EAPSI: A Method Connecting Radiation Diffusion and Radiation Transfer Along Resonant Lines

辐射传递方程可以用来描述光通过光学厚或光学薄区域的传输。因此，许多科学领域都对辐射传递方程的解产生了极大的兴趣。本研究的目的是建立一种将辐射扩散方程的解与光厚和光薄共存的辐射传递方程的解连接起来的方法。众所周知，在许多实际应用中，辐射传递方程很难求解。因此，通常使用扩散近似，因为得到的方程更容易求解。在光学厚区，扩散近似是合适的；然而，在光学薄区，需要求解辐射传递方程。与日本筑波大学计算科学中心的Masayuki Umemura教授合作，将开发一种连接光学厚区辐射扩散方程解与光学薄区辐射传递方程解的方法。这种方法将产生一种更有效的方法来解决光学厚和薄共存区域的辐射转移问题。更具体地说，辐射传递方程是一个积分微分方程，它在数学上描述了辐射在参与的介质中的传播。该方程的解析解只在一小部分情况下是已知的，这就需要它的数值解。然而在数值上，由于方程的高维和积分-微分形式，计算需求是显著的。扩散近似是辐射传递中一个极其重要的近似。相比之下，辐射扩散方程更容易求解，并且在光学厚的情况下是一个精确的近似。为了开发一种更有效的方法来解决光学厚区和薄区共存的辐射传递问题，我们的目标是开发一种能够耦合辐射扩散方程解和辐射传递方程解的方法。该奖项隶属于东亚和太平洋暑期研究所项目，由美国国家科学基金会和日本科学促进会共同资助，支持一名美国研究生进行暑期研究。

The radiative transfer equation can be used to describe the transport of light through optically thick or optically thin regimes. As a result, there is great interest in the solution of the radiative transfer equation from numerous scientific fields. The goal of this research is to develop a method connecting solutions of the radiative diffusion equation with solutions of the radiative transfer equation where optically thick and thin regimes coexist. The radiative transfer equation is notoriously difficult to solve in many real-world applications. As a result, the diffusion approximation is often used, as the resulting equation is easier to solve. In optically thick regimes, the diffusion approximation is appropriate; however, in optically thin regimes, the solution of the radiative transfer equation is needed. In collaboration with Professor Masayuki Umemura of the Center for Computational Sciences at the University of Tsukuba, Japan, a method connecting solutions of the radiative diffusion equation in optically thick regimes with solutions of the radiative transfer equation in optically thin regimes will be developed. This approach will result in a more efficient means of solving radiative transfer problems in regions where optically thick and thin regimes coexist.More specifically, the radiative transfer equation is an integro-differential equation which mathematically describes the propagation of radiation within a medium that participates. Analytical solutions for the equation are known only for a small subset of situations, which necessitates its numerical solution. Yet numerically, due to the high dimension and integro-differential form of the equation, the computational demands are significant. The diffusion approximation is an extremely important approximation in radiative transfer. In comparison, the radiative diffusion equation is much easier to solve and is an accurate approximation in optically thick regimes. In order to develop a more efficient method for solving radiative transfer problems where optically thick and thin regimes coexist, we aim to develop a method capable of coupling solutions of the radiative diffusion equation with solutions of the radiative transfer equation. This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the Japan Society for the Promotion of Science.