论文标题
使用离散的外部微积分对分数矢量计算的结构保存离散化
Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus
论文作者
论文摘要
分数矢量演算是模拟非本地或远程现象的分数偏微分方程的基础，例如，异常扩散，分数电磁磁性和分数对流分散。在这项工作中，我们重新准备了一种使用CAPUTO分数部分衍生物的分数矢量计算，并以结构性能的方式在立方体上使用离散的外部演算进行离散重新分配，这意味着连续级别的属性$ \ operatories $ \ operatorname { $ \ operatatorName {div}^α\ operatatorName {curl}^α= 0 $恰好在离散级别上保持。我们讨论了分数离散外部导数的重要属性，并在数值上验证了均方根误差中的二阶收敛。我们提出的离散化有可能为分数偏微分方程提供准确，稳定的数值解决方案，而无论网格大小如何，都可以在离散水平上保留基本物理法。
Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties $\operatorname{curl}^α\operatorname{grad}^α= \mathbf{0}$ and $\operatorname{div}^α\operatorname{curl}^α= 0$ hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.
