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Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus

Alon Jacobson, Xiaozhe Hu

TL;DR

The paper addresses the challenge of discretizing fractional vector calculus (FVC) while exactly preserving the fractional de Rham identities on a discrete grid. It achieves this by reformulating Tarasov's Caputo-based FVC as a composition of fractional integration with exterior derivatives and then discretizing using discrete exterior calculus on a 3D cubical complex, yielding structure-preserving operators. The main results define $\mathbb{D}_p^{\alpha}= {}_{} {\mathbb{I}}_{p+1}^{1-\alpha} \mathbb{D}_p \left({}_{}^{} {\mathbb{I}}_{p}^{1-\alpha}\right)^{-1}$, introduce a fractional de Rham map, and prove (numerically) that $\mathbb{D}_{p+1}^{\alpha} \mathbb{D}_p^{\alpha} = 0$ with second-order RMS convergence to the continuous operators. These structure-preserving, relatively sparse operators enable accurate numerical solutions of FPDEs and exactly enforce fundamental physics laws on the discrete level across mesh sizes, with plans to extend to arbitrary cell complexes and other FVC variants.

Abstract

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.

Structure-Preserving Discretization of Fractional Vector Calculus using Discrete Exterior Calculus

TL;DR

The paper addresses the challenge of discretizing fractional vector calculus (FVC) while exactly preserving the fractional de Rham identities on a discrete grid. It achieves this by reformulating Tarasov's Caputo-based FVC as a composition of fractional integration with exterior derivatives and then discretizing using discrete exterior calculus on a 3D cubical complex, yielding structure-preserving operators. The main results define , introduce a fractional de Rham map, and prove (numerically) that with second-order RMS convergence to the continuous operators. These structure-preserving, relatively sparse operators enable accurate numerical solutions of FPDEs and exactly enforce fundamental physics laws on the discrete level across mesh sizes, with plans to extend to arbitrary cell complexes and other FVC variants.

Abstract

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 and 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.
Paper Structure (20 sections, 7 theorems, 58 equations, 4 figures, 1 table)

This paper contains 20 sections, 7 theorems, 58 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Preliminaries
  5. Fractional calculus
  6. Fractional integrals and derivatives
  7. Fractional calculus identities
  8. Fractional vector calculus
  9. Tarasov's fractional vector calculus
  10. Other fractional vector calculus
  11. Discrete exterior calculus
  12. Fractional discrete exterior calculus via reformulation
  13. Reformulation and equivalence
  14. Definition of FDEC by discretization
  15. Discretization of ${}_{}^{} {\mathcal{I}}_{p}^{\beta}$
  16. ...and 5 more sections

Key Result

Lemma 1

Let $f : \Omega \subset \mathbb{R}^m \to \mathbb{R}$ be continuous and let $\alpha > 0$. Then at any point $(x^1,\dots,x^m) \in \Omega$ and any $j=1,\dots,m$,

Figures (4)

  • Figure 1: Diagram of a 3D regular cubical complex, with $n_x=3$, $n_y=2$, $n_z=2$
  • Figure 2: Error of discrete fractional gradient, curl, and divergence to their continuous counterparts against the number of subdivisions. Left column: $\alpha=0.25$. Right column: $\alpha=0.9$.
  • Figure 3: Error of discrete fractional gradient, curl, and divergence to their continuous counterparts against the fractional order $\alpha$. Left column: error versus $\alpha$ in linear scale. Right column: error versus $1-\alpha$ in log scale.
  • Figure 4: Numerical verification of $\mathbb{D}_{p+1}^\alpha \mathbb{D}_p^\alpha \textbf{c}^p = 0$

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Theorem 2.2, fracdiffeq
  • Lemma 4
  • proof
  • Definition 1
  • Proposition 1
  • Theorem 1
  • ...and 6 more