移问题既是理论挑战也是数值挑战，尤其是在频率相关和多维的情况下。虽然近似方法在数值上更容易处理，但它们通常无法准确描述复杂几何形状内的辐射场。对于这些情况，有必要直接对辐射转移方程进行数值求解。我们研究了间断Galerkin有限元方法（以下简称DGFEM）应用于轴对称星周包层内、与辐射平衡方程耦合的频率相关二维辐射转移方程的准确性。DGFEM是有限元方法的一种变体。它采用间断单元及其沿边界的通量积分，从而确保局部守恒。然而，与经典有限元方法不同，其解在单元边界处是不连续的。我们在一个代码中实现了该方法，并通过将我们的结果与文献中的基准进行比较来测试其准确性。在所有测试案例中，温度分布均在百分之一以内吻合。此外，随后通过光线追踪技术从DGFEM解中获得的出射光谱能量分布（SED）和图像，平均分别在5%和10%以内吻合。我们表明DGFEM能够准确描述轴对称星周包层内的温度分布。随后，出射的SED和图像也得到了很好的重现。间断Galerkin有限元方法提供了一种求解辐射转移方程的替代方法（例如除了蒙特卡罗方法之外），并且可用于其他方法更难处理的情况。
英文摘要：
The study of the continuum radiative transfer problem inside circumstellar envelopes is both a theoretical and numerical challenge, especially in the frequency-dependent and multi-dimensional case. While approximate methods are easier to handle numerically, they often fail to accurately describe the radiation field inside complex geometries. For these cases, it is necessary to directly solve numerically the radiative transfer equation. We investigate the accuracy of a discontinuous Galerkin finite element method (DGFEM hereafter) applied to the frequency-dependent two dimensional radiative transfer equation, coupled with the radiative equilibrium equation, inside axis-symmetric circumstellar envelopes. The DGFEM is a variant of finite element methods. It employs discontinuous elements and flux integrals along their boundaries, ensuring local conservation. However, as opposed to the classical finite-element methods, the solution is discontinuous across element edges. We implemented the method in a code and tested its accuracy by comparing our results with the benchmarks from the literature. For all the tested cases, the temperature profile agrees within one percent. Additionally, the emerging spectral energy distributions (SEDs) and images, obtained subsequently by ray-tracing techniques from the DGFEM solution, agree on average within $5~\mathrm{\%}$ and $10~\mathrm{\%}$, respectively. We show that the DGFEM can accurately describe the temperature profile inside axis-symmetric circumstellar envelopes. Consecutively the emerging SEDs and images are also well reproduced. The discontinuous Galerkin finite element method provides an alternative method (other than Monte Carlo methods for instance) for solving the radiative transfer equation, and could be used in cases that are more difficult to handle with the other methods.
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