We present a numerical method for handling the resolution of a general transport equation for radiative particles, aimed at physical problems with a general spherical geometry. Having in mind the computational time difficulties encountered in problems such as neutrino transport in astrophysical supernovae, we present a scheme based on full spectral methods in 6d spherical coordinates. This approach, known to be suited when the characteristic length of the dynamics is much smaller than the domain size, has the potential advantage of a global speedup with respect to usual finite difference schemes. An analysis of the properties of the Liouville operator expressed in our coordinates is necessary in order to handle correctly the numerical behaviour of the solution. This reflects on a specific (spherical) geometry of the computational domain. The numerical tests, performed under several different regimes for the equation, prove the robustness of the scheme: their performances also point out to the suitability of such an approach to large scale computations involving transport physics for mass less radiative particles.
Mathematical and numerical validation of the simplified spherical harmonics approach for time-dependent anisotropic-scattering transport problems in homogeneous media
Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity
Princeton University（普林斯顿大学）；Université Libre de Bruxelles（布鲁塞尔自由大学）；GéoAzur Université de Nice（尼斯大学地理与地球物理实验室）；Program in Applied and Computational Mathematics Princeton University（普林斯顿大学应用与计算数学项目）
A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions
最佳质量输运的问题出现在许多应用中，包括图像配准、网格生成、反射器设计和天体物理学。解决这个问题的一种方法是通过蒙日-安培方程。虽然近年来已经看到了很多工作的数值方法的发展，以解决这个方程，很少已经做了实施的运输边界条件。本文提出了一种求解迁移问题的方法，即迭代求解带有Neumann边界条件的蒙日-安培方程。为了实现可变密度之间的映射，我们扩展了方程的早期离散化以允许取决于解的梯度的右侧[Froese和Oberman，SIAM J. Numer.肛门，49（2011）1692- 1714]。该离散化可证明收敛于黏性解。用牛顿法有效地求解了所得系统。我们提供了几个具有挑战性的计算实例，证明了所提出的方法的有效性和效率（$O(M)-O(M^{1.3})$时间）。
