空间分数阶扩散-对流方程：解析解及数值解的批判性评估
The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions
AI总结：
本文对空间分数阶扩散-对流方程的数值解进行了批判性评估，通过三种方法对比揭示了随机微分方程在准确性上的优势。
摘要：
本文对空间分数阶扩散-对流方程的数值解进行了批判性评估，强调其在自然科学研究中的重要性。鉴于目前尚无标准方法和对应数值代码用于异常扩散问题，本文通过三种方法（有限差分法、矩阵转移技术及基于随机微分方程的蒙特卡洛方法）在三个测试问题上的应用，对比分析了其准确性和性能差异，结果显示随机微分方程在应用中具有优势。
英文摘要：
The present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and performance of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The accuracy and performance differences are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
