Numerical methods for fractional diffusion

Abstract

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.Date: Draft version of July 7, 2017.

Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

“…This approach is further applied to problems with fractional order boundary conditions M3 Integral representation of a sought solution of non‐local problem . Different quadrature formulas can be applied to evaluate numerically the related integrals.…”

“…In the literature, several different numerical quadrature formulas are proposed to approximate integral . In our previous studies, we have used the quadrature formula based on a β ‐dependent graded mesh.…”

“…There exist different definitions of fractional power of elliptic operators . Let us start with several comments on the integral definition of the fractional Laplacian in a bounded domain .…”

“…The research about the relations between these definitions is still ongoing. However, it is known that they are not equivalent and may produce different solutions over bounded domains.…”

“…This paper presents several important contributions to our previous research. First, for the pseudo‐parabolic method M2, we use the recently proposed geometrically graded time stepping scheme, which should resolve the singular behavior of the solution for pseudo‐time t close to 0 and give better convergence results compared to the previously used uniform time stepping scheme. Second, for the quadrature method M3, we employ a quadrature formula, which is different from the previously used formula with geometrically graded mesh and is expected to show superior convergence results. Third, for analysis of the BURA method M4, we include its recent modification R‐BURA, which has been recommended for the power coefficient β closer to 1. Finally, we compare the parallel solution times of the developed algorithms that are required to achieve the specified accuracy of the solution. We present and discuss the comparison plots, showing which parallel numerical algorithms are recommended for different levels of accuracy and different values of fractional power coefficient β . …”