论文标题
各向异性扩散的指数方法
Exponential methods for anisotropic diffusion
论文作者
论文摘要
各向异性扩散方程对于理解宇宙射线扩散跨银河系，地球层及其与环境磁场的相互作用至关重要。该扩散项有助于CR传输方程的高度僵硬性质。为了对时间依赖性宇宙射线传输进行数值模拟，传统上，隐式集成器受到CFL结合的显式积分器的青睐，以便能够采用很大的步骤尺寸。我们提出的指数方法可以直接计算矩阵的指数来求解线性各向异性扩散方程。这些方法使我们能够采用更大的步骤。在某些情况下，我们能够选择一个与模拟时间一样大的步长，即只有一个时间步。这可以实质上加速模拟，同时生成高度准确的解决方案（L2错误$ \ leq 10^{ - 10} $）。此外，我们测试了一种基于从各向异性扩散方程中提取恒定扩散系数的方法，在各向异性扩散方程中，恒定系数项被隐式或指数级求解，并使用某种显式方法对其进行处理。我们发现，对于均匀的线性问题，这种方法无法改善直接评估矩阵指数级的基于指数的方法。
The anisotropic diffusion equation is imperative in understanding cosmic ray diffusion across the Galaxy, the heliosphere, and its interplay with the ambient magnetic field. This diffusion term contributes to the highly stiff nature of the CR transport equation. In order to conduct numerical simulations of time-dependent cosmic ray transport, implicit integrators have been traditionally favoured over the CFL-bound explicit integrators in order to be able to take large step sizes. We propose exponential methods that directly compute the exponential of the matrix to solve the linear anisotropic diffusion equation. These methods allow us to take even larger step sizes; in certain cases, we are able to choose a step size as large as the simulation time, i.e., only one time step. This can substantially speed-up the simulations whilst generating highly accurate solutions (l2 error $\leq 10^{-10}$). Additionally, we test an approach based on extracting a constant diffusion coefficient from the anisotropic diffusion equation, where the constant coefficient term is solved implicitly or exponentially and the remainder is treated using some explicit method. We find that this approach, for homogeneous linear problems, is unable to improve on the exponential-based methods that directly evaluate the matrix exponential.
