Anisotropic diffusion is imperative in understanding cosmic ray diffusion across the Galaxy, the heliosphere, and the interplay of cosmic rays with the Galactic magnetic field. This diffusion term contributes to the highly stiff nature of the cosmic ray transport equation. To conduct numerical simulations of time-dependent cosmic ray transport, implicit integrators (namely, Crank-Nicolson (CN)) have been traditionally favoured over the CFL-bound explicit integrators in order to be able to take large step sizes. We propose exponential methods to treat the linear anisotropc diffusion equation in the presence of advection and time-independent and time-dependent sources. These methods allow us to take even larger step sizes that can substantially speed-up the simulations whilst generating highly accurate solutions. In or subsequent work, we will use these exponential solvers in the Picard code to study anisotropic cosmic ray diffusion and we will consider additional physical processes such as continuous momentum losses and reacceleration.

翻译：各向异性扩散对于理解银河系内宇宙线扩散、日球层及其与银河磁场相互作用至关重要。该扩散项导致宇宙线输运方程具有高度刚性特征。为进行时变宇宙线输运的数值模拟，传统上优先采用隐式积分器（即Crank-Nicolson方法）而非受CFL条件限制的显式积分器，以便采用大步长计算。我们提出指数型方法处理存在平流项及时间无关/相关源项的线性各向异性扩散方程。这些方法允许采用更大步长，在显著加速模拟的同时生成高精度解。在后续工作中，我们将把这些指数求解器集成至Picard代码中研究各向异性宇宙线扩散，并考虑连续动量损失与再加速等附加物理过程。